Handbook of integral equations:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2008
|
| Edition: | 2. ed. |
| Series: | Handbooks of mathematical equations
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXXIII, 1108 S. Ill. |
| ISBN: | 9781584885078 1584885076 |
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MARC
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| 035 | |a (OCoLC)167516078 | ||
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| 100 | 1 | |a Poljanin, Andrej D. |d 1951- |e Verfasser |0 (DE-588)128391251 |4 aut | |
| 245 | 1 | 0 | |a Handbook of integral equations |c Andrei D. Polyanin ; Alexander V. Manzhirov |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boca Raton [u.a.] |b Chapman & Hall/CRC |c 2008 | |
| 300 | |a XXXIII, 1108 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Handbooks of mathematical equations | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Integral equations |v Handbooks, manuals, etc | |
| 650 | 0 | 7 | |a Integralgleichung |0 (DE-588)4027229-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Exakte Lösung |0 (DE-588)4348289-2 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4155008-0 |a Formelsammlung |2 gnd-content | |
| 655 | 7 | |0 (DE-588)4173536-5 |a Patentschrift |2 gnd-content | |
| 689 | 0 | 0 | |a Integralgleichung |0 (DE-588)4027229-1 |D s |
| 689 | 0 | 1 | |a Exakte Lösung |0 (DE-588)4348289-2 |D s |
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| 700 | 1 | |a Manžirov, Aleksandr V. |e Verfasser |0 (DE-588)120620758 |4 aut | |
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| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016138406&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
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| _version_ | 1804137166557675520 |
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| adam_text | Mathematics, Physics, Mechanics, Control, Engineering Sciences
гЈАШЗОСЖ
0?
INTEGRAL
S OUATIONS
EDITION
-»
UPDATED,
ft£7B£D
AND EXTENDED
•
Represents a unique reference for engineers
and scientists that does not have any
counterpart in the literature
•
Contains over
2,500
linear and nonlinear
integral equations and their exact solutions
•
Outlines exact, approximate analytical,
and numerical methods for solving integral
equations
•
Illustrates the application of the methods with numerous examples
•
Considers equations that arise in elasticity, plasticity, creep, heat
and mass transfer, hydrodynamics, chemical engineering, and other
areas
•
Can be used as a database of test problems for numerical and
approximate methods for solving linear and nonlinear integral
equations
•
Presents many times more integral equations than any other book
currently available
New to the Second Edition
•
Additional material on
Volteira,
Fredholm,
singular, hypersingular,
mixed, multidimensional, dual, and nonlinear integral equations,
integral transforms, and special functions
•
Methods of integral equations for ODEs and PDEs
•
Over
300
added pages and more than
400
new integral equations
with exact solutions
To accommodate different mathematical backgrounds, the authors
avoid wherever possible the use of special terminology, outline some of
the methods in a schematic, simplified manner, and arrange the material
in increasing order of complexity.
CONTENTS
Authors
...................................................................xxix
Preface
....................................................................xxxi
Some Remarks and Notation
.................................................xxxiii
CONTENTS
Authors
...................................................................xxix
Preface
....................................................................xxxi
Some Remarks and Notation
................................................xxxiii
Part I. Exact Solutions of Integral Equations
1.
Linear Equations of the First Kind with Variable Limit of Integration
............ 3
1.1.
Equations Whose Kernels Contain Power-Law Functions
........................ 4
-1.
Kernels Linear in the Arguments
χ
and
t
................................ 4
-2.
Kernels Quadratic in the Arguments
x
and
t
............................. 4
-3.
Kernels Cubic in the Arguments
x
and
t
................................ 5
1-4.
Kernels Containing Higher-Order Polynomials in
x
and
t
.................. 6
1-5.
Kernels Containing Rational Functions
................................. 7
1 -6.
Kernels Containing Square Roots
..................................... 9
1-7.
Kernels Containing Arbitrary Powers
.................................. 12
1.
Two-Dimensional Equation of the Abel Type
............................ 15
1.2.
Equations Whose Kernels Contain Exponential Functions
........................ 15
1.2-1.
Kernels Containing Exponential Functions
.............................. 15
1.2-2.
Kernels Containing Power-Law and Exponential Functions
................. 19
1.3.
Equations Whose Kernels Contain Hyperbolic Functions
......................... 22
1.3-1.
Kernels Containing Hyperbolic Cosine
................................. 22
1.3-2.
Kernels Containing Hyperbolic Sine
................................... 28
1.3-3.
Kernels Containing Hyperbolic Tangent
................................ 36
1.3-4.
Kernels Containing Hyperbolic Cotangent
.............................. 38
1.3-5.
Kernels Containing Combinations of Hyperbolic Functions
................. 39
1.4.
Equations Whose Kernels Contain Logarithmic Functions
........................ 42
1.4-1.
Kernels Containing Logarithmic Functions
.............................. 42
1.4-2.
Kernels Containing Power-Law and Logarithmic Functions
................. 45
1.5.
Equations Whose Kernels Contain Trigonometric Functions
...................... 46
1.5-1.
Kernels Containing Cosine
.......................................... 46
1.5-2.
Kernels Containing Sine
............................................ 52
1.5-3.
Kernels Containing Tangent
.......................................... 60
1.5-4.
Kernels Containing Cotangent
........................................ 62
1.5-5.
Kernels Containing Combinations of Trigonometric Functions
.............. 63
1.6.
Equations Whose Kernels Contain Inverse Trigonometric Functions
................ 66
1.6-1.
Kernels Containing Arccosine
........................................ 66
1.6-2.
Kernels Containing Arcsine
.......................................... 68
1.6-3.
Kernels Containing Arctangent
....................................... 70
1.6-4.
Kernels Containing Arccotangent
..................................... 71
vi
Contents
1.7.
Equations
Whose
Kernels
Contain Combinations of Elementary Functions
.......... 73
1.7-1.
Kernels Containing Exponential and Hyperbolic Functions
................. 73
1.7-2.
Kernels Containing Exponential and Logarithmic Functions
................ 77
1.7-3.
Kernels Containing Exponential and Trigonometric Functions
............... 78
1
.7-4.
Kernels Containing Hyperbolic and Logarithmic Functions
................. 83
1.7-5.
Kernels Containing Hyperbolic and Trigonometric Functions
............... 84
1.7-6.
Kernels Containing Logarithmic and Trigonometric Functions
.............. 85
1.8.
Equations Whose Kernels Contain Special Functions
............................ 86
1.8-1.
Kernels Containing Error Function or Exponential Integral
................. 86
3-2.
Kernels Containing Sine and Cosine Integrals
........................... 87
3-3.
Kernels Containing Fresnel Integrals
................................... 87
1
.8-4.
Kernels Containing Incomplete Gamma Functions
........................ 88
i-5. Kernels Containing Bessel Functions
.................................. 88
1
.8-6.
Kernels Containing Modified Bessel Functions
.......................... 97
1.8-7.
Kernels Containing Legendre Polynomials
.............................. 105
1.8-8.
Kernels Containing Associated Legendre Functions
....................... 107
1.8-9.
Kernels Containing Confluent Hypergeometric Functions
.................. 107
1.8-10.
Kernels Containing Hermite Polynomials
.............................. 108
1.8-11.
Kernels Containing Chebyshev Polynomials
............................ 109
1.8-12.
Kernels Containing Laguerre Polynomials
............................. 110
1.8-13.
Kernels Containing Jacobi Theta Functions
............................ 110
1.8-14.
Kernels Containing Other Special Functions
............................
Ill
1.9.
Equations Whose Kernels Contain Arbitrary Functions
..........................
Ill
1.9-1.
Equations with Degenerate Kernel: K(x,t) = g {x)h (t) +
дг{х)Н2{Ь)
.........
Ill
1.9-2.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
..................... 114
1.9-3.
Other Equations
___............................................... 122
1.10.
Some Formulas and Transformations
....................................... 124
2.
Linear Equations of the Second Kind with Variable Limit of Integration
.......... 127
2.1.
Equations Whose Kernels Contain Power-Law Functions
........................ 127
2.1-1.
Kernels Linear in the Arguments
χ
and
t
................................ 127
2.1-2.
Kernels Quadratic in the Arguments
x
and
í
............................. 129
2.1-3.
Kernels Cubic in the Arguments
x
and
t
................................ 132
2.1-4.
Kernels Containing Higher-Order Polynomials in
x
and
t
.................. 133
2.1-5.
Kernels Containing Rational Functions
................................. 136
2.1-6.
Kernels Containing Square Roots and Fractional Powers
................... 138
2.1-7.
Kernels Containing Arbitrary Powers
.................................. 139
2.2.
Equations Whose Kernels Contain Exponential Functions
........................ 144
2.2-1.
Kernels Containing Exponential Functions
.............................. 144
2.2-2.
Kernels Containing Power-Law and Exponential Functions
................. 151
2.3.
Equations Whose Kernels Contain Hyperbolic Functions
......................... 154
2.3-1.
Kernels Containing Hyperbolic Cosine
................................. 154
2.3-2.
Kernels Containing Hyperbolic Sine
................................... 156
2.3-3.
Kernels Containing Hyperbolic Tangent
................................ 161
2.3-4.
Kernels Containing Hyperbolic Cotangent
.............................. 162
2.3-5.
Kernels Containing Combinations of Hyperbolic Functions
................. 164
2.4.
Equations Whose Kernels Contain Logarithmic Functions
........................ 164
2.4-1.
Kernels Containing Logarithmic Functions
.............................. 164
2.4-2.
Kernels Containing Power-Law and Logarithmic Functions
................. 165
CONTKNTS
VU
2.5.
Equations
Whose
Kernels
Contain Trigonometric Functions
...................... 166
2.5-1.
Kernels Containing Cosine
.......................................... 166
2.5-2.
Kernels Containing Sine
............................................ 169
2.5-3.
Kernels Containing Tangent
.......................................... 174
2.5-4.
Kernels Containing Cotangent
........................................ 175
2.5-5.
Kernels Containing Combinations of Trigonometric Functions
.............. 176
2.6.
Equations Whose Kernels Contain Inverse Trigonometric Functions
................ 176
2.6-1.
Kernels Containing Arccosine
........................................ 176
2.6-2.
Kernels Containing Arcsine
.......................................... 177
2.6-3.
Kernels Containing Arctangent
....................................... 178
2.6-4.
Kernels Containing Arccotangent
..................................... 178
2.7.
Equations Whose Kernels Contain Combinations of Elementary Functions
.......... 179
2.7-1.
Kernels Containing Exponential and Hyperbolic Functions
................. 179
2.7-2.
Kernels Containing Exponential and Logarithmic Functions
................ 180
2.7-3.
Kernels Containing Exponential and Trigonometric Functions
............... 181
2.7-4.
Kernels Containing Hyperbolic and Logarithmic Functions
................. 185
2.7-5.
Kernels Containing Hyperbolic and Trigonometric Functions
............... 186
2.7-6.
Kernels Containing Logarithmic and Trigonometric Functions
.............. 187
2.8.
Equations Whose Kernels Contain Special Functions
............................ 187
2.8-1.
Kernels Containing Bessel Functions
.................................. 187
2.8-2.
Kernels Containing Modified Bessel Functions
.......................... 189
2.9.
Equations Whose Kernels Contain Arbitrary Functions
.......................... 191
2.9-1.
Equations with Degenerate Kernel: K(x,t)
=
g¡(x)hi(t) +
■■■ +
gn(,x)hn(t)
■··· 191
2.9-2.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
..................... 203
2.9-3.
Other Equations
................................................... 212
2.10.
Some Formulas and Transformations
....................................... 215
3.
Linear Equations of the First Kind with Constant Limits of Integration
........... 217
3.1.
Equations Whose Kernels Contain Power-Law Functions
........................ 217
-1.
Kernels Linear in the Arguments
χ
and
t
................................ 217
-2.
Kernels Quadratic in the Arguments
x
and
t
............................. 219
-3.
Kernels Containing Integer Powers of
x
and
t
or Rational Functions
.......... 220
-4.
Kernels Containing Square Roots
..................................... 222
-5.
Kernels Containing Arbitrary Powers
.................................. 223
-6.
Equations Containing the Unknown Function of a Complicated Argument
..... 227
3.1-7.
Singular Equations
................................................. 228
3.2.
Equations Whose Kernels Contain Exponential Functions
........................ 231
3.2-1.
Kernels Containing Exponential Functions of the Form
ρλΙΓ
I
............... 231
3.2-2.
Kernels Containing Exponential Functions of the Forms eAr and
с
......... 234
3.2-3.
Kernels Containing Exponential Functions of the Form eXxt
................ 234
3.2-4.
Kernels Containing Power-Law and Exponential Functions
................. 236
3.2-5.
Kernels Containing Exponential Functions of the Form eMx±tr
............. 236
3.2-6.
Other Kernels
..................................................... 237
3.3.
Equations Whose Kernels Contain Hyperbolic Functions
......................... 238
3.3-1.
Kernels Containing Hyperbolic Cosine
................................. 238
3.3-2.
Kernels Containing Hyperbolic Sine
................................... 238
3.3-3.
Kernels Containing Hyperbolic Tangent
................................ 241
3.3-4.
Kernels Containing Hyperbolic Cotangent
.............................. 242
viii Contents
3.4.
Equations
Whose
Kernels
Contain Logarithmic Functions
........................ 242
3.4-1.
Kernels Containing Logarithmic Functions
.............................. 242
3.4-2.
Kernels Containing Power-Law and Logarithmic Functions
................. 244
3.4-3.
Equation Containing the Unknown Function of a Complicated Argument
...... 246
3.5.
Equations Whose Kernels Contain Trigonometric Functions
...................... 246
3.5-1.
Kernels Containing Cosine
.......................................... 246
3.5-2.
Kernels Containing Sine
............................................ 247
3.5-3.
Kernels Containing Tangent
.......................................... 251
3.5-4.
Kernels Containing Cotangent
........................................ 252
3.5-5.
Kernels Containing a Combination of Trigonometric Functions
.............. 252
3.5-6.
Equations Containing the Unknown Function of a Complicated Argument
..... 254
3.5-7.
Singular Equations
................................................. 255
3.6.
Equations Whose Kernels Contain Combinations of Elementary Functions
.......... 255
3.6-1.
Kernels Containing Hyperbolic and Logarithmic Functions
................. 255
3.6-2.
Kernels Containing Logarithmic and Trigonometric Functions
.............. 256
3.6-3.
Kernels Containing Combinations of Exponential and Other Elementary
Functions
........................................................ 257
3.7.
Equations Whose Kernels Contain Special Functions
............................ 258
3.7-1.
Kernels Containing Error Function, Exponential Integral or Logarithmic Integral
258
3.7-2.
Kernels Containing Sine Integrals, Cosine Integrals, or Fresnel Integrals
...... 258
3.7-3.
Kernels Containing Gamma Functions
................................. 260
3.7-4.
Kernels Containing Incomplete Gamma Functions
........................ 260
3.7-5.
Kernels Containing Bessel Functions of the First Kind
..................... 261
3.7-6.
Kernels Containing Bessel Functions of the Second Kind
.................. 264
3.7-7.
Kernels Containing Combinations of the Bessel Functions
................. 265
3.7-8.
Kernels Containing Modified Bessel Functions of the First Kind
............. 266
3.7-9.
Kernels Containing Modified Bessel Functions of the Second Kind
.......... 266
3.7-10.
Kernels Containing a Combination of Bessel and Modified Bessel Functions
. . 269
3.7-11.
Kernels Containing Legendre Functions
............................... 270
3.7-12.
Kernels Containing Associated Legendre Functions
...................... 271
3.7-13.
Kernels Containing
Kummer
Confluent Hypergeometric Functions
.......... 272
3.7-14.
Kernels Containing
Tricomi
Confluent Hypergeometric Functions
.......... 274
3.7-15.
Kernels Containing Whittaker Confluent Hypergeometric Functions
......... 274
3.7-16.
Kernels Containing Gauss Hypergeometric Functions
.................... 276
3.7-17.
Kernels Containing Parabolic Cylinder Functions
........................ 276
3.7-18.
Kernels Containing Other Special Functions
............................ 277
3.8.
Equations Whose Kernels Contain Arbitrary Functions
.......................... 278
3.8-1.
Equations with Degenerate Kernel
.................................... 278
3.8-2.
Equations Containing Modulus
....................................... 279
3.8-3.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
..................... 284
3.8-4.
Other Equations of the Form
ƒ
Ь
K(x, t)y(t) dt
=
F{x)
..................... 285
Ja
eb
3.8-5.
Equations of the Form
Ц
K(x, t)y{- ■■)dt = F(x)
........................ 289
3.9.
Dual Integral Equations of the First Kind
..................................... 295
3.9-1.
Kernels Containing Trigonometric Functions
............................ 295
3.9-2.
Kernels Containing Bessel Functions of the First Kind
..................... 297
3.9-3.
Kernels Containing Bessel Functions of the Second Kind
.................. 299
3.9-4.
Kernels Containing Legendre Spherical Functions of the First Kind, i2
= -1 ... 299
Contents
ix
4.
Linear Equations of the Second Kind with Constant Limits of Integration
......... 301
4.1.
Equations Whose Kernels Contain Power-Law Functions
........................ 301
4.1 -1.
Kernels Linear in the Arguments
χ
and
t
................................ 301
4.1-2.
Kernels Quadratic in the Arguments
x
and
t
............................. 304
4.1-3.
Kernels Cubic in the Arguments
x
and
t
................................ 307
4.1-4.
Kernels Containing Higher-Order Polynomials in
x
and
t
.................. 311
4.1-5.
Kernels Containing Rational Functions
................................. 314
4.1-6.
Kernels Containing Arbitrary Powers
.................................. 317
4.1-7.
Singular Equations
................................................. 319
4.2.
Equations Whose Kernels Contain Exponential Functions
........................ 320
4.2-1.
Kernels Containing Exponential Functions
.............................. 320
4.2-2.
Kernels Containing Power-Law and Exponential Functions
................. 326
4.3.
Equations Whose Kernels Contain Hyperbolic Functions
......................... 327
4.3-1.
Kernels Containing Hyperbolic Cosine
................................. 327
4.3-2.
Kernels Containing Hyperbolic Sine
................................... 329
4.3-3.
Kernels Containing Hyperbolic Tangent
................................ 332
4.3-4.
Kernels Containing Hyperbolic Cotangent
.............................. 333
4.3-5.
Kernels Containing Combination of Hyperbolic Functions
................. 334
4.4.
Equations Whose Kernels Contain Logarithmic Functions
........................ 334
4.4-1.
Kernels Containing Logarithmic Functions
.............................. 334
4.4-2.
Kernels Containing Power-Law and Logarithmic Functions
................. 335
4.5.
Equations Whose Kernels Contain Trigonometric Functions
...................... 335
4.5-1.
Kernels Containing Cosine
.......................................... 335
4.5-2.
Kernels Containing Sine
............................................ 337
4.5-3.
Kernels Containing Tangent
.......................................... 342
4.5-4.
Kernels Containing Cotangent
........................................ 343
4.5-5.
Kernels Containing Combinations of Trigonometric Functions
.............. 344
4.5-6.
Singular Equation
.................................................. 344
4.6.
Equations Whose Kernels Contain Inverse Trigonometric Functions
................ 344
4.6-1.
Kernels Containing Arccosine
........................................ 344
4.6-2.
Kernels Containing Arcsine
.......................................... 345
4.6-3.
Kernels Containing Arctangent
....................................... 346
4.6-4.
Kernels Containing Arccotangent
..................................... 347
4.7.
Equations Whose Kernels Contain Combinations of Elementary Functions
.......... 348
4.7-1.
Kernels Containing Exponential and Hyperbolic Functions
................. 348
4.7-2.
Kernels Containing Exponential and Logarithmic Functions
................ 349
4.7-3.
Kernels Containing Exponential and Trigonometric Functions
............... 349
4.7-4.
Kernels Containing Hyperbolic and Logarithmic Functions
................. 351
4.7-5.
Kernels Containing Hyperbolic and Trigonometric Functions
............... 352
4.7-6.
Kernels Containing Logarithmic and Trigonometric Functions
.............. 353
4.8.
Equations Whose Kernels Contain Special Functions
............................ 353
4.8-1.
Kernels Containing Bessel Functions
.................................. 353
4.8-2.
Kernels Containing Modified Bessel Functions
.......................... 355
4.9.
Equations Whose Kernels Contain Arbitrary Functions
.......................... 357
4.9-1.
Equations with Degenerate Kernel: K{x,t) = g (x)h {t) +
■ ■ ■ +
gn{x)hn{t)
----- 357
4.9-2.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
..................... 372
4.9-3.
Other Equations of the Form y(x)
+
f* K(x, t)y(t) dt
-
F(x)
............... 374
4.9-4.
Equations of the Form y(x)
+
¡I K(x, t)y(- ■■)dt = F(x)
................... 381
4.10.
Some Formulas and Transformations
....................................... 390
Contents
5.
Nonlinear Equations of the First Kind with Variable Limit of Integration
......... 393
5.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
........... 393
5.1-1.
Equations of the Form ¡* y(t)y(x
-
1) dt
=
f(x)
.......................... 393
5.1-2.
Equations of the Form
ƒ„*
K(x, t)y(t)y(x -t)dt = f(x)
.................... 395
5.1-3.
Equations of the Form f* y(t)y(- ■•)dt = f(x)
........................... 396
5.2.
Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
............ 397
5.2-1.
Equations of the Form
ƒ*
K(x, t)[Ay(t)
+
By2(t)] dt
=
f(x)
................ 397
5.2-2.
Equations of the Form
ƒƒ
K(x, t)y(t)y(ax
+
bt) dt
= ƒ
(x)
.................. 398
5.3.
Equations with Nonlinearity of General Form
.................................. 399
5.3-1.
Equations of the Form
ƒ*
K(x, t)f(t, y(t)) dt
=
g(x)
...................... 399
5.3-2.
Other Equations
..... ............................................. 401
6.
Nonlinear Equations of the Second Kind with Variable Limit of Integration
....... 403
6.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
........... 403
6.1 -1.
Equations of the Form y(x)
+ ƒ*
K(x, t)y t) dt
=
F(x)
................... 403
6.1-2.
Equations of the Form y(x)
+
j* K(x, t)y(t)y(x -t)dt = F(x)
.............. 406
6.2.
Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
............ 406
6.2-1.
Equations of the Form y{x)
+ ƒ*
K{x, t)y t) dt
=
F(x)
................... 406
6.2-2.
Other Equations
................................................... 407
6.3.
Equations with Power-Law Nonlinearity
...................................... 408
6.3-1.
Equations Containing Arbitrary Parameters
............................. 408
6.3-2.
Equations Containing Arbitrary Functions
.............................. 410
6.4.
Equations with Exponential Nonlinearity
..................................... 411
6.4-1.
Equations Containing Arbitrary Parameters
............................. 411
6.4-2.
Equations Containing Arbitrary Functions
.............................. 413
6.5.
Equations with Hyperbolic Nonlinearity
...................................... 414
6.5-1.
Integrands with Nonlinearity of the Form
cosh[/3y(í)]
..................... 414
6.5-2.
Integrands with Nonlinearity of the Form sinh[/3y(i)]
..................... 415
6.5-3.
Integrands with Nonlinearity of the Form
tanh[ßy(t)] ..................... 416
6.5-4.
Integrands with Nonlinearity of the Form
coth[ßy(t)] ..................... 418
6.6.
Equations with Logarithmic Nonlinearity
..................................... 419
6.6-1.
Integrands Containing Power-Law Functions of
x
and
í
.................... 419
6.6-2.
Integrands Containing Exponential Functions of
x
and
t
................... 419
6.6-3.
Other Integrands
................................................... 420
6.7.
Equations with Trigonometric Nonlinearity
................................... 420
6.7-1.
Integrands with Nonlinearity of the Form
cos[ßy(t)] ...................... 420
6.7-2.
Integrands with Nonlinearity of the Form
sin[ßy(t)] ...................... 422
6.7-3.
Integrands with Nonlinearity of the Form
tan[ßy(t)] ...................... 423
6.7-4.
Integrands with Nonlinearity of the Form
cot[ßy(t)] ...................... 424
6.8.
Equations with Nonlinearity of General Form
.................................. 425
6.8-1.
Equations of the Form y(x)
+ ƒƒ
K(x, t)G(y(t)) dt
=
F(x)
................. 425
6.8-2.
Equations of the Form y{x)
+
f* K(x
-
t)G(t, y(t)) dt
=
F{x)
.............. 428
6.8-3.
Other Equations
................................................... 431
7.
Nonlinear Equations of the First Kind with Constant Limits of Integration
........ 433
7.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
........... 433
7.1-1.
Equations of the Form ¡^ K(t)y(x)y(t) dt
=
F(x)
........................ 433
7.1-2.
Equations of the Form
£
K(t)y(t)y(xt) dt
=
F(x)
....................... 435
7.1-3.
Other Equations
................................................... 436
CONTKNTS
X¡
7.2.
Equations
with Quadratic Nonlinearity That Contain Arbitrary Functions
............ 437
7.2-1.
Equations of the Form ¡^ K(t)y(t)y(- ■■)dt = F(x)
....................... 437
7.2-2.
Equations of the Form J^[K(x, t)y(t)
+
M(x, t)y2(t)] dt
=
F{x)
............. 443
7.3.
Equations with Power-Law Nonlinearity That Contain Arbitrary Functions
.......... 444
7.3-1.
Equations of the Form f* K(t)y^l(x)y l(t) dt
=
F(x)
...................... 444
7.3-2.
Equations of the Form /j* K(t)yi(t)y{xt) dt
-
F(x)
...................... 444
7.3-3.
Equations of the Form /J* K(t)y^(t)y(x
+ ßt)dt =
F (x)
................... 445
7.3-4.
Equations of the Form J^[K(x, t)y(t)
+
M(x, t)y~<(t)} dt
=
f(x)
............. 446
7.3-5.
Other Equations
................................................... 446
7.4.
Equations with Nonlinearity of General Form
.................................. 447
7.4-1.
Equations of the Form j^<p{ij{x))K(t,y(t)) dt
=
F(x)
.................... 447
7.4-2.
Equations of the Form
£
y(xt)K(t, y(t)) dt
-
F(x)
...................... 447
7.4-3.
Equations of the Form J* y(x
+ ßt)K(t,
y(t)) dt
=
F(x)
................... 449
7.4-4.
Equations of the Form tf[K(x, t)y(t)
+
ψ(χ)^{ί,
у(Щ
dt =
F (x)
........... 450
7.4-5.
Other Equations
................................................... 451
8.
Nonlinear Equations of the Second Kind with Constant Limits of Integration
...... 453
8.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
........... 453
8.1-1.
Equations of the Form y(x)
+
J^ K(x, t)y t) dt
=
F(x)
................... 453
8.
1
-2.
Equations of the Form y{x)
+
/j K(x, t)y(x)y(t) dt
=
F(x)
................. 454
8.1-3.
Equations of the Form y(x)
+
¡^ K(t)y(t)y(- --)dt = F(x)
................. 455
8.2.
Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
............ 456
8.2-1.
Equations of the Form y(x)
+ £
K(x, t)y2(t) dt
=
F(x)
................... 456
8.2-2.
Equations of the Form y(x)
+
¡^
Σ
Knm{x, t)yn(x)ym(t) dt
=
Fix),
η
+
m
< 2 457
8.2-3.
Equations of the Form y(x)
+
¡^ K(t)y(t)y(- --)dt = F(x)
................. 460
8.3.
Equations with Power-Law Nonlinearity
...................................... 464
8.3-1.
Equations of the Form y(x)
+
f£ K(x, t)i/(t) dt
=
F(x)
................... 464
8.3-2.
Other Equations
................................................... 465
8.4.
Equations with Exponential Nonlinearity
..................................... 467
8.4-1.
Integrands with Nonlinearity of the Form
e p[ßy(t)] ...................... 467
8.4-2.
Other Integrands
................................................... 468
8.5.
Equations with Hyperbolic Nonlinearity
...................................... 468
8.5-1.
Integrands with Nonlinearity of the Form
cosh[ßy(t)] ..................... 468
8.5-2.
Integrands with Nonlinearity of the Form
sinh[ßy(t)] ..................... 469
8.5-3.
Integrands with Nonlinearity of the Form
liinh[ßy(t) ..................... 469
8.5-4.
Integrands with Nonlinearity of the Form
coih[ßy(t)] ..................... 470
8.5-5.
Other Integrands
................................................... 471
8.6.
Equations with Logarithmic Nonlinearity
..................................... 472
8.6-1.
Integrands with Nonlinearity of the Form
n[ßy(t) ....................... 472
8.6-2.
Other Integrands
................................................... 473
8.7.
Equations with Trigonometric Nonlinearity
................................... 473
8.7-1.
Integrands with Nonlinearity of the Form
cos[ßy(t)} ...................... 473
8.7-2.
Integrands with Nonlinearity of the Form
un[ßy{t)} ...................... 474
8.7-3.
Integrands with Nonlinearity of the Form
ian[ßy(t)] ...................... 475
8.7-4.
Integrands with Nonlinearity of the Form
coi[ßy(t)] ...................... 475
8.7-5.
Other Integrands
................................................... 476
xii Contents
8.8.
Equations
with Nonlinearity of General Form
.................................. 477
8.8-1.
Equations of the Form y(x)
+
J^K( x-t )G (y(t)) dt
=
F(x)
............... 477
8.8-2.
Equations of the Form y(x)
+
¡^ K{x, t)G(t, y(t)) dt
=
F(x)
............... 479
8.8-3.
Equations of the Form y(x)
+
J* G(x, t, y(t)) dt
=
F(x)
................... 483
8.8-4.
Equations of the Form y(x)
+ £
y(xt)G(t, y(t)) dt
=
F(x)
................. 485
8.8-5.
Equations of the Form y(x)
+
¡* y(x
+ ßt)G
(t,
y(t))
dt =
F(x)
.............. 487
8.8-6.
Other Equations
................................................... 494
Part II. Methods for Solving Integral Equations
9.
Main Definitions and Formulas. Integral Transforms
.......................... 501
9.1.
Some Definitions, Remarks, and Formulas
.................................... 501
9.1-1.
Some Definitions
.................................................. 501
9.1-2.
Structure of Solutions to Linear Integral Equations
....................... 502
9.1-3.
Integral Transforms
................................................ 503
9.1-4.
Residues. Calculation Formulas. Cauchy s Residue Theorem
............... 504
9.1-5.
Jordan Lemma
.................................................... 505
9.2.
Laplace Transform
....................................................... 505
9.2-1.
Definition. Inversion Formula
........................................ 505
9.2-2.
Inverse Transforms of Rational Functions
............................... 506
9.2-3.
Inversion of Functions with Finitely Many Singular Points
................. 507
9.2-4.
Convolution Theorem. Main Properties of the Laplace Transform
............ 507
9.2-5.
Limit Theorems
................................................... 507
9.2-6.
Representation of Inverse Transforms as Convergent Series
................. 509
9.2-7.
Representation of Inverse Transforms as Asymptotic Expansions as
x
—>
oo
... 509
9.2-8.
Post-Widder Formula
.............................................. 510
9.3.
Mellin Transform
........................................................ 510
9.3-1.
Definition. Inversion Formula
........................................ 510
9.3-2.
Main Properties of the Mellin Transform
............................... 511
9.3-3.
Relation Among the Mellin, Laplace, and Fourier Transforms
............... 511
9.4.
Fourier Transform
....................................................... 512
9.4-1.
Definition. Inversion Formula
........................................ 512
9.4-2.
Asymmetric Form of the Transform
................................... 512
9.4-3.
Alternative Fourier Transform
........................................ 512
9.4-4.
Convolution Theorem. Main Properties of the Fourier Transforms
........... 513
9.5.
Fourier Cosine and Sine Transforms
......................................... 514
9.5-1.
Fourier Cosine Transform
........................................... 514
9.5-2.
Fourier Sine Transform
............................................. 514
9.6.
Other Integral Transforms
................................................. 515
9.6-1.
Hankel Transform
................................................. 515
9.6-2.
Meijer
Transform
.................................................. 516
9.6-3.
Kontorovich-Lebedev Transform
..................................... 516
9.6-4.
r-transform
...................................................... 516
9.6-5.
Summary Table of Integral Transforms
................................. 517
10.
Methods for Solving Linear Equations of the Form f* K(x,t)y(t) dt
=
f(x)
..... 519
10.1.
Volteira
Equations of the First Kind
........................................ 519
10.1-1.
Equations of the First Kind. Function and Kernel Classes
................ 519
10.1-2.
Existence and Uniqueness of a Solution
.............................. 520
10.1 -3.
Some Problems Leading to Volterra Integral Equations of the First Kind
.... 520
Contents xiii
10.2.
Equations
with Degenerate Kernel: K(x, t)
=
gx(x)h (t)
+ ■■■ +
gn(x)hn(t)
......... 522
10.2-1.
Equations with Kernel of the Form K(x,t) = gdx)hdt) +g2(x)h2(t)
....... 522
10.2-2.
Equations with General Degenerate Kernel
............................ 523
10.3.
Reduction of Volterra Equations of the First Kind to Volterra Equations of the Second
Kind
................................................................. 524
10.3-1.
First Method
.................................................... 524
10.3-2.
Second Method
................................................. 524
10.4.
Equations with Difference Kernel: K(x, t)
=
K(x
-ť)
.......................... 524
10.4-1.
Solution Method Based on the Laplace Transform
...................... 524
10.4-2.
Case in Which the Transform of the Solution is a Rational Function
........ 525
10.4-3.
Convolution Representation of a Solution
............................. 526
10.4-4.
Application of an Auxiliary Equation
................................ 527
10.4-5.
Reduction to Ordinary Differential Equations
.......................... 527
10.4-6.
Reduction of a Volterra Equation to
a
Wiener-Hopf
Equation
............. 528
10.5.
Method of Fractional Differentiation
........................................ 529
10.5-1.
Definition of Fractional Integrals
.................................... 529
10.5-2.
Definition of Fractional Derivatives
.................................. 529
10.5-3.
Main Properties
................................................. 530
10.5-4.
Solution of the Generalized Abel Equation
............................ 531
10.5-5.
Erdélyi-Kober
Operators
.......................................... 532
10.6.
Equations with Weakly Singular Kernel
..................................... 532
10.6-1.
Method of Transformation of the Kernel
.............................. 532
10.6-2.
Kernel with Logarithmic Singularity
................................. 533
10.7.
Method of Quadratures
.................................................. 534
10.7-1.
Quadrature Formulas
............................................. 534
10.7-2.
General Scheme of the Method
..................................... 535
10.7-3.
Algorithm Based on the Trapezoidal Rule
............................. 536
10.7-4.
Algorithm for an Equation with Degenerate Kernel
..................... 536
10.8.
Equations with Infinite Integration Limit
.................................... 537
10.8-1.
Equation of the First Kind with Variable Lower Limit of Integration
........ 537
10.8-2.
Reduction
toa
Wiener-Hopf
Equation of the First Kind
................. 538
11.
Methods for Solving Linear Equations of the Form y(x)
-
f* K(x, t)y(t) dt
=
f(x)
539
11.1.
Volterra Integral Equations of the Second Kind
............................... 539
1.1-1.
Preliminary Remarks. Equations for the Resolvent
..................... 539
1.1-2.
Relationship Between Solutions of Some Integral Equations
.............. 540
11.2.
Equations with Degenerate Kernel: K(x,t)
=
g (x)h (t)
+ ■ ■ ■ +
gn(x)hn(t)
......... 540
1.2-1.
Equations with Kernel of the Form K(x, t)
=
ψ(χ)
+
ψ(χ)(χ
-t)
........... 540
1.2-2.
Equations with Kernel of the Form K(x, t)
=
ψ(ί)
+
tp(t)(t -x)
............ 541
1.2-3.
Equations with Kernel of the Form K(x,t) = T,m= ^rn(x)(x-t)m-]
....... 542
1
.2-4.
Equations with Kernel of the Form K(x, t)
=
^Li V™(W
-
x)m~x
....... 543
1.2-5.
Equations with Degenerate Kernel of the General Form
.................. 543
11.3.
Equations with Difference Kernel: K(x, t)
=
K(x
-í)
.......................... 544
1
.3-1.
Solution Method Based on the Laplace Transform
...................... 544
.3-2.
Method Based on the Solution of an Auxiliary Equation
................. 546
.3-3.
Reduction to Ordinary Differential Equations
.......................... 547
.3-4.
Reduction to
a
Wiener-Hopf
Equation of the Second Kind
............... 547
.3-5.
Method of Fractional Integration for the Generalized Abel Equation
........ 548
.3-6.
Systems of Volterra Integral Equations
............................... 549
xiv Contents
11.4. Operator
Methods for Solving Linear Integral Equations
........................ 549
11.4-1.
Application of a Solution of a Truncated Equation of the First Kind
...... 549
11.4-2.
Application of the Auxiliary Equation of the Second Kind
................ 551
11.4-3.
Method for Solving Quadratic Operator Equations
.................... 552
11.4-4.
Solution of Operator Equations of Polynomial Form
.................... 553
11.4-5.
Some Generalizations
............................................ 554
11.5.
Construction of Solutions of Integral Equations with Special Right-Hand Side
....... 555
11.5-1.
General Scheme
................................................. 555
11.5-2.
Generating Function of Exponential Form
............................ 555
11.5-3.
Power-Law Generating Function
.................................... 557
11.5-4.
Generating Function Containing Sines and Cosines
..................... 558
11.6.
Method of Model Solutions
............................................... 559
11.6-1.
Preliminary Remarks
.............................................. 559
11.6-2.
Description of the Method
......................................... 560
11.6-3.
Model Solution in the Case of an Exponential Right-Hand Side
........... 561
11.6-4.
Model Solution in the Case of a Power-Law Right-Hand Side
............. 562
11.6-5.
Model Solution in the Case of a Sine-Shaped Right-Hand Side
............ 562
11.6-6.
Model Solution in the Case of a Cosine-Shaped Right-Hand Side
.......... 563
11.6-7.
Some Generalizations
............................................ 563
11.7.
Method of Differentiation for Integral Equations
.............................. 564
11.7-1.
Equations with Kernel Containing a Sum of Exponential Functions
........ 564
11.7-2.
Equations with Kernel Containing a Sum of Hyperbolic Functions
......... 564
11.7-3.
Equations with Kernel Containing a Sum of Trigonometric Functions
....... 564
11.7-4.
Equations Whose Kernels Contain Combinations of Various Functions
...... 565
11.8.
Reduction of Volterra Equations of the Second Kind to Volterra Equations of the First
Kind
................................................................. 565
11.8-1.
First Method
.................................................... 565
11.8-2.
Second Method
................................................. 566
11.9.
Successive Approximation Method
......................................... 566
11.9-1.
General Scheme
................................................. 566
11.9-2.
Formula for the Resolvent
......................................... 567
11.10.
Method of Quadratures
................................................. 568
11.10-1.
General Scheme of the Method
................................... 568
11.10-2.
Application of the Trapezoidal Rule
............................... 568
11.10-3.
Case of a Degenerate Kernel
..................................... 569
11.11.
Equations with Infinite Integration Limit
................................... 569
11.11-1.
Equation of the Second Kind with Variable Lower Integration Limit
...... 570
11.11-2.
Reduction to
a
Wiener-Hopf
Equation of the Second Kind
............. 571
12.
Methods for Solving Linear Equations of the Form ¡^ K(x, t)y(t) dt
= ƒ
(ж)
..... 573
12.1.
Some Definition and Remarks
............................................. 573
12.1-1.
Fredholm
Integral Equations of the First Kind
......................... 573
12.1-2.
Integral Equations of the First Kind with Weak Singularity
............... 574
12.1-3.
Integral Equations of Convolution Type
.............................. 574
12.1-4.
Dual Integral Equations of the First Kind
............................. 575
12.1-5.
Some Problems Leading to Integral Equations of the First Kind
........... 575
12.2.
Integral Equations of the First Kind with Symmetric Kernel
..................... 577
12.2-1.
Solution of an Integral Equation in Terms of Series in Eigenfunctions of Its
Kernel
......................................................... 577
12.2-2.
Method of Successive Approximations
............................... 579
CONTKNTS
XV
12.3. Integral
Equations
of the First Kind with Nonsymmetric Kernel
.................. 580
12.3-1.
Representation of a Solution in the Form of Series. General Description
.... 580
12.3-2.
Special Case of a Kernel That is a Generating Function
.................. 580
12.3-3.
Special Case of the Right-Hand Side Represented in Terms of Orthogonal
Functions
...................................................... 582
12.3-4.
General Case. Galerkin s Method
................................... 582
12.3-5.
Utilization of the Schmidt Kernels for the Construction of Solutions of
Equations
...................................................... 582
12.4.
Method of Differentiation for Integral Equations
.............................. 583
12.4-1.
Equations with Modulus
.......................................... 583
12.4-2.
Other Equations. Some Generalizations
.............................. 585
12.5.
Method of Integral Transforms
............................................ 586
12.5-1.
Equation with Difference Kernel on the Entire Axis
..................... 586
12.5-2.
Equations with Kernel K(x, t)
=
K(x/t) on the Semiaxis
................ 587
12.5-3.
Equation with Kernel K(x, t)
=
K(xt) and Some Generalizations
.......... 587
12.6.
Krein s Method and Some Other Exact Methods for Integral Equations of Special Types
588
12.6-1.
Krein s Method for an Equation with Difference Kernel with a Weak Singularity
588
12.6-2.
Kernel is the Sum of
a Nondegenerate
Kernel and an Arbitrary Degenerate
Kernel
......................................................... 589
12.6-3.
Reduction of Integral Equations of the First Kind to Equations of the Second
Kind
.......................................................... 591
12.7.
Riemann Problem for the Real Axis
........................................ 592
12.7-1.
Relationships Between the Fourier Integral and the Cauchy Type Integral
.... 592
12.7-2.
One-Sided Fourier Integrals
........................................ 593
12.7-3.
Analytic Continuation Theorem and the Generalized Liouville Theorem
.... 595
12.7-4.
Riemann Boundary Value Problem
.................................. 595
12.7-5.
Problems with Rational Coefficients
................................. 601
12.7-6.
Exceptional Cases. The Homogeneous Problem
........................ 602
12.7-7.
Exceptional Cases. The Nonhomogeneous Problem
..................... 604
12.8.
Carleman
Method for Equations of the Convolution Type of the First Kind
......... 606
12.8-1. Wiener-Hopf
Equation of the First Kind
.............................. 606
12.8-2.
Integral Equations of the First Kind with Two Kernels
................... 607
12.9.
Dual Integral Equations of the First Kind
.................................... 610
12.9-1.
Carleman
Method for Equations with Difference Kernels
................ 610
12.9-2.
General Scheme of Finding Solutions of Dual Integral Equations
.......... 611
12.9-3.
Exact Solutions of Some Dual Equations of the First Kind
................ 613
12.9-4.
Reduction of Dual Equations to
a Fredholm
Equation
................... 615
12.10.
Asymptotic Methods for Solving Equations with Logarithmic Singularity
......... 618
12.10-1.
Preliminary Remarks
........................................... 618
12.10-2.
Solution for Large A
............................................ 619
12.10-3.
Solution for Small A
............................................ 620
12.10-4.
Integral Equation of Elasticity
.................................... 621
12.11.
Regularization Methods
................................................. 621
12.11-1.
Lavrentiev Regularization Method
................................ 621
12.11-2.
Tikhonov Regularization Method
................................. 622
12.12.
Fredholm
Integral Equation of the First Kind as an Ill-Posed Problem
............ 623
12.12-1.
General Notions of Well-Posed and Ill-Posed Problems
................ 623
12.12-2.
Integral Equation of the First Kind
isan
Ill-Posed Problem
............. 624
xvi Contents
13.
Methods for Solving Linear Equations of the Form y(x)
-
f£ K(x, t)y(t) dt
= ƒ
(ж)
625
13.1.
Some Definition and Remarks
............................................. 625
13.1-1.
Fredholm
Equations and Equations with Weak Singularity of the Second Kind
625
13.1-2.
Structure of the Solution
.......................................... 626
13.1-3.
Integral Equations of Convolution Type of the Second Kind
.............. 626
13.1-4.
Dual Integral Equations of the Second Kind
........................... 627
13.2.
Fredholm
Equations of the Second Kind with Degenerate Kernel. Some Generalizations
627
13.2-1.
Simplest Degenerate Kernel
........................................ 627
13.2-2.
Degenerate Kernel in the General Case
............................... 628
13.2-3.
Kernel is the Sum of
a Nondegenerate
Kernel and an Arbitrary Degenerate
Kernel
......................................................... 631
13.3.
Solution as a Power Series in the Parameter. Method of Successive Approximations
.. 632
13.3-1.
Iterated Kernels
................................................. 632
13.3-2.
Method of Successive Approximations
............................... 633
13.3-3.
Construction of the Resolvent
...................................... 633
13.3-4.
Orthogonal Kernels
.............................................. 634
13.4.
Method of
Fredholm
Determinants
......................................... 635
13.4-1.
Formula for the Resolvent
......................................... 635
13.4-2.
Recurrent Relations
.............................................. 636
13.5.
Fredholm
Theorems and the
Fredholm
Alternative
............................. 637
13.5-1.
Fredholm
Theorems
.............................................. 637
13.5-2.
Fredholm
Alternative
............................................. 638
13.6.
Fredholm
Integral Equations of the Second Kind with Symmetric Kernel
........... 639
13.6-1.
Characteristic Values and Eigenfunctions
............................. 639
13.6-2.
Bilinear Series
.................................................. 640
13.6-3.
Hilbert-Schmidt Theorem
......................................... 641
13.6-4.
Bilinear Series of Iterated Kernels
................................... 642
13.6-5.
Solution of the Nonhomogeneous Equation
........................... 642
13.6-6.
Fredholm
Alternative for Symmetric Equations
........................ 643
13.6-7.
Resolvent of a Symmetric Kernel
................................... 644
13.6-8.
Extremal Properties of Characteristic Values and Eigenfunctions
.......... 644
13.6-9.
Kellog s Method for Finding Characteristic Values in the Case of Symmetric
Kernel
......................................................... 645
13.6-10.
Trace Method for the Approximation of Characteristic Values
............ 646
13.6-11.
Integral Equations Reducible to Symmetric Equations
.................. 647
13.6-12.
Skew-Symmetric Integral Equations
................................ 647
13.6-13.
Remark on Nonsymmetric Kernels
................................. 647
13.7.
Integral Equations with
Nonnegative
Kernels
................................. 648
13.7-1.
Positive Principal Eigenvalues. Generalized Jentzch Theorem
............. 648
13.7-2.
Positive Solutions of a Nonhomogeneous Integral Equation
............... 649
13.7-3.
Estimates for the Spectral Radius
................................... 649
13.7-4.
Basic Definition and Theorems for Oscillating Kernels
.................. 651
13.7-5.
Stochastic Kernels
............................................... 654
13.8.
Operator Method for Solving Integral Equations of the Second Kind
.............. 655
13.8-1.
Simplest Scheme
................................................ 655
13.8-2.
Solution of Equations of the Second Kind on the Semiaxis
............... 655
CONTKNTS
XVII
13.
1-І
13.
1-2
13.
1-3
13.
1-4
13.
1-5
13.9.
Methods of Integral Transforms and Model Solutions
.......................... 656
13.9-1.
Equation with Difference Kernel on the Entire Axis
..................... 656
13.9-2.
Equation with the Kernel K(x, t)
=
t~xQ(x/t) on the Semiaxis
............ 657
13.9-3.
Equation with the Kernel K(x, t)
=
t0Q(xt) on the Semiaxis
............. 658
13.9-4.
Method of Model Solutions for Equations on the Entire Axis
............. 659
13.10.
Carleman
Method for Integral Equations of Convolution Type of the Second Kind
. . 660
13.10-1. Wiener-Hopf
Equation of the Second Kind
......................... 660
13.10-2.
Integral Equation of the Second Kind with Two Kernels
............... 664
13.10-3.
Equations of Convolution Type with Variable Integration Limit
.......... 668
13.10-4.
Dual Equation of Convolution Type of the Second Kind
............... 670
13.11. Wiener-Hopf
Method
.................................................. 671
Some Remarks
................................................ 671
Homogeneous
Wiener-Hopf
Equation of the Second Kind
............. 673
General Scheme of the Method. The Factorization Problem
............ 676
Nonhomogeneous
Wiener-Hopf
Equation of the Second Kind
.......... 677
Exceptional Case of
a
Wiener-Hopf
Equation of the Second Kind
....... 678
13.12.
Krein s Method for
Wiener-Hopf
Equations
................................. 679
13.12-1.
Some Remarks. The Factorization Problem
......................... 679
13.12-2.
Solution of the
Wiener-Hopf
Equations of the Second Kind
............ 681
13.12-3.
Hopf-Fock Formula
............................................ 683
13.13.
Methods for Solving Equations with Difference Kernels on a Finite Interval
....... 683
13.13-1.
Krein s Method
............................................... 683
13.13-2.
Kernels with Rational Fourier Transforms
.......................... 685
13.13-3.
Reduction to Ordinary Differential Equations
........................ 686
13.14.
Method of Approximating a Kernel by a Degenerate One
...................... 687
13.14-1.
Approximation of the Kernel
..................................... 687
13.14-2.
Approximate Solution
.......................................... 688
13.15.
Bateman Method
...................................................... 689
13.15-1.
General Scheme of the Method
................................... 689
13.15-2.
Some Special Cases
............................................ 690
13.16.
Collocation Method
.................................................... 692
13.16-1.
General Remarks
.............................................. 692
13.16-2.
Approximate Solution
.......................................... 693
13.16-3.
Eigenfunctions of the Equation
................................... 694
13.17.
Method of Least Squares
................................................ 695
13.17-1.
Description of the Method
....................................... 695
13.17-2.
Construction of Eigenfunctions
................................... 696
13.18.
Bubnov-Galerkin Method
............................................... 697
13.18-1.
Description of the Method
....................................... 697
13.18-2.
Characteristic Values
........................................... 697
13.19.
Quadrature Method
.................................................... 698
13.19-1.
General Scheme for
Fredholm
Equations of the Second Kind
........... 698
13.19-2.
Construction of the Eigenfunctions
................................ 699
13.19-3.
Specific Features of the Application of Quadrature Formulas
............ 700
13.20.
Systems of
Fredholm
Integral Equations of the Second Kind
.................... 701
13.20-1.
Some Remarks
................................................ 701
13.20-2.
Method of Reducing a System of Equations to a Single Equation
........ 701
xviii Contents
13.21. Regularization
Method for Equations with Infinite Limits of Integration
........... 702
13.21-1.
Basic Equation and
Fredholm
Theorems
............................ 702
13.21-2.
Regularizing Operators
......................................... 703
13.21-3.
Regularization Method
.......................................... 704
14.
Methods for Solving Singular Integral Equations of the First Kind
.............. 707
14.1.
Some Definitions and Remarks
............................................ 707
14.1-1.
Integral Equations of the First Kind with Cauchy Kernel
................. 707
14.1-2.
Integral Equations of the First Kind with Hubert Kernel
................. 707
14.2.
Cauchy Type Integral
.................................................... 708
14.2-1.
Definition of the Cauchy Type Integral
............................... 708
14.2-2.
Holder Condition
................................................ 709
14.2-3.
Principal Value of a Singular Integral
................................ 709
14.2-4.
Multivalued Functions
............................................ 711
14.2-5.
Principal Value of a Singular Curvilinear Integral
....................... 712
14.2-6.
Poincaré-Bertrand
Formula
........................................ 714
14.3.
Riemann Boundary Value Problem
......................................... 714
14.3-1.
Principle of Argument. The Generalized Liouville Theorem
.............. 714
14.3-2.
Hermite Interpolation Polynomial
................................... 716
14.3-3.
Notion of the Index
.............................................. 716
14.3-4.
Statement of the Riemann Problem
.................................. 718
14.3-5.
Solution of the Homogeneous Problem
............................... 720
14.3-6.
Solution of the Nonhomogeneous Problem
............................ 721
14.3-7.
Riemann Problem with Rational Coefficients
.......................... 723
14.3-8.
Riemann Problem for a Half-Plane
.................................. 725
14.3-9.
Exceptional Cases of the Riemann Problem
........................... 727
14.3-10.
Riemann Problem for a Multiply Connected Domain
................... 731
14.3-11.
Riemann Problem for Open Curves
................................. 734
14.3-12.
Riemann Problem with a Discontinuous Coefficient
.................... 739
14.3-13.
Riemann Problem in the General Case
.............................. 741
14.3-14.
Hubert Boundary Value Problem
................................... 742
14.4.
Singular Integral Equations of the First Kind
................................. 743
14.4-1.
Simplest Equation with Cauchy Kernel
............................... 743
14.4-2.
Equation with Cauchy Kernel on the Real Axis
........................ 743
14.4-3.
Equation of the First Kind on a Finite Interval
......................... 744
14.4-4.
General Equation of the First Kind with Cauchy Kernel
.................. 745
14.4-5.
Equations of the First Kind with Hubert Kernel
........................ 746
14.5.
Multhopp-Kalandiya Method
............................................. 747
14.5-1.
Solution That is Unbounded at the
Endpoints
of the Interval
.............. 747
14.5-2.
Solution Bounded at One
Endpoint
of the Interval
...................... 749
14.5-3.
Solution Bounded at Both
Endpoints
of the Interval
..................... 750
14.6.
Hypersingular Integral Equations
.......................................... 751
14.6-1.
Hypersingular Integral Equations with Cauchy- and Hilbert-Type Kernels
... 751
14.6-2.
Definition of Hypersingular Integrals
................................ 751
14.6-3.
Exact Solution of the Simplest Hypersingular Equation with Cauchy-Type
Kernel
......................................................... 753
14.6-4.
Exact Solution of the Simplest Hypersingular Equation with Hilbert-Type
Kernel
......................................................... 754
14.6-5.
Numerical Methods for Hypersingular Equations
....................... 754
CONTKNTS
ХІХ
15.
Methods for Solving Complete Singular Integral Equations
.................... 757
15.1.
Some Definitions and Remarks
............................................ 757
15.1-1.
Integral Equations with Cauchy Kernel
............................... 757
15.1-2.
Integral Equations with Hubert Kernel
............................... 759
15.1-3.
Fredholm
Equations of the Second Kind on a Contour
................... 759
15.2.
Carleman
Method for Characteristic Equations
................................ 761
15.2-1.
Characteristic Equation with Cauchy Kernel
........................... 761
15.2-2.
Transposed Equation of a Characteristic Equation
...................... 764
15.2-3.
Characteristic Equation on the Real Axis
............................. 765
15.2-4.
Exceptional Case of a Characteristic Equation
......................... 767
15.2-5.
Characteristic Equation with Hubert Kernel
........................... 769
15.2-6.
Tricomi
Equation
................................................ 769
15.3.
Complete Singular Integral Equations Solvable in a Closed Form
................. 770
15.3-1.
Closed-Form Solutions in the Case of Constant Coefficients
.............. 770
15.3-2.
Closed-Form Solutions in the General Case
........................... 77
1
15.4.
Regularization Method for Complete Singular Integral Equations
................. 772
15.4-1.
Certain Properties of Singular Operators
.............................. 772
15.4-2.
Regularizer
..................................................... 774
15.4-3.
Methods of Left and Right Regularization
............................ 775
15.4-4.
Problem of Equivalent Regularization
................................ 776
15.4-5.
Fredholm
Theorems
.............................................. 777
15.4-6.
Carleman-Vekua Approach to the Regularization
....................... 778
15.4-7.
Regularization in Exceptional Cases
................................. 779
15.4-8.
Complete Equation with Hubert Kernel
.............................. 780
15.5.
Analysis of Solutions Singularities for Complete Integral Equations with Generalized
Cauchy Kernels
........................................................ 783
15.5-1.
Statement of the Problem and Preliminary Remarks
..................... 783
15.5-2.
Auxiliary Results
................................................ 784
15.5-3.
Equations for the Exponents of Singularity of a Solution
................. 787
15.5-4.
Analysis of Equations for Singularity Exponents
....................... 789
15.5-5.
Application to an Equation Arising in Fracture Mechanics
................ 791
15.6.
Direct Numerical Solution of Singular Integral Equations with Generalized Kernels
.. 792
15.6-1.
Preliminary Remarks
............................................. 792
15.6-2.
Quadrature Formulas for Integrals with the Jacobi Weight Function
........ 793
15.6-3.
Approximation of Solutions in Terms of a System of Orthogonal Polynomials
795
15.6-4.
Some Special Functions and Their Calculations
........................ 797
15.6-5.
Numerical Solution of Singular Integral Equations
...................... 799
15.6-6.
Numerical Solutions of Singular Integral Equations of Bueckner Type
...... 801
16.
Methods for Solving Nonlinear Integral Equations
805
16.1.
Some Definitions and Remarks
............................................ 805
16.1-1.
Nonlinear Equations with Variable Limit of Integration (Volterra Equations)
. 805
16.1-2.
Nonlinear Equations with Constant Integration Limits (Urysohn Equations)
.. 806
16.1-3.
Some Special Features of Nonlinear Integral Equations
.................. 807
16.2.
Exact Methods for Nonlinear Equations with Variable Limit of Integration
.......... 809
16.2-1.
Method of Integral Transforms
..................................... 809
16.2-2.
Method of Differentiation for Nonlinear Equations with Degenerate Kernel
.. 810
xx Contents
16.3.
Approximate and Numerical Methods for Nonlinear Equations with Variable Limit of
Integration
............................................................ 811
16.3-1.
Successive Approximation Method
.................................. 811
16.3-2.
Newton-Kantorovich Method
...................................... 813
16.3-3.
Collocation Method
.............................................. 815
16.3-4.
Quadrature Method
.............................................. 816
16.4.
Exact Methods for Nonlinear Equations with Constant Integration Limits
.......... 817
16.4-1.
Nonlinear Equations with Degenerate Kernels
......................... 817
16.4-2.
Method of Integral Transforms
..................................... 819
16.4-3.
Method of Differentiating for Integral Equations
....................... 820
16.4-4.
Method for Special Urysohn Equations of the First Kind
................. 821
16.4-5.
Method for Special Urysohn Equations of the Second Kind
............... 822
16.4-6.
Some Generalizations
............................................ 824
16.5.
Approximate and Numerical Methods for Nonlinear Equations with Constant Integration
Limits
................................................................ 826
16.5-1.
Successive Approximation Method
.................................. 826
16.5-2.
Newton-Kantorovich Method
...................................... 827
16.5-3.
Quadrature Method
.............................................. 829
16.5-4.
Tikhonov Regularization Method
................................... 829
16.6
Existence and Uniqueness Theorems for Nonlinear Equations
.................... 830
16.6-1. Hammerstein
Equations
........................................... 830
16.6-2.
Urysohn Equations
............................................... 832
16.7.
Nonlinear Equations with a Parameter: Eigenfunctions, Eigenvalues, Bifurcation Points
834
16.7-1.
Eigenfunctions and Eigenvalues of Nonlinear Integral Equations
........... 834
16.7-2.
Local Solutions of a Nonlinear Integral Equation with a Parameter
......... 835
16.7-3.
Bifurcation Points of Nonlinear Integral Equations
...................... 835
17.
Methods for Solving Multidimensional Mixed Integral Equations
............... 839
17.1.
Some Definition and Remarks
............................................. 839
17.1-1.
Basic Classes of Functions
......................................... 839
17.1-2.
Mixed Equations on a Finite Interval
................................. 840
17.1-3.
Mixed Equation on a Ring-Shaped (Circular) Domain
................... 841
17.1-4.
Mixed Equations on a Closed Bounded Set
............................ 842
17.2.
Methods of Solution of Mixed Integral Equations on a Finite Interval
.............. 843
17.2-1.
Equation with a Hilbert-Schmidt Kernel and a Given Right-Hand Side
...... 843
17.2-2.
Equation with Hilbert-Schmidt Kernel and Auxiliary Conditions
.......... 845
17.2-3.
Equation with a Schmidt Kernel and a Given Right-Hand Side on an Interval
. 848
17.2-4.
Equation with a Schmidt Kernel and Auxiliary Conditions
............... 851
17.3.
Methods of Solving Mixed Integral Equations on a Ring-Shaped Domain
.......... 855
17.3-1.
Equation with a Hilbert-Schmidt Kernel and a Given Right-Hand Side
...... 855
17.3-2.
Equation with a Hilbert-Schmidt Kernel and Auxiliary Conditions
......... 856
17.3-3.
Equation with a Schmidt Kernel and a Given Right-Hand Side
............ 859
17.3-4.
Equation with a Schmidt Kernel and Auxiliary Conditions on Ring-Shaped
Domain
........................................................ 862
17.4.
Projection Method for Solving Mixed Equations on a Bounded Set
................ 866
17.4-1.
Mixed Operator Equation with a Given Right-Hand Side
................. 866
Π
.4-2.
Mixed Operator Equations with Auxiliary Conditions
................... 869
17.4-3.
General Projection Problem for Operator Equation
...................... 873
Contents
xxi
18.
Application of Integral Equations for the Investigation of Differential Equations
.. 875
18.1.
Reduction of the Cauchy Problem for ODEs to Integral Equations
................ 875
18.1-1.
Cauchy Problem for First-Order ODEs. Uniqueness and Existence Theorems
875
18.1-2.
Cauchy Problem for First-Order ODEs. Method of Successive Approximations
876
18.1-3.
Cauchy Problem for Second-Order ODEs. Method of Successive
Approximations
................................................. 876
18.1-4.
Cauchy Problem for a Special
η
-Order
Linear ODE
..................... 876
18.2.
Reduction of Boundary Value Problems for ODEs to Volterra Integral Equations.
Calculation of Eigenvalues
............................................... 877
18.2-1.
Reduction of Differential Equations to Volterra Integral Equations
......... 877
18.2-2.
Application of Volterra Equations to the Calculation of Eigenvalues
........ 879
18.3.
Reduction of Boundary Value Problems for ODEs to
Fredholm
Integral Equations with
the Help of the Green s Function
........................................... 881
18.3-1.
Linear Ordinary Differential Equations. Fundamental Solutions
........... 881
18.3-2.
Boundary Value Problems for nth Order Differential Equations. Green s
Function
....................................................... 882
18.3-3.
Boundary Value Problems for Second-Order Differential Equations. Green s
Function
....................................................... 883
18.3-4.
Nonlinear Problem of Nonisothermal Flow in Plane Channel
............. 884
18.4.
Reduction of PDEs with Boundary Conditions of the Third Kind to Integral Equations
887
18.4-1.
Usage of Particular Solutions of PDEs for the Construction of Other Solutions
887
18.4-2.
Mass Transfer to a Particle in Fluid Flow Complicated by a Surface Reaction
888
18.4-3.
Integral Equations for Surface Concentration and Diffusion Flux
.......... 890
18.4-4.
Method of Numerical Integration of the Equation for Surface Concentration
. 891
18.5.
Representation of Linear Boundary Value Problems in Terms of Potentials
.......... 892
18.5-1.
Basic Types of Potentials for the Laplace Equation and Their Properties
..... 892
18.5-2.
Integral Identities. Green s Formula
................................. 895
18.5-3.
Reduction of Interior Dirichlet and Neumann Problems to Integral Equations
. 895
18.5-4.
Reduction of Exterior Dirichlet and Neumann Problems to Integral Equations
896
18.6.
Representation of Solutions of Nonlinear PDEs in Terms of Solutions of Linear Integral
Equations (Inverse Scattering)
............................................. 898
18.6-1.
Description of the Zakharov-Shabat Method
.......................... 898
18.6-2.
Korteweg-de
Vries
Equation and Other Nonlinear Equations
............. 899
Supplements
Supplement
1.
Elementary Functions and Their Properties
....................... 905
1.1.
Power, Exponential, and Logarithmic Functions
................................ 905
1.1-1.
Properties of the Power Function
...................................... 905
1.1-2.
Properties of the Exponential Function
................................. 905
1.1-3.
Properties of the Logarithmic Function
................................. 906
1.2.
Trigonometric Functions
.................................................. 907
1.2-1.
Simplest Relations
................................................. 907
1.2-2.
Reduction Formulas
................................................ 907
1.2-3.
Relations Between Trigonometric Functions of Single Argument
............ 908
1.2-4.
Addition and Subtraction of Trigonometric Functions
..................... 908
1.2-5.
Products of Trigonometric Functions
.................................. 908
1.2-6.
Powers of Trigonometric Functions
.................................... 908
1.2-7.
Addition Formulas
................................................. 909
xxii Contents
1.2-8.
Trigonometric
Functions of Multiple Arguments
......................... 909
1.2-9.
Trigonometric Functions of Half Argument
............................. 909
1.2-10.
Differentiation Formulas
........................................... 910
1.2-11.
Integration Formulas
.............................................. 910
1.2-12.
Expansion in Power Series
.......................................... 910
1.2-13.
Representation in the Form of Infinite Products
......................... 910
1.2-14.
Euler
and
de Moivre
Formulas. Relationship with Hyperbolic Functions
..... 911
1.3.
Inverse Trigonometric Functions
............................................ 911
1.3-1.
Definitions of Inverse Trigonometric Functions
.......................... 911
1.3-2.
Simplest Formulas
................................................. 912
1.3-3.
Some Properties
................................................... 912
1.3-4.
Relations Between Inverse Trigonometric Functions
...................... 912
1.3-5.
Addition and Subtraction of Inverse Trigonometric Functions
............... 912
1.3-6.
Differentiation Formulas
............................................ 913
1.3-7.
Integration Formulas
............................................... 913
1.3-8.
Expansion in Power Series
........................................... 913
1.4.
Hyperbolic Functions
..................................................... 913
1.4-1.
Definitions of Hyperbolic Functions
................................... 913
1.4-2.
Simplest Relations
................................................. 913
1.4-3.
Relations Between Hyperbolic Functions of Single Argument (x
> 0) ........ 914
1.4-4.
Addition and Subtraction of Hyperbolic Functions
........................ 914
1.4-5.
Products of Hyperbolic Functions
..................................... 914
1.4-6.
Powers of Hyperbolic Functions
...................................... 914
1.4-7.
Addition Formulas
................................................. 915
1.4-8.
Hyperbolic Functions of Multiple Argument
............................ 915
1.4-9.
Hyperbolic Functions of Half Argument
................................ 915
1.4-10.
Differentiation Formulas
........................................... 916
1.4-11.
Integration Formulas
.............................................. 916
1.4-12.
Expansion in Power Series
.......................................... 916
1.4-13.
Representation in the Form of Infinite Products
......................... 916
1.4-14.
Relationship with Trigonometric Functions
............................ 916
1.5.
Inverse Hyperbolic Functions
.............................................. 917
1.5-1.
Definitions of Inverse Hyperbolic Functions
............................. 917
1.5-2.
Simplest Relations
................................................. 917
1.5-3.
Relations Between Inverse Hyperbolic Functions
......................... 917
1.5-4.
Addition and Subtraction of Inverse Hyperbolic Functions
................. 917
1.5-5.
Differentiation Formulas
............................................ 917
1.5-6.
Integration Formulas
............................................... 918
1.5-7.
Expansion in Power Series
........................................... 918
Supplement
2.
Finite Sums and Infinite Series
................................... 919
2.1.
Finite Numerical Sums
........................ ...... 919
-1.
Progressions
...................................................... 919
-2.
Sums of Powers of Natural Numbers Having the Form
£
km
............... 919
-3.
Alternating Sums of Powers of Natural Numbers, ]T(-l)feA;m
............... 920
-4.
Other Sums Containing Integers
...................................... 920
-5.
Sums Containing Binomial Coefficients
................................ 920
-6.
Other Numerical Sums
................. ..... 921
CONTKNTS
ХХІІІ
2.2.
Finite
Functional Sums
................................................... 922
2.2-1.
Sums Involving Hyperbolic Functions
.................................. 922
2.2-2.
Sums Involving Trigonometric Functions
............................... 922
2.3.
Infinite Numerical Series
.................................................. 924
2.3-1.
Progressions
...................................................... 924
2.3-2.
Other Numerical Series
............................................. 924
2.4.
Infinite Functional Series
.................................................. 925
2.4-1.
Power Series
...................................................... 925
2.4-2.
Trigonometric Series in One Variable Involving Sine
...................... 927
2.4-3.
Trigonometric Series in One Variable Involving Cosine
.................... 928
2.4-4.
Trigonometric Series in Two Variables
................................. 930
Supplement
3.
Tables of Indefinite Integrals
.................................... 933
3.1.
Integrals Involving Rational Functions
....................................... 933
3.1-1.
Integrals Involving a + bx
........................................... 933
1-2.
Integrals Involving a
+
χ
and
b + x ....................................
933
-3.
Integrals Involving a~
+
x
.......................................... 934
-4.
Integrals Involving a2
-
x2
.......................................... 935
-5.
Integrals Involving a?
+
Xі
.......................................... 936
1-6.
Integrals Involving a?
-
x3
.......................................... 936
1-7.
Integrals Involving a4 ±x4
.......................................... 937
3.2.
Integrals Involving Irrational Functions
....................................... 937
3.2-1.
Integrals Involving x /2
............................................. 937
3.2-2.
Integrals Involving (a
+
bx)p 2
....................................... 938
3.2-3.
Integrals Involving (x2
+
a2)1/2
....................................... 938
3.2-4.
Integrals Involving (x2
-
a2)i/2
....................................... 938
3.2-5.
Integrals Involving (a2 -x2) /2
....................................... 939
3.2-6.
Integrals Involving Arbitrary Powers. Reduction Formulas
................. 939
3.3.
Integrals Involving Exponential Functions
.................................... 940
3.4.
Integrals Involving Hyperbolic Functions
..................................... 940
3.4-1.
Integrals Involving cosh
x
........................................... 940
3.4-2.
Integrals Involving sinh
x
............................................ 94
1
3.4-3.
Integrals Involving tanh
x
or coth
x
................................... 942
3.5.
Integrals Involving Logarithmic Functions
.................................... 943
3.6.
Integrals Involving Trigonometric Functions
................................... 944
3.6-1.
Integrals Involving cos
x (n
=1,2,...) ................................ 944
3.6-2.
Integrals Involving sin
x (n
=
1
, 2, ... ) ................................ 945
3.6-3.
Integrals Involving sin.
г
and cos.
r
..................................... 947
3.6-4.
Reduction Formulas
................................................ 947
3.6-5.
Integrals Involving tan
x
and cot
r
..................................... 947
3.7.
Integrals Involving Inverse Trigonometric Functions
............................ 948
Supplement
4.
Tables of Definite Integrals
951
4.1.
Integrals Involving Power-Law Functions
..................................... 951
4.1-1.
Integrals Over a Finite Interval
....................................... 951
4.1-2.
Integrals Over an Infinite Interval
..................................... 952
4.2.
Integrals Involving Exponential Functions
.................................... 954
4.3.
Integrals Involving Hyperbolic Functions
..................................... 955
4.4.
Integrals Involving Logarithmic Functions
.................................... 955
xxiv Contents
4.5. Integrals
Involving
Trigonometrie
Functions...................................
956
4.5-1. Integrals
Over a Finite Interval
....................................... 956
4.5-2.
Integrals Over an Infinite Interval
..................................... 957
4.6.
Integrals Involving Bessel Functions
......................................... 958
4.6-1.
Integrals Over an Infinite Interval
..................................... 958
4.6-2.
Other Integrals
.................................................... 959
Supplement
5.
Tables of Laplace Transforms
................................... 961
5.1.
General Formulas
........................................................ 961
5.2.
Expressions with Power-Law Functions
...................................... 963
5.3.
Expressions with Exponential Functions
...................................... 963
5.4.
Expressions with Hyperbolic Functions
...................................... 964
5.5.
Expressions with Logarithmic Functions
...................................... 965
5.6.
Expressions with Trigonometric Functions
.................................... 966
5.7.
Expressions with Special Functions
.......................................... 967
Supplement
6.
Tables of Inverse Laplace Transforms
............................. 969
6.1.
General Formulas
........................................................ 969
6.2.
Expressions with Rational Functions
......................................... 971
6.3.
Expressions with Square Roots
............................................. 975
6.4.
Expressions with Arbitrary Powers
.......................................... 977
6.5.
Expressions with Exponential Functions
...................................... 978
6.6.
Expressions with Hyperbolic Functions
...................................... 979
6.7.
Expressions with Logarithmic Functions
...................................... 980
6.8.
Expressions with Trigonometric Functions
.................................... 981
6.9.
Expressions with Special Functions
.......................................... 981
Supplement
7.
Tables of Fourier Cosine Transforms
............................. 983
7.1.
General Formulas
........................................................ 983
7.2.
Expressions with Power-Law Functions
...................................... 983
7.3.
Expressions with Exponential Functions
...................................... 984
7.4.
Expressions with Hyperbolic Functions
...................................... 985
7.5.
Expressions with Logarithmic Functions
...................................... 985
7.6.
Expressions with Trigonometric Functions
.................................... 986
7.7.
Expressions with Special Functions
.......................................... 987
Supplement
8.
Tables of Fourier Sine Transforms
................................ 989
8.1.
General Formulas
........................................................ 989
8.2.
Expressions with Power-Law Functions
...................................... 989
8.3.
Expressions with Exponential Functions
...................................... 990
8.4.
Expressions with Hyperbolic Functions
...................................... 991
8.5.
Expressions with Logarithmic Functions
...................................... 992
8.6.
Expressions with Trigonometric Functions
.................................... 992
8.7.
Expressions with Special Functions
.......................................... 993
Contents
xxv
Supplement
9.
Tables of Mellin Transforms
..................................... 997
9.1.
General Formulas
........................................................ 997
9.2.
Expressions with Power-Law Functions
...................................... 998
9.3.
Expressions with Exponential Functions
...................................... 998
9.4.
Expressions with Logarithmic Functions
...................................... 999
9.5.
Expressions with Trigonometric Functions
.................................... 999
9.6.
Expressions with Special Functions
.......................................... 1000
Supplement
10.
Tables of Inverse Mellin Transforms
.............................1001
10.1.
Expressions with Power-Law Functions
.....................................1001
10.2.
Expressions with Exponential and Logarithmic Functions
.......................1002
10.3.
Expressions with Trigonometric Functions
...................................1003
10.4.
Expressions with Special Functions
.........................................1004
Supplement
11.
Special Functions and Their Properties
..........................1007
11.1.
Some Coefficients, Symbols, and Numbers
...................................1007
11.1-1.
Binomial Coefficients
............................................1007
11.1-2.
Pochhammer Symbol
.............................................1007
11.1-3.
Bernoulli Numbers
...............................................1008
11.1-4.
Euler
Numbers
..................................................1008
11.2.
Error Functions. Exponential and Logarithmic Integrals
........................1009
11.2-1.
Error Function and Complementary Error Function
.....................1009
11.2-2.
Exponential Integral
..............................................1010
11.2-3.
Logarithmic Integral
.............................................1010
11.3.
Sine Integral and Cosine Integral. Fresnel Integrals
............................ 1011
11.3-1.
Sine Integral
.................................................... 1011
11.3-2.
Cosine Integral
.................................................. 1011
11.3-3.
Fresnel Integrals and Generalized Fresnel Integrals
..................... 1012
11.4.
Gamma Function,
Psi
Function, and Beta Function
............................1012
11.4-1.
Gamma Function
................................................1012
11.4-2.
Psi
Function
(Digamma
Function)
...................................1013
11.4-3.
Beta Function
...................................................1014
11.5.
Incomplete Gamma and Beta Functions
.....................................1014
11.5-1.
Incomplete Gamma Function
.......................................1014
11.5-2.
Incomplete Beta Function
.........................................1015
11.6.
Bessel Functions (Cylindrical Functions)
....................................1016
11.6-1.
Definitions and Basic Formulas
.....................................1016
11.6-2.
Integral Representations and Asymptotic Expansions
....................1017
11.6-3.
Zeros of Bessel Functions
.........................................1019
11.6-4.
Orthogonality Properties of Bessel Functions
..........................1019
11.6-5.
Hankel Functions (Bessel Functions of the Third Kind)
..................1020
11.7.
Modified Bessel Functions
................................................1021
11.7-1.
Definitions. Basic Formulas
.......................................1021
11.7-2.
Integral Representations and Asymptotic Expansions
....................1022
11.8.
Airy Functions
.........................................................
Ю23
11.8-1.
Definition and Basic Formulas
......................................1023
11.8-2.
Power Series and Asymptotic Expansions
.............................1023
xxvi Contents
11.9. Confluent Hypergeometric
Functions
.......................................1024
11.9-1. Kummer and
Tricomi Confluent Hypergeometric
Functions
..............1024
11.9-2.
Integral Representations and Asymptotic Expansions
....................1027
11.9-3.
Whittaker Confluent Hypergeometric Functions
........................1027
11.10.
Gauss Hypergeometric Functions
.........................................1028
11.10-1.
Various Representations of the Gauss Hypergeometric Function
.........1028
11.10-2.
Basic Properties
...............................................1028
11.11.
Legendre Polynomials, Legendre Functions, and Associated Legendre Functions
.. . 1030
11.11-1.
Legendre Polynomials and Legendre Functions
......................1030
11.11-2.
Associated Legendre Functions with Integer Indices and Real Argument
. . 1031
11.11-3.
Associated Legendre Functions. General Case
.......................1032
11.12.
Parabolic Cylinder Functions
.............................................1034
11.12-1.
Definitions. Basic Formulas
.....................................1034
11.12-2.
Integral Representations, Asymptotic Expansions, and Linear Relations
.. . 1035
11.13.
Elliptic Integrals
.......................................................1035
11.13-1.
Complete Elliptic Integrals
......................................1035
11.13-2.
Incomplete Elliptic Integrals (Elliptic Integrals)
......................1037
11.14.
Elliptic Functions
......................................................1038
11.14-1.
Jacobi Elliptic Functions
........................................1039
11.14-2.
Weierstrass
Elliptic Function
.....................................1042
11.15.
Jacobi Theta Functions
.................................................1043
11.15-1.
Series Representation of the Jacobi Theta Functions. Simplest Properties
.. 1043
11.15-2.
Various Relations and Formulas. Connection with Jacobi Elliptic Functions
1044
11.16.
Mathieu
Functions and Modified
Mathieu
Functions
..........................1045
11.16-1.
Mathieu
Functions
.............................................1045
11.16-2.
Modified
Mathieu
Functions
.....................................1046
11.17.
Orthogonal Polynomials
................................................1047
11.17-1.
Laguerre Polynomials and Generalized Laguerre Polynomials
...........1047
11.17-2.
Chebyshev Polynomials and Functions
.............................1048
11.17-3.
Hermite Polynomials and Functions
...............................1050
11.17-4.
Jacobi Polynomials
............................................1051
11.17-5. Gegenbauer
Polynomials
........................................1051
11.18. Nonorthogonal
Polynomials
.............................................1052
11.18-1.
Bernoulli Polynomials
..........................................1052
11.18-2.
Euler
Polynomials
.............................................1053
Supplement
12.
Some Notions of Functional Analysis
............................1055
12.1.
Functions of Bounded Variation
...........................................1055
-1.
Definition of a Function of Bounded Variation
.........................1055
-2.
Classes of Functions of Bounded Variation
............................1056
-3.
Properties of Functions of Bounded Variation
..........................1056
-4.
Criteria for Functions to Have Bounded Variation
......................1057
12.
12.
12.
12.
12.
-5.
Properties of Continuous Functions of Bounded Variation
................1057
12.2.
Stieltjes
Integral
........................................................1057
12.2-1.
Basic Definitions
................................................1057
12.2-2.
Properties of the
Stieltjes
Integral
...................................1058
12.2-3.
Existence Theorems for the
Stieltjes
Integral
..........................1058
CONTKNTS
ХХУ!!
12.3.
Lebesgue
Integral.......................................................1059
12.3-1.
Riemann
Integral
and the Lebesgue Integral
...........................1059
12.3-2.
Sets of Zero Measure. Notion of Almost Everywhere
..................1060
12.3-3.
Step Functions and Measurable Functions
............................1060
12.3-4.
Definition and Properties of the Lebesgue Integral
......................1061
12.3-5.
Measurable Sets
.................................................1062
12.3-6.
Integration Over Measurable Sets
...................................1063
12.3-7.
Case of an Infinite Interval
.........................................1063
12.3-8.
Case of Several Variables
..........................................1064
12.3-9.
Spaces Lp
......................................................1064
12.4.
Linear Normed Spaces
...................................................1065
12.4-1.
Linear Spaces
...................................................1065
12.4-2.
Linear Normed Spaces
............................................1065
12.4-3.
Space of Continuous Functions C(a,b)
...............................1066
12.4-4.
Lebesgue Space Lp(a, b)
..........................................1066
12.4-5.
Holder Space Ca(0,
1) ............................................1066
12.4-6.
Space of Functions of Bounded Variation V(0,
1).......................1066
12.5.
Euclidean and Hubert Spaces. Linear Operators in Hubert Spaces
................1067
12.5-1.
Preliminary Remarks
.............................................1067
12.5-2.
Euclidean and Hubert Spaces
......................................1067
12.5-3.
Linear Operators in Hubert Spaces
..................................1068
References
.................................................................1071
Index
.....................................................................1081
|
| adam_txt |
Mathematics, Physics, Mechanics, Control, Engineering Sciences
гЈАШЗОСЖ
0?
INTEGRAL
S OUATIONS
EDITION
-»
UPDATED,
ft£7B£D
AND EXTENDED
•
Represents a unique reference for engineers
and scientists that does not have any
counterpart in the literature
•
Contains over
2,500
linear and nonlinear
integral equations and their exact solutions
•
Outlines exact, approximate analytical,
and numerical methods for solving integral
equations
•
Illustrates the application of the methods with numerous examples
•
Considers equations that arise in elasticity, plasticity, creep, heat
and mass transfer, hydrodynamics, chemical engineering, and other
areas
•
Can be used as a database of test problems for numerical and
approximate methods for solving linear and nonlinear integral
equations
•
Presents many times more integral equations than any other book
currently available
New to the Second Edition
•
Additional material on
Volteira,
Fredholm,
singular, hypersingular,
mixed, multidimensional, dual, and nonlinear integral equations,
integral transforms, and special functions
•
Methods of integral equations for ODEs and PDEs
•
Over
300
added pages and more than
400
new integral equations
with exact solutions
To accommodate different mathematical backgrounds, the authors
avoid wherever possible the use of special terminology, outline some of
the methods in a schematic, simplified manner, and arrange the material
in increasing order of complexity.
CONTENTS
Authors
.xxix
Preface
.xxxi
Some Remarks and Notation
.xxxiii
CONTENTS
Authors
.xxix
Preface
.xxxi
Some Remarks and Notation
.xxxiii
Part I. Exact Solutions of Integral Equations
1.
Linear Equations of the First Kind with Variable Limit of Integration
. 3
1.1.
Equations Whose Kernels Contain Power-Law Functions
. 4
-1.
Kernels Linear in the Arguments
χ
and
t
. 4
-2.
Kernels Quadratic in the Arguments
x
and
t
. 4
-3.
Kernels Cubic in the Arguments
x
and
t
. 5
1-4.
Kernels Containing Higher-Order Polynomials in
x
and
t
. 6
1-5.
Kernels Containing Rational Functions
. 7
1 -6.
Kernels Containing Square Roots
. 9
1-7.
Kernels Containing Arbitrary Powers
. 12
1.
Two-Dimensional Equation of the Abel Type
. 15
1.2.
Equations Whose Kernels Contain Exponential Functions
. 15
1.2-1.
Kernels Containing Exponential Functions
. 15
1.2-2.
Kernels Containing Power-Law and Exponential Functions
. 19
1.3.
Equations Whose Kernels Contain Hyperbolic Functions
. 22
1.3-1.
Kernels Containing Hyperbolic Cosine
. 22
1.3-2.
Kernels Containing Hyperbolic Sine
. 28
1.3-3.
Kernels Containing Hyperbolic Tangent
. 36
1.3-4.
Kernels Containing Hyperbolic Cotangent
. 38
1.3-5.
Kernels Containing Combinations of Hyperbolic Functions
. 39
1.4.
Equations Whose Kernels Contain Logarithmic Functions
. 42
1.4-1.
Kernels Containing Logarithmic Functions
. 42
1.4-2.
Kernels Containing Power-Law and Logarithmic Functions
. 45
1.5.
Equations Whose Kernels Contain Trigonometric Functions
. 46
1.5-1.
Kernels Containing Cosine
. 46
1.5-2.
Kernels Containing Sine
. 52
1.5-3.
Kernels Containing Tangent
. 60
1.5-4.
Kernels Containing Cotangent
. 62
1.5-5.
Kernels Containing Combinations of Trigonometric Functions
. 63
1.6.
Equations Whose Kernels Contain Inverse Trigonometric Functions
. 66
1.6-1.
Kernels Containing Arccosine
. 66
1.6-2.
Kernels Containing Arcsine
. 68
1.6-3.
Kernels Containing Arctangent
. 70
1.6-4.
Kernels Containing Arccotangent
. 71
vi
Contents
1.7.
Equations
Whose
Kernels
Contain Combinations of Elementary Functions
. 73
1.7-1.
Kernels Containing Exponential and Hyperbolic Functions
. 73
1.7-2.
Kernels Containing Exponential and Logarithmic Functions
. 77
1.7-3.
Kernels Containing Exponential and Trigonometric Functions
. 78
1
.7-4.
Kernels Containing Hyperbolic and Logarithmic Functions
. 83
1.7-5.
Kernels Containing Hyperbolic and Trigonometric Functions
. 84
1.7-6.
Kernels Containing Logarithmic and Trigonometric Functions
. 85
1.8.
Equations Whose Kernels Contain Special Functions
. 86
1.8-1.
Kernels Containing Error Function or Exponential Integral
. 86
3-2.
Kernels Containing Sine and Cosine Integrals
. 87
3-3.
Kernels Containing Fresnel Integrals
. 87
1
.8-4.
Kernels Containing Incomplete Gamma Functions
. 88
i-5. Kernels Containing Bessel Functions
. 88
1
.8-6.
Kernels Containing Modified Bessel Functions
. 97
1.8-7.
Kernels Containing Legendre Polynomials
. 105
1.8-8.
Kernels Containing Associated Legendre Functions
. 107
1.8-9.
Kernels Containing Confluent Hypergeometric Functions
. 107
1.8-10.
Kernels Containing Hermite Polynomials
. 108
1.8-11.
Kernels Containing Chebyshev Polynomials
. 109
1.8-12.
Kernels Containing Laguerre Polynomials
. 110
1.8-13.
Kernels Containing Jacobi Theta Functions
. 110
1.8-14.
Kernels Containing Other Special Functions
.
Ill
1.9.
Equations Whose Kernels Contain Arbitrary Functions
.
Ill
1.9-1.
Equations with Degenerate Kernel: K(x,t) = g\{x)h\(t) +
дг{х)Н2{Ь)
.
Ill
1.9-2.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
. 114
1.9-3.
Other Equations
_. 122
1.10.
Some Formulas and Transformations
. 124
2.
Linear Equations of the Second Kind with Variable Limit of Integration
. 127
2.1.
Equations Whose Kernels Contain Power-Law Functions
. 127
2.1-1.
Kernels Linear in the Arguments
χ
and
t
. 127
2.1-2.
Kernels Quadratic in the Arguments
x
and
í
. 129
2.1-3.
Kernels Cubic in the Arguments
x
and
t
. 132
2.1-4.
Kernels Containing Higher-Order Polynomials in
x
and
t
. 133
2.1-5.
Kernels Containing Rational Functions
. 136
2.1-6.
Kernels Containing Square Roots and Fractional Powers
. 138
2.1-7.
Kernels Containing Arbitrary Powers
. 139
2.2.
Equations Whose Kernels Contain Exponential Functions
. 144
2.2-1.
Kernels Containing Exponential Functions
. 144
2.2-2.
Kernels Containing Power-Law and Exponential Functions
. 151
2.3.
Equations Whose Kernels Contain Hyperbolic Functions
. 154
2.3-1.
Kernels Containing Hyperbolic Cosine
. 154
2.3-2.
Kernels Containing Hyperbolic Sine
. 156
2.3-3.
Kernels Containing Hyperbolic Tangent
. 161
2.3-4.
Kernels Containing Hyperbolic Cotangent
. 162
2.3-5.
Kernels Containing Combinations of Hyperbolic Functions
. 164
2.4.
Equations Whose Kernels Contain Logarithmic Functions
. 164
2.4-1.
Kernels Containing Logarithmic Functions
. 164
2.4-2.
Kernels Containing Power-Law and Logarithmic Functions
. 165
CONTKNTS
VU
2.5.
Equations
Whose
Kernels
Contain Trigonometric Functions
. 166
2.5-1.
Kernels Containing Cosine
. 166
2.5-2.
Kernels Containing Sine
. 169
2.5-3.
Kernels Containing Tangent
. 174
2.5-4.
Kernels Containing Cotangent
. 175
2.5-5.
Kernels Containing Combinations of Trigonometric Functions
. 176
2.6.
Equations Whose Kernels Contain Inverse Trigonometric Functions
. 176
2.6-1.
Kernels Containing Arccosine
. 176
2.6-2.
Kernels Containing Arcsine
. 177
2.6-3.
Kernels Containing Arctangent
. 178
2.6-4.
Kernels Containing Arccotangent
. 178
2.7.
Equations Whose Kernels Contain Combinations of Elementary Functions
. 179
2.7-1.
Kernels Containing Exponential and Hyperbolic Functions
. 179
2.7-2.
Kernels Containing Exponential and Logarithmic Functions
. 180
2.7-3.
Kernels Containing Exponential and Trigonometric Functions
. 181
2.7-4.
Kernels Containing Hyperbolic and Logarithmic Functions
. 185
2.7-5.
Kernels Containing Hyperbolic and Trigonometric Functions
. 186
2.7-6.
Kernels Containing Logarithmic and Trigonometric Functions
. 187
2.8.
Equations Whose Kernels Contain Special Functions
. 187
2.8-1.
Kernels Containing Bessel Functions
. 187
2.8-2.
Kernels Containing Modified Bessel Functions
. 189
2.9.
Equations Whose Kernels Contain Arbitrary Functions
. 191
2.9-1.
Equations with Degenerate Kernel: K(x,t)
=
g¡(x)hi(t) +
■■■ +
gn(,x)hn(t)
■··· 191
2.9-2.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
. 203
2.9-3.
Other Equations
. 212
2.10.
Some Formulas and Transformations
. 215
3.
Linear Equations of the First Kind with Constant Limits of Integration
. 217
3.1.
Equations Whose Kernels Contain Power-Law Functions
. 217
-1.
Kernels Linear in the Arguments
χ
and
t
. 217
-2.
Kernels Quadratic in the Arguments
x
and
t
. 219
-3.
Kernels Containing Integer Powers of
x
and
t
or Rational Functions
. 220
-4.
Kernels Containing Square Roots
. 222
-5.
Kernels Containing Arbitrary Powers
. 223
-6.
Equations Containing the Unknown Function of a Complicated Argument
. 227
3.1-7.
Singular Equations
. 228
3.2.
Equations Whose Kernels Contain Exponential Functions
. 231
3.2-1.
Kernels Containing Exponential Functions of the Form
ρλΙΓ
'I
. 231
3.2-2.
Kernels Containing Exponential Functions of the Forms eAr and
с"'
. 234
3.2-3.
Kernels Containing Exponential Functions of the Form eXxt
. 234
3.2-4.
Kernels Containing Power-Law and Exponential Functions
. 236
3.2-5.
Kernels Containing Exponential Functions of the Form eMx±tr
. 236
3.2-6.
Other Kernels
. 237
3.3.
Equations Whose Kernels Contain Hyperbolic Functions
. 238
3.3-1.
Kernels Containing Hyperbolic Cosine
. 238
3.3-2.
Kernels Containing Hyperbolic Sine
. 238
3.3-3.
Kernels Containing Hyperbolic Tangent
. 241
3.3-4.
Kernels Containing Hyperbolic Cotangent
. 242
viii Contents
3.4.
Equations
Whose
Kernels
Contain Logarithmic Functions
. 242
3.4-1.
Kernels Containing Logarithmic Functions
. 242
3.4-2.
Kernels Containing Power-Law and Logarithmic Functions
. 244
3.4-3.
Equation Containing the Unknown Function of a Complicated Argument
. 246
3.5.
Equations Whose Kernels Contain Trigonometric Functions
. 246
3.5-1.
Kernels Containing Cosine
. 246
3.5-2.
Kernels Containing Sine
. 247
3.5-3.
Kernels Containing Tangent
. 251
3.5-4.
Kernels Containing Cotangent
. 252
3.5-5.
Kernels Containing a Combination of Trigonometric Functions
. 252
3.5-6.
Equations Containing the Unknown Function of a Complicated Argument
. 254
3.5-7.
Singular Equations
. 255
3.6.
Equations Whose Kernels Contain Combinations of Elementary Functions
. 255
3.6-1.
Kernels Containing Hyperbolic and Logarithmic Functions
. 255
3.6-2.
Kernels Containing Logarithmic and Trigonometric Functions
. 256
3.6-3.
Kernels Containing Combinations of Exponential and Other Elementary
Functions
. 257
3.7.
Equations Whose Kernels Contain Special Functions
. 258
3.7-1.
Kernels Containing Error Function, Exponential Integral or Logarithmic Integral
258
3.7-2.
Kernels Containing Sine Integrals, Cosine Integrals, or Fresnel Integrals
. 258
3.7-3.
Kernels Containing Gamma Functions
. 260
3.7-4.
Kernels Containing Incomplete Gamma Functions
. 260
3.7-5.
Kernels Containing Bessel Functions of the First Kind
. 261
3.7-6.
Kernels Containing Bessel Functions of the Second Kind
. 264
3.7-7.
Kernels Containing Combinations of the Bessel Functions
. 265
3.7-8.
Kernels Containing Modified Bessel Functions of the First Kind
. 266
3.7-9.
Kernels Containing Modified Bessel Functions of the Second Kind
. 266
3.7-10.
Kernels Containing a Combination of Bessel and Modified Bessel Functions
. . 269
3.7-11.
Kernels Containing Legendre Functions
. 270
3.7-12.
Kernels Containing Associated Legendre Functions
. 271
3.7-13.
Kernels Containing
Kummer
Confluent Hypergeometric Functions
. 272
3.7-14.
Kernels Containing
Tricomi
Confluent Hypergeometric Functions
. 274
3.7-15.
Kernels Containing Whittaker Confluent Hypergeometric Functions
. 274
3.7-16.
Kernels Containing Gauss Hypergeometric Functions
. 276
3.7-17.
Kernels Containing Parabolic Cylinder Functions
. 276
3.7-18.
Kernels Containing Other Special Functions
. 277
3.8.
Equations Whose Kernels Contain Arbitrary Functions
. 278
3.8-1.
Equations with Degenerate Kernel
. 278
3.8-2.
Equations Containing Modulus
. 279
3.8-3.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
. 284
3.8-4.
Other Equations of the Form
ƒ
Ь
K(x, t)y(t) dt
=
F{x)
. 285
Ja
eb
3.8-5.
Equations of the Form
Ц
K(x, t)y{- ■■)dt = F(x)
. 289
3.9.
Dual Integral Equations of the First Kind
. 295
3.9-1.
Kernels Containing Trigonometric Functions
. 295
3.9-2.
Kernels Containing Bessel Functions of the First Kind
. 297
3.9-3.
Kernels Containing Bessel Functions of the Second Kind
. 299
3.9-4.
Kernels Containing Legendre Spherical Functions of the First Kind, i2
= -1 . 299
Contents
ix
4.
Linear Equations of the Second Kind with Constant Limits of Integration
. 301
4.1.
Equations Whose Kernels Contain Power-Law Functions
. 301
4.1 -1.
Kernels Linear in the Arguments
χ
and
t
. 301
4.1-2.
Kernels Quadratic in the Arguments
x
and
t
. 304
4.1-3.
Kernels Cubic in the Arguments
x
and
t
. 307
4.1-4.
Kernels Containing Higher-Order Polynomials in
x
and
t
. 311
4.1-5.
Kernels Containing Rational Functions
. 314
4.1-6.
Kernels Containing Arbitrary Powers
. 317
4.1-7.
Singular Equations
. 319
4.2.
Equations Whose Kernels Contain Exponential Functions
. 320
4.2-1.
Kernels Containing Exponential Functions
. 320
4.2-2.
Kernels Containing Power-Law and Exponential Functions
. 326
4.3.
Equations Whose Kernels Contain Hyperbolic Functions
. 327
4.3-1.
Kernels Containing Hyperbolic Cosine
. 327
4.3-2.
Kernels Containing Hyperbolic Sine
. 329
4.3-3.
Kernels Containing Hyperbolic Tangent
. 332
4.3-4.
Kernels Containing Hyperbolic Cotangent
. 333
4.3-5.
Kernels Containing Combination of Hyperbolic Functions
. 334
4.4.
Equations Whose Kernels Contain Logarithmic Functions
. 334
4.4-1.
Kernels Containing Logarithmic Functions
. 334
4.4-2.
Kernels Containing Power-Law and Logarithmic Functions
. 335
4.5.
Equations Whose Kernels Contain Trigonometric Functions
. 335
4.5-1.
Kernels Containing Cosine
. 335
4.5-2.
Kernels Containing Sine
. 337
4.5-3.
Kernels Containing Tangent
. 342
4.5-4.
Kernels Containing Cotangent
. 343
4.5-5.
Kernels Containing Combinations of Trigonometric Functions
. 344
4.5-6.
Singular Equation
. 344
4.6.
Equations Whose Kernels Contain Inverse Trigonometric Functions
. 344
4.6-1.
Kernels Containing Arccosine
. 344
4.6-2.
Kernels Containing Arcsine
. 345
4.6-3.
Kernels Containing Arctangent
. 346
4.6-4.
Kernels Containing Arccotangent
. 347
4.7.
Equations Whose Kernels Contain Combinations of Elementary Functions
. 348
4.7-1.
Kernels Containing Exponential and Hyperbolic Functions
. 348
4.7-2.
Kernels Containing Exponential and Logarithmic Functions
. 349
4.7-3.
Kernels Containing Exponential and Trigonometric Functions
. 349
4.7-4.
Kernels Containing Hyperbolic and Logarithmic Functions
. 351
4.7-5.
Kernels Containing Hyperbolic and Trigonometric Functions
. 352
4.7-6.
Kernels Containing Logarithmic and Trigonometric Functions
. 353
4.8.
Equations Whose Kernels Contain Special Functions
. 353
4.8-1.
Kernels Containing Bessel Functions
. 353
4.8-2.
Kernels Containing Modified Bessel Functions
. 355
4.9.
Equations Whose Kernels Contain Arbitrary Functions
. 357
4.9-1.
Equations with Degenerate Kernel: K{x,t) = g\(x)h\{t) +
■ ■ ■ +
gn{x)hn{t)
----- 357
4.9-2.
Equations with Difference Kernel: K(x, t)
=
K(x -t)
. 372
4.9-3.
Other Equations of the Form y(x)
+
f* K(x, t)y(t) dt
-
F(x)
. 374
4.9-4.
Equations of the Form y(x)
+
¡I K(x, t)y(- ■■)dt = F(x)
. 381
4.10.
Some Formulas and Transformations
. 390
Contents
5.
Nonlinear Equations of the First Kind with Variable Limit of Integration
. 393
5.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
. 393
5.1-1.
Equations of the Form ¡* y(t)y(x
-
1) dt
=
f(x)
. 393
5.1-2.
Equations of the Form
ƒ„*
K(x, t)y(t)y(x -t)dt = f(x)
. 395
5.1-3.
Equations of the Form f* y(t)y(- ■•)dt = f(x)
. 396
5.2.
Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
. 397
5.2-1.
Equations of the Form
ƒ*
K(x, t)[Ay(t)
+
By2(t)] dt
=
f(x)
. 397
5.2-2.
Equations of the Form
ƒƒ
K(x, t)y(t)y(ax
+
bt) dt
= ƒ
(x)
. 398
5.3.
Equations with Nonlinearity of General Form
. 399
5.3-1.
Equations of the Form
ƒ*
K(x, t)f(t, y(t)) dt
=
g(x)
. 399
5.3-2.
Other Equations
.". 401
6.
Nonlinear Equations of the Second Kind with Variable Limit of Integration
. 403
6.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
. 403
6.1 -1.
Equations of the Form y(x)
+ ƒ*
K(x, t)y\t) dt
=
F(x)
. 403
6.1-2.
Equations of the Form y(x)
+
j* K(x, t)y(t)y(x -t)dt = F(x)
. 406
6.2.
Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
. 406
6.2-1.
Equations of the Form y{x)
+ ƒ*
K{x, t)y\t) dt
=
F(x)
. 406
6.2-2.
Other Equations
. 407
6.3.
Equations with Power-Law Nonlinearity
. 408
6.3-1.
Equations Containing Arbitrary Parameters
. 408
6.3-2.
Equations Containing Arbitrary Functions
. 410
6.4.
Equations with Exponential Nonlinearity
. 411
6.4-1.
Equations Containing Arbitrary Parameters
. 411
6.4-2.
Equations Containing Arbitrary Functions
. 413
6.5.
Equations with Hyperbolic Nonlinearity
. 414
6.5-1.
Integrands with Nonlinearity of the Form
cosh[/3y(í)]
. 414
6.5-2.
Integrands with Nonlinearity of the Form sinh[/3y(i)]
. 415
6.5-3.
Integrands with Nonlinearity of the Form
tanh[ßy(t)] . 416
6.5-4.
Integrands with Nonlinearity of the Form
coth[ßy(t)] . 418
6.6.
Equations with Logarithmic Nonlinearity
. 419
6.6-1.
Integrands Containing Power-Law Functions of
x
and
í
. 419
6.6-2.
Integrands Containing Exponential Functions of
x
and
t
. 419
6.6-3.
Other Integrands
. 420
6.7.
Equations with Trigonometric Nonlinearity
. 420
6.7-1.
Integrands with Nonlinearity of the Form
cos[ßy(t)] . 420
6.7-2.
Integrands with Nonlinearity of the Form
sin[ßy(t)] . 422
6.7-3.
Integrands with Nonlinearity of the Form
tan[ßy(t)] . 423
6.7-4.
Integrands with Nonlinearity of the Form
cot[ßy(t)] . 424
6.8.
Equations with Nonlinearity of General Form
. 425
6.8-1.
Equations of the Form y(x)
+ ƒƒ
K(x, t)G(y(t)) dt
=
F(x)
. 425
6.8-2.
Equations of the Form y{x)
+
f* K(x
-
t)G(t, y(t)) dt
=
F{x)
. 428
6.8-3.
Other Equations
. 431
7.
Nonlinear Equations of the First Kind with Constant Limits of Integration
. 433
7.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
. 433
7.1-1.
Equations of the Form ¡^ K(t)y(x)y(t) dt
=
F(x)
. 433
7.1-2.
Equations of the Form
£
K(t)y(t)y(xt) dt
=
F(x)
. 435
7.1-3.
Other Equations
. 436
CONTKNTS
X¡
7.2.
Equations
with Quadratic Nonlinearity That Contain Arbitrary Functions
. 437
7.2-1.
Equations of the Form ¡^ K(t)y(t)y(- ■■)dt = F(x)
. 437
7.2-2.
Equations of the Form J^[K(x, t)y(t)
+
M(x, t)y2(t)] dt
=
F{x)
. 443
7.3.
Equations with Power-Law Nonlinearity That Contain Arbitrary Functions
. 444
7.3-1.
Equations of the Form f* K(t)y^l(x)y'l(t) dt
=
F(x)
. 444
7.3-2.
Equations of the Form /j* K(t)yi(t)y{xt) dt
-
F(x)
. 444
7.3-3.
Equations of the Form /J* K(t)y^(t)y(x
+ ßt)dt =
F (x)
. 445
7.3-4.
Equations of the Form J^[K(x, t)y(t)
+
M(x, t)y~<(t)} dt
=
f(x)
. 446
7.3-5.
Other Equations
. 446
7.4.
Equations with Nonlinearity of General Form
. 447
7.4-1.
Equations of the Form j^<p{ij{x))K(t,y(t)) dt
=
F(x)
. 447
7.4-2.
Equations of the Form
£
y(xt)K(t, y(t)) dt
-
F(x)
. 447
7.4-3.
Equations of the Form J* y(x
+ ßt)K(t,
y(t)) dt
=
F(x)
. 449
7.4-4.
Equations of the Form tf[K(x, t)y(t)
+
ψ(χ)^{ί,
у(Щ
dt =
F (x)
. 450
7.4-5.
Other Equations
. 451
8.
Nonlinear Equations of the Second Kind with Constant Limits of Integration
. 453
8.1.
Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
. 453
8.1-1.
Equations of the Form y(x)
+
J^ K(x, t)y\t) dt
=
F(x)
. 453
8.
1
-2.
Equations of the Form y{x)
+
/j" K(x, t)y(x)y(t) dt
=
F(x)
. 454
8.1-3.
Equations of the Form y(x)
+
¡^ K(t)y(t)y(- --)dt = F(x)
. 455
8.2.
Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
. 456
8.2-1.
Equations of the Form y(x)
+ £
K(x, t)y2(t) dt
=
F(x)
. 456
8.2-2.
Equations of the Form y(x)
+
¡^
Σ
Knm{x, t)yn(x)ym(t) dt
=
Fix),
η
+
m
< 2 457
8.2-3.
Equations of the Form y(x)
+
¡^ K(t)y(t)y(- --)dt = F(x)
. 460
8.3.
Equations with Power-Law Nonlinearity
. 464
8.3-1.
Equations of the Form y(x)
+
f£ K(x, t)i/(t) dt
=
F(x)
. 464
8.3-2.
Other Equations
. 465
8.4.
Equations with Exponential Nonlinearity
. 467
8.4-1.
Integrands with Nonlinearity of the Form
e\p[ßy(t)] . 467
8.4-2.
Other Integrands
. 468
8.5.
Equations with Hyperbolic Nonlinearity
. 468
8.5-1.
Integrands with Nonlinearity of the Form
cosh[ßy(t)] . 468
8.5-2.
Integrands with Nonlinearity of the Form
sinh[ßy(t)] . 469
8.5-3.
Integrands with Nonlinearity of the Form
liinh[ßy(t)\ . 469
8.5-4.
Integrands with Nonlinearity of the Form
coih[ßy(t)] . 470
8.5-5.
Other Integrands
. 471
8.6.
Equations with Logarithmic Nonlinearity
. 472
8.6-1.
Integrands with Nonlinearity of the Form
\n[ßy(t)\ . 472
8.6-2.
Other Integrands
. 473
8.7.
Equations with Trigonometric Nonlinearity
. 473
8.7-1.
Integrands with Nonlinearity of the Form
cos[ßy(t)} . 473
8.7-2.
Integrands with Nonlinearity of the Form
un[ßy{t)} . 474
8.7-3.
Integrands with Nonlinearity of the Form
ian[ßy(t)] . 475
8.7-4.
Integrands with Nonlinearity of the Form
coi[ßy(t)] . 475
8.7-5.
Other Integrands
. 476
xii Contents
8.8.
Equations
with Nonlinearity of General Form
. 477
8.8-1.
Equations of the Form y(x)
+
J^K(\x-t\)G (y(t)) dt
=
F(x)
. 477
8.8-2.
Equations of the Form y(x)
+
¡^ K{x, t)G(t, y(t)) dt
=
F(x)
. 479
8.8-3.
Equations of the Form y(x)
+
J* G(x, t, y(t)) dt
=
F(x)
. 483
8.8-4.
Equations of the Form y(x)
+ £
y(xt)G(t, y(t)) dt
=
F(x)
. 485
8.8-5.
Equations of the Form y(x)
+
¡* y(x
+ ßt)G
(t,
y(t))
dt =
F(x)
. 487
8.8-6.
Other Equations
. 494
Part II. Methods for Solving Integral Equations
9.
Main Definitions and Formulas. Integral Transforms
. 501
9.1.
Some Definitions, Remarks, and Formulas
. 501
9.1-1.
Some Definitions
. 501
9.1-2.
Structure of Solutions to Linear Integral Equations
. 502
9.1-3.
Integral Transforms
. 503
9.1-4.
Residues. Calculation Formulas. Cauchy's Residue Theorem
. 504
9.1-5.
Jordan Lemma
. 505
9.2.
Laplace Transform
. 505
9.2-1.
Definition. Inversion Formula
. 505
9.2-2.
Inverse Transforms of Rational Functions
. 506
9.2-3.
Inversion of Functions with Finitely Many Singular Points
. 507
9.2-4.
Convolution Theorem. Main Properties of the Laplace Transform
. 507
9.2-5.
Limit Theorems
. 507
9.2-6.
Representation of Inverse Transforms as Convergent Series
. 509
9.2-7.
Representation of Inverse Transforms as Asymptotic Expansions as
x
—>
oo
. 509
9.2-8.
Post-Widder Formula
. 510
9.3.
Mellin Transform
. 510
9.3-1.
Definition. Inversion Formula
. 510
9.3-2.
Main Properties of the Mellin Transform
. 511
9.3-3.
Relation Among the Mellin, Laplace, and Fourier Transforms
. 511
9.4.
Fourier Transform
. 512
9.4-1.
Definition. Inversion Formula
. 512
9.4-2.
Asymmetric Form of the Transform
. 512
9.4-3.
Alternative Fourier Transform
. 512
9.4-4.
Convolution Theorem. Main Properties of the Fourier Transforms
. 513
9.5.
Fourier Cosine and Sine Transforms
. 514
9.5-1.
Fourier Cosine Transform
. 514
9.5-2.
Fourier Sine Transform
. 514
9.6.
Other Integral Transforms
. 515
9.6-1.
Hankel Transform
. 515
9.6-2.
Meijer
Transform
. 516
9.6-3.
Kontorovich-Lebedev Transform
. 516
9.6-4.
r-transform
. 516
9.6-5.
Summary Table of Integral Transforms
. 517
10.
Methods for Solving Linear Equations of the Form f* K(x,t)y(t) dt
=
f(x)
. 519
10.1.
Volteira
Equations of the First Kind
. 519
10.1-1.
Equations of the First Kind. Function and Kernel Classes
. 519
10.1-2.
Existence and Uniqueness of a Solution
. 520
10.1 -3.
Some Problems Leading to Volterra Integral Equations of the First Kind
. 520
Contents xiii
10.2.
Equations
with Degenerate Kernel: K(x, t)
=
gx(x)h\(t)
+ ■■■ +
gn(x)hn(t)
. 522
10.2-1.
Equations with Kernel of the Form K(x,t) = gdx)hdt) +g2(x)h2(t)
. 522
10.2-2.
Equations with General Degenerate Kernel
. 523
10.3.
Reduction of Volterra Equations of the First Kind to Volterra Equations of the Second
Kind
. 524
10.3-1.
First Method
. 524
10.3-2.
Second Method
. 524
10.4.
Equations with Difference Kernel: K(x, t)
=
K(x
-ť)
. 524
10.4-1.
Solution Method Based on the Laplace Transform
. 524
10.4-2.
Case in Which the Transform of the Solution is a Rational Function
. 525
10.4-3.
Convolution Representation of a Solution
. 526
10.4-4.
Application of an Auxiliary Equation
. 527
10.4-5.
Reduction to Ordinary Differential Equations
. 527
10.4-6.
Reduction of a Volterra Equation to
a
Wiener-Hopf
Equation
. 528
10.5.
Method of Fractional Differentiation
. 529
10.5-1.
Definition of Fractional Integrals
. 529
10.5-2.
Definition of Fractional Derivatives
. 529
10.5-3.
Main Properties
. 530
10.5-4.
Solution of the Generalized Abel Equation
. 531
10.5-5.
Erdélyi-Kober
Operators
. 532
10.6.
Equations with Weakly Singular Kernel
. 532
10.6-1.
Method of Transformation of the Kernel
. 532
10.6-2.
Kernel with Logarithmic Singularity
. 533
10.7.
Method of Quadratures
. 534
10.7-1.
Quadrature Formulas
. 534
10.7-2.
General Scheme of the Method
. 535
10.7-3.
Algorithm Based on the Trapezoidal Rule
. 536
10.7-4.
Algorithm for an Equation with Degenerate Kernel
. 536
10.8.
Equations with Infinite Integration Limit
. 537
10.8-1.
Equation of the First Kind with Variable Lower Limit of Integration
. 537
10.8-2.
Reduction
toa
Wiener-Hopf
Equation of the First Kind
. 538
11.
Methods for Solving Linear Equations of the Form y(x)
-
f* K(x, t)y(t) dt
=
f(x)
539
11.1.
Volterra Integral Equations of the Second Kind
. 539
1.1-1.
Preliminary Remarks. Equations for the Resolvent
. 539
1.1-2.
Relationship Between Solutions of Some Integral Equations
. 540
11.2.
Equations with Degenerate Kernel: K(x,t)
=
g\(x)h\(t)
+ ■ ■ ■ +
gn(x)hn(t)
. 540
1.2-1.
Equations with Kernel of the Form K(x, t)
=
ψ(χ)
+
ψ(χ)(χ
-t)
. 540
1.2-2.
Equations with Kernel of the Form K(x, t)
=
ψ(ί)
+
tp(t)(t -x)
. 541
1.2-3.
Equations with Kernel of the Form K(x,t) = T,m=\^rn(x)(x-t)m-]
. 542
1
.2-4.
Equations with Kernel of the Form K(x, t)
=
^Li V™(W
-
x)m~x
. 543
1.2-5.
Equations with Degenerate Kernel of the General Form
. 543
11.3.
Equations with Difference Kernel: K(x, t)
=
K(x
-í)
. 544
1
.3-1.
Solution Method Based on the Laplace Transform
. 544
.3-2.
Method Based on the Solution of an Auxiliary Equation
. 546
.3-3.
Reduction to Ordinary Differential Equations
. 547
.3-4.
Reduction to
a
Wiener-Hopf
Equation of the Second Kind
. 547
.3-5.
Method of Fractional Integration for the Generalized Abel Equation
. 548
.3-6.
Systems of Volterra Integral Equations
. 549
xiv Contents
11.4. Operator
Methods for Solving Linear Integral Equations
. 549
11.4-1.
Application of a Solution of a "Truncated" Equation of the First Kind
. 549
11.4-2.
Application of the Auxiliary Equation of the Second Kind
. 551
11.4-3.
Method for Solving "Quadratic" Operator Equations
. 552
11.4-4.
Solution of Operator Equations of Polynomial Form
. 553
11.4-5.
Some Generalizations
. 554
11.5.
Construction of Solutions of Integral Equations with Special Right-Hand Side
. 555
11.5-1.
General Scheme
. 555
11.5-2.
Generating Function of Exponential Form
. 555
11.5-3.
Power-Law Generating Function
. 557
11.5-4.
Generating Function Containing Sines and Cosines
. 558
11.6.
Method of Model Solutions
. 559
11.6-1.
Preliminary Remarks
. 559
11.6-2.
Description of the Method
. 560
11.6-3.
Model Solution in the Case of an Exponential Right-Hand Side
. 561
11.6-4.
Model Solution in the Case of a Power-Law Right-Hand Side
. 562
11.6-5.
Model Solution in the Case of a Sine-Shaped Right-Hand Side
. 562
11.6-6.
Model Solution in the Case of a Cosine-Shaped Right-Hand Side
. 563
11.6-7.
Some Generalizations
. 563
11.7.
Method of Differentiation for Integral Equations
. 564
11.7-1.
Equations with Kernel Containing a Sum of Exponential Functions
. 564
11.7-2.
Equations with Kernel Containing a Sum of Hyperbolic Functions
. 564
11.7-3.
Equations with Kernel Containing a Sum of Trigonometric Functions
. 564
11.7-4.
Equations Whose Kernels Contain Combinations of Various Functions
. 565
11.8.
Reduction of Volterra Equations of the Second Kind to Volterra Equations of the First
Kind
. 565
11.8-1.
First Method
. 565
11.8-2.
Second Method
. 566
11.9.
Successive Approximation Method
. 566
11.9-1.
General Scheme
. 566
11.9-2.
Formula for the Resolvent
. 567
11.10.
Method of Quadratures
. 568
11.10-1.
General Scheme of the Method
. 568
11.10-2.
Application of the Trapezoidal Rule
. 568
11.10-3.
Case of a Degenerate Kernel
. 569
11.11.
Equations with Infinite Integration Limit
. 569
11.11-1.
Equation of the Second Kind with Variable Lower Integration Limit
. 570
11.11-2.
Reduction to
a
Wiener-Hopf
Equation of the Second Kind
. 571
12.
Methods for Solving Linear Equations of the Form ¡^ K(x, t)y(t) dt
= ƒ
(ж)
. 573
12.1.
Some Definition and Remarks
. 573
12.1-1.
Fredholm
Integral Equations of the First Kind
. 573
12.1-2.
Integral Equations of the First Kind with Weak Singularity
. 574
12.1-3.
Integral Equations of Convolution Type
. 574
12.1-4.
Dual Integral Equations of the First Kind
. 575
12.1-5.
Some Problems Leading to Integral Equations of the First Kind
. 575
12.2.
Integral Equations of the First Kind with Symmetric Kernel
. 577
12.2-1.
Solution of an Integral Equation in Terms of Series in Eigenfunctions of Its
Kernel
. 577
12.2-2.
Method of Successive Approximations
. 579
CONTKNTS
XV
12.3. Integral
Equations
of the First Kind with Nonsymmetric Kernel
. 580
12.3-1.
Representation of a Solution in the Form of Series. General Description
. 580
12.3-2.
Special Case of a Kernel That is a Generating Function
. 580
12.3-3.
Special Case of the Right-Hand Side Represented in Terms of Orthogonal
Functions
. 582
12.3-4.
General Case. Galerkin's Method
. 582
12.3-5.
Utilization of the Schmidt Kernels for the Construction of Solutions of
Equations
. 582
12.4.
Method of Differentiation for Integral Equations
. 583
12.4-1.
Equations with Modulus
. 583
12.4-2.
Other Equations. Some Generalizations
. 585
12.5.
Method of Integral Transforms
. 586
12.5-1.
Equation with Difference Kernel on the Entire Axis
. 586
12.5-2.
Equations with Kernel K(x, t)
=
K(x/t) on the Semiaxis
. 587
12.5-3.
Equation with Kernel K(x, t)
=
K(xt) and Some Generalizations
. 587
12.6.
Krein's Method and Some Other Exact Methods for Integral Equations of Special Types
588
12.6-1.
Krein's Method for an Equation with Difference Kernel with a Weak Singularity
588
12.6-2.
Kernel is the Sum of
a Nondegenerate
Kernel and an Arbitrary Degenerate
Kernel
. 589
12.6-3.
Reduction of Integral Equations of the First Kind to Equations of the Second
Kind
. 591
12.7.
Riemann Problem for the Real Axis
. 592
12.7-1.
Relationships Between the Fourier Integral and the Cauchy Type Integral
. 592
12.7-2.
One-Sided Fourier Integrals
. 593
12.7-3.
Analytic Continuation Theorem and the Generalized Liouville Theorem
. 595
12.7-4.
Riemann Boundary Value Problem
. 595
12.7-5.
Problems with Rational Coefficients
. 601
12.7-6.
Exceptional Cases. The Homogeneous Problem
. 602
12.7-7.
Exceptional Cases. The Nonhomogeneous Problem
. 604
12.8.
Carleman
Method for Equations of the Convolution Type of the First Kind
. 606
12.8-1. Wiener-Hopf
Equation of the First Kind
. 606
12.8-2.
Integral Equations of the First Kind with Two Kernels
. 607
12.9.
Dual Integral Equations of the First Kind
. 610
12.9-1.
Carleman
Method for Equations with Difference Kernels
. 610
12.9-2.
General Scheme of Finding Solutions of Dual Integral Equations
. 611
12.9-3.
Exact Solutions of Some Dual Equations of the First Kind
. 613
12.9-4.
Reduction of Dual Equations to
a Fredholm
Equation
. 615
12.10.
Asymptotic Methods for Solving Equations with Logarithmic Singularity
. 618
12.10-1.
Preliminary Remarks
. 618
12.10-2.
Solution for Large A
. 619
12.10-3.
Solution for Small A
. 620
12.10-4.
Integral Equation of Elasticity
. 621
12.11.
Regularization Methods
. 621
12.11-1.
Lavrentiev Regularization Method
. 621
12.11-2.
Tikhonov Regularization Method
. 622
12.12.
Fredholm
Integral Equation of the First Kind as an Ill-Posed Problem
. 623
12.12-1.
General Notions of Well-Posed and Ill-Posed Problems
. 623
12.12-2.
Integral Equation of the First Kind
isan
Ill-Posed Problem
. 624
xvi Contents
13.
Methods for Solving Linear Equations of the Form y(x)
-
f£ K(x, t)y(t) dt
= ƒ
(ж)
625
13.1.
Some Definition and Remarks
. 625
13.1-1.
Fredholm
Equations and Equations with Weak Singularity of the Second Kind
625
13.1-2.
Structure of the Solution
. 626
13.1-3.
Integral Equations of Convolution Type of the Second Kind
. 626
13.1-4.
Dual Integral Equations of the Second Kind
. 627
13.2.
Fredholm
Equations of the Second Kind with Degenerate Kernel. Some Generalizations
627
13.2-1.
Simplest Degenerate Kernel
. 627
13.2-2.
Degenerate Kernel in the General Case
. 628
13.2-3.
Kernel is the Sum of
a Nondegenerate
Kernel and an Arbitrary Degenerate
Kernel
. 631
13.3.
Solution as a Power Series in the Parameter. Method of Successive Approximations
. 632
13.3-1.
Iterated Kernels
. 632
13.3-2.
Method of Successive Approximations
. 633
13.3-3.
Construction of the Resolvent
. 633
13.3-4.
Orthogonal Kernels
. 634
13.4.
Method of
Fredholm
Determinants
. 635
13.4-1.
Formula for the Resolvent
. 635
13.4-2.
Recurrent Relations
. 636
13.5.
Fredholm
Theorems and the
Fredholm
Alternative
. 637
13.5-1.
Fredholm
Theorems
. 637
13.5-2.
Fredholm
Alternative
. 638
13.6.
Fredholm
Integral Equations of the Second Kind with Symmetric Kernel
. 639
13.6-1.
Characteristic Values and Eigenfunctions
. 639
13.6-2.
Bilinear Series
. 640
13.6-3.
Hilbert-Schmidt Theorem
. 641
13.6-4.
Bilinear Series of Iterated Kernels
. 642
13.6-5.
Solution of the Nonhomogeneous Equation
. 642
13.6-6.
Fredholm
Alternative for Symmetric Equations
. 643
13.6-7.
Resolvent of a Symmetric Kernel
. 644
13.6-8.
Extremal Properties of Characteristic Values and Eigenfunctions
. 644
13.6-9.
Kellog's Method for Finding Characteristic Values in the Case of Symmetric
Kernel
. 645
13.6-10.
Trace Method for the Approximation of Characteristic Values
. 646
13.6-11.
Integral Equations Reducible to Symmetric Equations
. 647
13.6-12.
Skew-Symmetric Integral Equations
. 647
13.6-13.
Remark on Nonsymmetric Kernels
. 647
13.7.
Integral Equations with
Nonnegative
Kernels
. 648
13.7-1.
Positive Principal Eigenvalues. Generalized Jentzch Theorem
. 648
13.7-2.
Positive Solutions of a Nonhomogeneous Integral Equation
. 649
13.7-3.
Estimates for the Spectral Radius
. 649
13.7-4.
Basic Definition and Theorems for Oscillating Kernels
. 651
13.7-5.
Stochastic Kernels
. 654
13.8.
Operator Method for Solving Integral Equations of the Second Kind
. 655
13.8-1.
Simplest Scheme
. 655
13.8-2.
Solution of Equations of the Second Kind on the Semiaxis
. 655
CONTKNTS
XVII
13.
1-І
13.
1-2
13.
1-3
13.
1-4
13.
1-5
13.9.
Methods of Integral Transforms and Model Solutions
. 656
13.9-1.
Equation with Difference Kernel on the Entire Axis
. 656
13.9-2.
Equation with the Kernel K(x, t)
=
t~xQ(x/t) on the Semiaxis
. 657
13.9-3.
Equation with the Kernel K(x, t)
=
t0Q(xt) on the Semiaxis
. 658
13.9-4.
Method of Model Solutions for Equations on the Entire Axis
. 659
13.10.
Carleman
Method for Integral Equations of Convolution Type of the Second Kind
. . 660
13.10-1. Wiener-Hopf
Equation of the Second Kind
. 660
13.10-2.
Integral Equation of the Second Kind with Two Kernels
. 664
13.10-3.
Equations of Convolution Type with Variable Integration Limit
. 668
13.10-4.
Dual Equation of Convolution Type of the Second Kind
. 670
13.11. Wiener-Hopf
Method
. 671
Some Remarks
. 671
Homogeneous
Wiener-Hopf
Equation of the Second Kind
. 673
General Scheme of the Method. The Factorization Problem
. 676
Nonhomogeneous
Wiener-Hopf
Equation of the Second Kind
. 677
Exceptional Case of
a
Wiener-Hopf
Equation of the Second Kind
. 678
13.12.
Krein's Method for
Wiener-Hopf
Equations
. 679
13.12-1.
Some Remarks. The Factorization Problem
. 679
13.12-2.
Solution of the
Wiener-Hopf
Equations of the Second Kind
. 681
13.12-3.
Hopf-Fock Formula
. 683
13.13.
Methods for Solving Equations with Difference Kernels on a Finite Interval
. 683
13.13-1.
Krein's Method
. 683
13.13-2.
Kernels with Rational Fourier Transforms
. 685
13.13-3.
Reduction to Ordinary Differential Equations
. 686
13.14.
Method of Approximating a Kernel by a Degenerate One
. 687
13.14-1.
Approximation of the Kernel
. 687
13.14-2.
Approximate Solution
. 688
13.15.
Bateman Method
. 689
13.15-1.
General Scheme of the Method
. 689
13.15-2.
Some Special Cases
. 690
13.16.
Collocation Method
. 692
13.16-1.
General Remarks
. 692
13.16-2.
Approximate Solution
. 693
13.16-3.
Eigenfunctions of the Equation
. 694
13.17.
Method of Least Squares
. 695
13.17-1.
Description of the Method
. 695
13.17-2.
Construction of Eigenfunctions
. 696
13.18.
Bubnov-Galerkin Method
. 697
13.18-1.
Description of the Method
. 697
13.18-2.
Characteristic Values
. 697
13.19.
Quadrature Method
. 698
13.19-1.
General Scheme for
Fredholm
Equations of the Second Kind
. 698
13.19-2.
Construction of the Eigenfunctions
. 699
13.19-3.
Specific Features of the Application of Quadrature Formulas
. 700
13.20.
Systems of
Fredholm
Integral Equations of the Second Kind
. 701
13.20-1.
Some Remarks
. 701
13.20-2.
Method of Reducing a System of Equations to a Single Equation
. 701
xviii Contents
13.21. Regularization
Method for Equations with Infinite Limits of Integration
. 702
13.21-1.
Basic Equation and
Fredholm
Theorems
. 702
13.21-2.
Regularizing Operators
. 703
13.21-3.
Regularization Method
. 704
14.
Methods for Solving Singular Integral Equations of the First Kind
. 707
14.1.
Some Definitions and Remarks
. 707
14.1-1.
Integral Equations of the First Kind with Cauchy Kernel
. 707
14.1-2.
Integral Equations of the First Kind with Hubert Kernel
. 707
14.2.
Cauchy Type Integral
. 708
14.2-1.
Definition of the Cauchy Type Integral
. 708
14.2-2.
Holder Condition
. 709
14.2-3.
Principal Value of a Singular Integral
. 709
14.2-4.
Multivalued Functions
. 711
14.2-5.
Principal Value of a Singular Curvilinear Integral
. 712
14.2-6.
Poincaré-Bertrand
Formula
. 714
14.3.
Riemann Boundary Value Problem
. 714
14.3-1.
Principle of Argument. The Generalized Liouville Theorem
. 714
14.3-2.
Hermite Interpolation Polynomial
. 716
14.3-3.
Notion of the Index
. 716
14.3-4.
Statement of the Riemann Problem
. 718
14.3-5.
Solution of the Homogeneous Problem
. 720
14.3-6.
Solution of the Nonhomogeneous Problem
. 721
14.3-7.
Riemann Problem with Rational Coefficients
. 723
14.3-8.
Riemann Problem for a Half-Plane
. 725
14.3-9.
Exceptional Cases of the Riemann Problem
. 727
14.3-10.
Riemann Problem for a Multiply Connected Domain
. 731
14.3-11.
Riemann Problem for Open Curves
. 734
14.3-12.
Riemann Problem with a Discontinuous Coefficient
. 739
14.3-13.
Riemann Problem in the General Case
. 741
14.3-14.
Hubert Boundary Value Problem
. 742
14.4.
Singular Integral Equations of the First Kind
. 743
14.4-1.
Simplest Equation with Cauchy Kernel
. 743
14.4-2.
Equation with Cauchy Kernel on the Real Axis
. 743
14.4-3.
Equation of the First Kind on a Finite Interval
. 744
14.4-4.
General Equation of the First Kind with Cauchy Kernel
. 745
14.4-5.
Equations of the First Kind with Hubert Kernel
. 746
14.5.
Multhopp-Kalandiya Method
. 747
14.5-1.
Solution That is Unbounded at the
Endpoints
of the Interval
. 747
14.5-2.
Solution Bounded at One
Endpoint
of the Interval
. 749
14.5-3.
Solution Bounded at Both
Endpoints
of the Interval
. 750
14.6.
Hypersingular Integral Equations
. 751
14.6-1.
Hypersingular Integral Equations with Cauchy- and Hilbert-Type Kernels
. 751
14.6-2.
Definition of Hypersingular Integrals
. 751
14.6-3.
Exact Solution of the Simplest Hypersingular Equation with Cauchy-Type
Kernel
. 753
14.6-4.
Exact Solution of the Simplest Hypersingular Equation with Hilbert-Type
Kernel
. 754
14.6-5.
Numerical Methods for Hypersingular Equations
. 754
CONTKNTS
ХІХ
15.
Methods for Solving Complete Singular Integral Equations
. 757
15.1.
Some Definitions and Remarks
. 757
15.1-1.
Integral Equations with Cauchy Kernel
. 757
15.1-2.
Integral Equations with Hubert Kernel
. 759
15.1-3.
Fredholm
Equations of the Second Kind on a Contour
. 759
15.2.
Carleman
Method for Characteristic Equations
. 761
15.2-1.
Characteristic Equation with Cauchy Kernel
. 761
15.2-2.
Transposed Equation of a Characteristic Equation
. 764
15.2-3.
Characteristic Equation on the Real Axis
. 765
15.2-4.
Exceptional Case of a Characteristic Equation
. 767
15.2-5.
Characteristic Equation with Hubert Kernel
. 769
15.2-6.
Tricomi
Equation
. 769
15.3.
Complete Singular Integral Equations Solvable in a Closed Form
. 770
15.3-1.
Closed-Form Solutions in the Case of Constant Coefficients
. 770
15.3-2.
Closed-Form Solutions in the General Case
. 77
1
15.4.
Regularization Method for Complete Singular Integral Equations
. 772
15.4-1.
Certain Properties of Singular Operators
. 772
15.4-2.
Regularizer
. 774
15.4-3.
Methods of Left and Right Regularization
. 775
15.4-4.
Problem of Equivalent Regularization
. 776
15.4-5.
Fredholm
Theorems
. 777
15.4-6.
Carleman-Vekua Approach to the Regularization
. 778
15.4-7.
Regularization in Exceptional Cases
. 779
15.4-8.
Complete Equation with Hubert Kernel
. 780
15.5.
Analysis of Solutions Singularities for Complete Integral Equations with Generalized
Cauchy Kernels
. 783
15.5-1.
Statement of the Problem and Preliminary Remarks
. 783
15.5-2.
Auxiliary Results
. 784
15.5-3.
Equations for the Exponents of Singularity of a Solution
. 787
15.5-4.
Analysis of Equations for Singularity Exponents
. 789
15.5-5.
Application to an Equation Arising in Fracture Mechanics
. 791
15.6.
Direct Numerical Solution of Singular Integral Equations with Generalized Kernels
. 792
15.6-1.
Preliminary Remarks
. 792
15.6-2.
Quadrature Formulas for Integrals with the Jacobi Weight Function
. 793
15.6-3.
Approximation of Solutions in Terms of a System of Orthogonal Polynomials
795
15.6-4.
Some Special Functions and Their Calculations
. 797
15.6-5.
Numerical Solution of Singular Integral Equations
. 799
15.6-6.
Numerical Solutions of Singular Integral Equations of Bueckner Type
. 801
16.
Methods for Solving Nonlinear Integral Equations
805
16.1.
Some Definitions and Remarks
. 805
16.1-1.
Nonlinear Equations with Variable Limit of Integration (Volterra Equations)
. 805
16.1-2.
Nonlinear Equations with Constant Integration Limits (Urysohn Equations)
. 806
16.1-3.
Some Special Features of Nonlinear Integral Equations
. 807
16.2.
Exact Methods for Nonlinear Equations with Variable Limit of Integration
. 809
16.2-1.
Method of Integral Transforms
. 809
16.2-2.
Method of Differentiation for Nonlinear Equations with Degenerate Kernel
. 810
xx Contents
16.3.
Approximate and Numerical Methods for Nonlinear Equations with Variable Limit of
Integration
. 811
16.3-1.
Successive Approximation Method
. 811
16.3-2.
Newton-Kantorovich Method
. 813
16.3-3.
Collocation Method
. 815
16.3-4.
Quadrature Method
. 816
16.4.
Exact Methods for Nonlinear Equations with Constant Integration Limits
. 817
16.4-1.
Nonlinear Equations with Degenerate Kernels
. 817
16.4-2.
Method of Integral Transforms
. 819
16.4-3.
Method of Differentiating for Integral Equations
. 820
16.4-4.
Method for Special Urysohn Equations of the First Kind
. 821
16.4-5.
Method for Special Urysohn Equations of the Second Kind
. 822
16.4-6.
Some Generalizations
. 824
16.5.
Approximate and Numerical Methods for Nonlinear Equations with Constant Integration
Limits
. 826
16.5-1.
Successive Approximation Method
. 826
16.5-2.
Newton-Kantorovich Method
. 827
16.5-3.
Quadrature Method
. 829
16.5-4.
Tikhonov Regularization Method
. 829
16.6
Existence and Uniqueness Theorems for Nonlinear Equations
. 830
16.6-1. Hammerstein
Equations
. 830
16.6-2.
Urysohn Equations
. 832
16.7.
Nonlinear Equations with a Parameter: Eigenfunctions, Eigenvalues, Bifurcation Points
834
16.7-1.
Eigenfunctions and Eigenvalues of Nonlinear Integral Equations
. 834
16.7-2.
Local Solutions of a Nonlinear Integral Equation with a Parameter
. 835
16.7-3.
Bifurcation Points of Nonlinear Integral Equations
. 835
17.
Methods for Solving Multidimensional Mixed Integral Equations
. 839
17.1.
Some Definition and Remarks
. 839
17.1-1.
Basic Classes of Functions
. 839
17.1-2.
Mixed Equations on a Finite Interval
. 840
17.1-3.
Mixed Equation on a Ring-Shaped (Circular) Domain
. 841
17.1-4.
Mixed Equations on a Closed Bounded Set
. 842
17.2.
Methods of Solution of Mixed Integral Equations on a Finite Interval
. 843
17.2-1.
Equation with a Hilbert-Schmidt Kernel and a Given Right-Hand Side
. 843
17.2-2.
Equation with Hilbert-Schmidt Kernel and Auxiliary Conditions
. 845
17.2-3.
Equation with a Schmidt Kernel and a Given Right-Hand Side on an Interval
. 848
17.2-4.
Equation with a Schmidt Kernel and Auxiliary Conditions
. 851
17.3.
Methods of Solving Mixed Integral Equations on a Ring-Shaped Domain
. 855
17.3-1.
Equation with a Hilbert-Schmidt Kernel and a Given Right-Hand Side
. 855
17.3-2.
Equation with a Hilbert-Schmidt Kernel and Auxiliary Conditions
. 856
17.3-3.
Equation with a Schmidt Kernel and a Given Right-Hand Side
. 859
17.3-4.
Equation with a Schmidt Kernel and Auxiliary Conditions on Ring-Shaped
Domain
. 862
17.4.
Projection Method for Solving Mixed Equations on a Bounded Set
. 866
17.4-1.
Mixed Operator Equation with a Given Right-Hand Side
. 866
Π
.4-2.
Mixed Operator Equations with Auxiliary Conditions
. 869
17.4-3.
General Projection Problem for Operator Equation
. 873
Contents
xxi
18.
Application of Integral Equations for the Investigation of Differential Equations
. 875
18.1.
Reduction of the Cauchy Problem for ODEs to Integral Equations
. 875
18.1-1.
Cauchy Problem for First-Order ODEs. Uniqueness and Existence Theorems
875
18.1-2.
Cauchy Problem for First-Order ODEs. Method of Successive Approximations
876
18.1-3.
Cauchy Problem for Second-Order ODEs. Method of Successive
Approximations
. 876
18.1-4.
Cauchy Problem for a Special
η
-Order
Linear ODE
. 876
18.2.
Reduction of Boundary Value Problems for ODEs to Volterra Integral Equations.
Calculation of Eigenvalues
. 877
18.2-1.
Reduction of Differential Equations to Volterra Integral Equations
. 877
18.2-2.
Application of Volterra Equations to the Calculation of Eigenvalues
. 879
18.3.
Reduction of Boundary Value Problems for ODEs to
Fredholm
Integral Equations with
the Help of the Green's Function
. 881
18.3-1.
Linear Ordinary Differential Equations. Fundamental Solutions
. 881
18.3-2.
Boundary Value Problems for nth Order Differential Equations. Green's
Function
. 882
18.3-3.
Boundary Value Problems for Second-Order Differential Equations. Green's
Function
. 883
18.3-4.
Nonlinear Problem of Nonisothermal Flow in Plane Channel
. 884
18.4.
Reduction of PDEs with Boundary Conditions of the Third Kind to Integral Equations
887
18.4-1.
Usage of Particular Solutions of PDEs for the Construction of Other Solutions
887
18.4-2.
Mass Transfer to a Particle in Fluid Flow Complicated by a Surface Reaction
888
18.4-3.
Integral Equations for Surface Concentration and Diffusion Flux
. 890
18.4-4.
Method of Numerical Integration of the Equation for Surface Concentration
. 891
18.5.
Representation of Linear Boundary Value Problems in Terms of Potentials
. 892
18.5-1.
Basic Types of Potentials for the Laplace Equation and Their Properties
. 892
18.5-2.
Integral Identities. Green's Formula
. 895
18.5-3.
Reduction of Interior Dirichlet and Neumann Problems to Integral Equations
. 895
18.5-4.
Reduction of Exterior Dirichlet and Neumann Problems to Integral Equations
896
18.6.
Representation of Solutions of Nonlinear PDEs in Terms of Solutions of Linear Integral
Equations (Inverse Scattering)
. 898
18.6-1.
Description of the Zakharov-Shabat Method
. 898
18.6-2.
Korteweg-de
Vries
Equation and Other Nonlinear Equations
. 899
Supplements
Supplement
1.
Elementary Functions and Their Properties
. 905
1.1.
Power, Exponential, and Logarithmic Functions
. 905
1.1-1.
Properties of the Power Function
. 905
1.1-2.
Properties of the Exponential Function
. 905
1.1-3.
Properties of the Logarithmic Function
. 906
1.2.
Trigonometric Functions
. 907
1.2-1.
Simplest Relations
. 907
1.2-2.
Reduction Formulas
. 907
1.2-3.
Relations Between Trigonometric Functions of Single Argument
. 908
1.2-4.
Addition and Subtraction of Trigonometric Functions
. 908
1.2-5.
Products of Trigonometric Functions
. 908
1.2-6.
Powers of Trigonometric Functions
. 908
1.2-7.
Addition Formulas
. 909
xxii Contents
1.2-8.
Trigonometric
Functions of Multiple Arguments
. 909
1.2-9.
Trigonometric Functions of Half Argument
. 909
1.2-10.
Differentiation Formulas
. 910
1.2-11.
Integration Formulas
. 910
1.2-12.
Expansion in Power Series
. 910
1.2-13.
Representation in the Form of Infinite Products
. 910
1.2-14.
Euler
and
de Moivre
Formulas. Relationship with Hyperbolic Functions
. 911
1.3.
Inverse Trigonometric Functions
. 911
1.3-1.
Definitions of Inverse Trigonometric Functions
. 911
1.3-2.
Simplest Formulas
. 912
1.3-3.
Some Properties
. 912
1.3-4.
Relations Between Inverse Trigonometric Functions
. 912
1.3-5.
Addition and Subtraction of Inverse Trigonometric Functions
. 912
1.3-6.
Differentiation Formulas
. 913
1.3-7.
Integration Formulas
. 913
1.3-8.
Expansion in Power Series
. 913
1.4.
Hyperbolic Functions
. 913
1.4-1.
Definitions of Hyperbolic Functions
. 913
1.4-2.
Simplest Relations
. 913
1.4-3.
Relations Between Hyperbolic Functions of Single Argument (x
> 0) . 914
1.4-4.
Addition and Subtraction of Hyperbolic Functions
. 914
1.4-5.
Products of Hyperbolic Functions
. 914
1.4-6.
Powers of Hyperbolic Functions
. 914
1.4-7.
Addition Formulas
. 915
1.4-8.
Hyperbolic Functions of Multiple Argument
. 915
1.4-9.
Hyperbolic Functions of Half Argument
. 915
1.4-10.
Differentiation Formulas
. 916
1.4-11.
Integration Formulas
. 916
1.4-12.
Expansion in Power Series
. 916
1.4-13.
Representation in the Form of Infinite Products
. 916
1.4-14.
Relationship with Trigonometric Functions
. 916
1.5.
Inverse Hyperbolic Functions
. 917
1.5-1.
Definitions of Inverse Hyperbolic Functions
. 917
1.5-2.
Simplest Relations
. 917
1.5-3.
Relations Between Inverse Hyperbolic Functions
. 917
1.5-4.
Addition and Subtraction of Inverse Hyperbolic Functions
. 917
1.5-5.
Differentiation Formulas
. 917
1.5-6.
Integration Formulas
. 918
1.5-7.
Expansion in Power Series
. 918
Supplement
2.
Finite Sums and Infinite Series
. 919
2.1.
Finite Numerical Sums
. . 919
-1.
Progressions
. 919
-2.
Sums of Powers of Natural Numbers Having the Form
£
km
. 919
-3.
Alternating Sums of Powers of Natural Numbers, ]T(-l)feA;m
. 920
-4.
Other Sums Containing Integers
. 920
-5.
Sums Containing Binomial Coefficients
. 920
-6.
Other Numerical Sums
. . 921
CONTKNTS
ХХІІІ
2.2.
Finite
Functional Sums
. 922
2.2-1.
Sums Involving Hyperbolic Functions
. 922
2.2-2.
Sums Involving Trigonometric Functions
. 922
2.3.
Infinite Numerical Series
. 924
2.3-1.
Progressions
. 924
2.3-2.
Other Numerical Series
. 924
2.4.
Infinite Functional Series
. 925
2.4-1.
Power Series
. 925
2.4-2.
Trigonometric Series in One Variable Involving Sine
. 927
2.4-3.
Trigonometric Series in One Variable Involving Cosine
. 928
2.4-4.
Trigonometric Series in Two Variables
. 930
Supplement
3.
Tables of Indefinite Integrals
. 933
3.1.
Integrals Involving Rational Functions
. 933
3.1-1.
Integrals Involving a + bx
. 933
1-2.
Integrals Involving a
+
χ
and
b + x .
933
-3.
Integrals Involving a~
+
x
. 934
-4.
Integrals Involving a2
-
x2
. 935
-5.
Integrals Involving a?
+
Xі
. 936
1-6.
Integrals Involving a?
-
x3
. 936
1-7.
Integrals Involving a4 ±x4
. 937
3.2.
Integrals Involving Irrational Functions
. 937
3.2-1.
Integrals Involving x'/2
. 937
3.2-2.
Integrals Involving (a
+
bx)p'2
. 938
3.2-3.
Integrals Involving (x2
+
a2)1/2
. 938
3.2-4.
Integrals Involving (x2
-
a2)i/2
. 938
3.2-5.
Integrals Involving (a2 -x2)'/2
. 939
3.2-6.
Integrals Involving Arbitrary Powers. Reduction Formulas
. 939
3.3.
Integrals Involving Exponential Functions
. 940
3.4.
Integrals Involving Hyperbolic Functions
. 940
3.4-1.
Integrals Involving cosh
x
. 940
3.4-2.
Integrals Involving sinh
x
. 94
1
3.4-3.
Integrals Involving tanh
x
or coth
x
. 942
3.5.
Integrals Involving Logarithmic Functions
. 943
3.6.
Integrals Involving Trigonometric Functions
. 944
3.6-1.
Integrals Involving cos
x (n
=1,2,.) . 944
3.6-2.
Integrals Involving sin
x (n
=
1
, 2, . ) . 945
3.6-3.
Integrals Involving sin.
г
and cos.
r
. 947
3.6-4.
Reduction Formulas
. 947
3.6-5.
Integrals Involving tan
x
and cot
r
. 947
3.7.
Integrals Involving Inverse Trigonometric Functions
. 948
Supplement
4.
Tables of Definite Integrals
951
4.1.
Integrals Involving Power-Law Functions
. 951
4.1-1.
Integrals Over a Finite Interval
. 951
4.1-2.
Integrals Over an Infinite Interval
. 952
4.2.
Integrals Involving Exponential Functions
. 954
4.3.
Integrals Involving Hyperbolic Functions
. 955
4.4.
Integrals Involving Logarithmic Functions
. 955
xxiv Contents
4.5. Integrals
Involving
Trigonometrie
Functions.
956
4.5-1. Integrals
Over a Finite Interval
. 956
4.5-2.
Integrals Over an Infinite Interval
. 957
4.6.
Integrals Involving Bessel Functions
. 958
4.6-1.
Integrals Over an Infinite Interval
. 958
4.6-2.
Other Integrals
. 959
Supplement
5.
Tables of Laplace Transforms
. 961
5.1.
General Formulas
. 961
5.2.
Expressions with Power-Law Functions
. 963
5.3.
Expressions with Exponential Functions
. 963
5.4.
Expressions with Hyperbolic Functions
. 964
5.5.
Expressions with Logarithmic Functions
. 965
5.6.
Expressions with Trigonometric Functions
. 966
5.7.
Expressions with Special Functions
. 967
Supplement
6.
Tables of Inverse Laplace Transforms
. 969
6.1.
General Formulas
. 969
6.2.
Expressions with Rational Functions
. 971
6.3.
Expressions with Square Roots
. 975
6.4.
Expressions with Arbitrary Powers
. 977
6.5.
Expressions with Exponential Functions
. 978
6.6.
Expressions with Hyperbolic Functions
. 979
6.7.
Expressions with Logarithmic Functions
. 980
6.8.
Expressions with Trigonometric Functions
. 981
6.9.
Expressions with Special Functions
. 981
Supplement
7.
Tables of Fourier Cosine Transforms
. 983
7.1.
General Formulas
. 983
7.2.
Expressions with Power-Law Functions
. 983
7.3.
Expressions with Exponential Functions
. 984
7.4.
Expressions with Hyperbolic Functions
. 985
7.5.
Expressions with Logarithmic Functions
. 985
7.6.
Expressions with Trigonometric Functions
. 986
7.7.
Expressions with Special Functions
. 987
Supplement
8.
Tables of Fourier Sine Transforms
. 989
8.1.
General Formulas
. 989
8.2.
Expressions with Power-Law Functions
. 989
8.3.
Expressions with Exponential Functions
. 990
8.4.
Expressions with Hyperbolic Functions
. 991
8.5.
Expressions with Logarithmic Functions
. 992
8.6.
Expressions with Trigonometric Functions
. 992
8.7.
Expressions with Special Functions
. 993
Contents
xxv
Supplement
9.
Tables of Mellin Transforms
. 997
9.1.
General Formulas
. 997
9.2.
Expressions with Power-Law Functions
. 998
9.3.
Expressions with Exponential Functions
. 998
9.4.
Expressions with Logarithmic Functions
. 999
9.5.
Expressions with Trigonometric Functions
. 999
9.6.
Expressions with Special Functions
. 1000
Supplement
10.
Tables of Inverse Mellin Transforms
.1001
10.1.
Expressions with Power-Law Functions
.1001
10.2.
Expressions with Exponential and Logarithmic Functions
.1002
10.3.
Expressions with Trigonometric Functions
.1003
10.4.
Expressions with Special Functions
.1004
Supplement
11.
Special Functions and Their Properties
.1007
11.1.
Some Coefficients, Symbols, and Numbers
.1007
11.1-1.
Binomial Coefficients
.1007
11.1-2.
Pochhammer Symbol
.1007
11.1-3.
Bernoulli Numbers
.1008
11.1-4.
Euler
Numbers
.1008
11.2.
Error Functions. Exponential and Logarithmic Integrals
.1009
11.2-1.
Error Function and Complementary Error Function
.1009
11.2-2.
Exponential Integral
.1010
11.2-3.
Logarithmic Integral
.1010
11.3.
Sine Integral and Cosine Integral. Fresnel Integrals
. 1011
11.3-1.
Sine Integral
. 1011
11.3-2.
Cosine Integral
. 1011
11.3-3.
Fresnel Integrals and Generalized Fresnel Integrals
. 1012
11.4.
Gamma Function,
Psi
Function, and Beta Function
.1012
11.4-1.
Gamma Function
.1012
11.4-2.
Psi
Function
(Digamma
Function)
.1013
11.4-3.
Beta Function
.1014
11.5.
Incomplete Gamma and Beta Functions
.1014
11.5-1.
Incomplete Gamma Function
.1014
11.5-2.
Incomplete Beta Function
.1015
11.6.
Bessel Functions (Cylindrical Functions)
.1016
11.6-1.
Definitions and Basic Formulas
.1016
11.6-2.
Integral Representations and Asymptotic Expansions
.1017
11.6-3.
Zeros of Bessel Functions
.1019
11.6-4.
Orthogonality Properties of Bessel Functions
.1019
11.6-5.
Hankel Functions (Bessel Functions of the Third Kind)
.1020
11.7.
Modified Bessel Functions
.1021
11.7-1.
Definitions. Basic Formulas
.1021
11.7-2.
Integral Representations and Asymptotic Expansions
.1022
11.8.
Airy Functions
.
Ю23
11.8-1.
Definition and Basic Formulas
.1023
11.8-2.
Power Series and Asymptotic Expansions
.1023
xxvi Contents
11.9. Confluent Hypergeometric
Functions
.1024
11.9-1. Kummer and
Tricomi Confluent Hypergeometric
Functions
.1024
11.9-2.
Integral Representations and Asymptotic Expansions
.1027
11.9-3.
Whittaker Confluent Hypergeometric Functions
.1027
11.10.
Gauss Hypergeometric Functions
.1028
11.10-1.
Various Representations of the Gauss Hypergeometric Function
.1028
11.10-2.
Basic Properties
.1028
11.11.
Legendre Polynomials, Legendre Functions, and Associated Legendre Functions
. . 1030
11.11-1.
Legendre Polynomials and Legendre Functions
.1030
11.11-2.
Associated Legendre Functions with Integer Indices and Real Argument
. . 1031
11.11-3.
Associated Legendre Functions. General Case
.1032
11.12.
Parabolic Cylinder Functions
.1034
11.12-1.
Definitions. Basic Formulas
.1034
11.12-2.
Integral Representations, Asymptotic Expansions, and Linear Relations
. . 1035
11.13.
Elliptic Integrals
.1035
11.13-1.
Complete Elliptic Integrals
.1035
11.13-2.
Incomplete Elliptic Integrals (Elliptic Integrals)
.1037
11.14.
Elliptic Functions
.1038
11.14-1.
Jacobi Elliptic Functions
.1039
11.14-2.
Weierstrass
Elliptic Function
.1042
11.15.
Jacobi Theta Functions
.1043
11.15-1.
Series Representation of the Jacobi Theta Functions. Simplest Properties
. 1043
11.15-2.
Various Relations and Formulas. Connection with Jacobi Elliptic Functions
1044
11.16.
Mathieu
Functions and Modified
Mathieu
Functions
.1045
11.16-1.
Mathieu
Functions
.1045
11.16-2.
Modified
Mathieu
Functions
.1046
11.17.
Orthogonal Polynomials
.1047
11.17-1.
Laguerre Polynomials and Generalized Laguerre Polynomials
.1047
11.17-2.
Chebyshev Polynomials and Functions
.1048
11.17-3.
Hermite Polynomials and Functions
.1050
11.17-4.
Jacobi Polynomials
.1051
11.17-5. Gegenbauer
Polynomials
.1051
11.18. Nonorthogonal
Polynomials
.1052
11.18-1.
Bernoulli Polynomials
.1052
11.18-2.
Euler
Polynomials
.1053
Supplement
12.
Some Notions of Functional Analysis
.1055
12.1.
Functions of Bounded Variation
.1055
-1.
Definition of a Function of Bounded Variation
.1055
-2.
Classes of Functions of Bounded Variation
.1056
-3.
Properties of Functions of Bounded Variation
.1056
-4.
Criteria for Functions to Have Bounded Variation
.1057
12.
12.
12.
12.
12.
-5.
Properties of Continuous Functions of Bounded Variation
.1057
12.2.
Stieltjes
Integral
.1057
12.2-1.
Basic Definitions
.1057
12.2-2.
Properties of the
Stieltjes
Integral
.1058
12.2-3.
Existence Theorems for the
Stieltjes
Integral
.1058
CONTKNTS
ХХУ!!
12.3.
Lebesgue
Integral.1059
12.3-1.
Riemann
Integral
and the Lebesgue Integral
.1059
12.3-2.
Sets of Zero Measure. Notion of "Almost Everywhere"
.1060
12.3-3.
Step Functions and Measurable Functions
.1060
12.3-4.
Definition and Properties of the Lebesgue Integral
.1061
12.3-5.
Measurable Sets
.1062
12.3-6.
Integration Over Measurable Sets
.1063
12.3-7.
Case of an Infinite Interval
.1063
12.3-8.
Case of Several Variables
.1064
12.3-9.
Spaces Lp
.1064
12.4.
Linear Normed Spaces
.1065
12.4-1.
Linear Spaces
.1065
12.4-2.
Linear Normed Spaces
.1065
12.4-3.
Space of Continuous Functions C(a,b)
.1066
12.4-4.
Lebesgue Space Lp(a, b)
.1066
12.4-5.
Holder Space Ca(0,
1) .1066
12.4-6.
Space of Functions of Bounded Variation V(0,
1).1066
12.5.
Euclidean and Hubert Spaces. Linear Operators in Hubert Spaces
.1067
12.5-1.
Preliminary Remarks
.1067
12.5-2.
Euclidean and Hubert Spaces
.1067
12.5-3.
Linear Operators in Hubert Spaces
.1068
References
.1071
Index
.1081 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Poljanin, Andrej D. 1951- Manžirov, Aleksandr V. |
| author_GND | (DE-588)128391251 (DE-588)120620758 |
| author_facet | Poljanin, Andrej D. 1951- Manžirov, Aleksandr V. |
| author_role | aut aut |
| author_sort | Poljanin, Andrej D. 1951- |
| author_variant | a d p ad adp a v m av avm |
| building | Verbundindex |
| bvnumber | BV022933618 |
| callnumber-first | Q - Science |
| callnumber-label | QA431 |
| callnumber-raw | QA431 |
| callnumber-search | QA431 |
| callnumber-sort | QA 3431 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SH 500 SK 640 |
| ctrlnum | (OCoLC)167516078 (DE-599)GBV543235092 |
| dewey-full | 515/.45 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.45 |
| dewey-search | 515/.45 |
| dewey-sort | 3515 245 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| genre | (DE-588)4155008-0 Formelsammlung gnd-content (DE-588)4173536-5 Patentschrift gnd-content |
| genre_facet | Formelsammlung Patentschrift |
| id | DE-604.BV022933618 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:55:23Z |
| indexdate | 2024-07-09T21:07:58Z |
| institution | BVB |
| isbn | 9781584885078 1584885076 |
| language | English |
| lccn | 007035725 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016138406 |
| oclc_num | 167516078 |
| open_access_boolean | |
| owner | DE-703 DE-824 DE-92 DE-19 DE-BY-UBM DE-188 |
| owner_facet | DE-703 DE-824 DE-92 DE-19 DE-BY-UBM DE-188 |
| physical | XXXIII, 1108 S. Ill. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| series2 | Handbooks of mathematical equations |
| spelling | Poljanin, Andrej D. 1951- Verfasser (DE-588)128391251 aut Handbook of integral equations Andrei D. Polyanin ; Alexander V. Manzhirov 2. ed. Boca Raton [u.a.] Chapman & Hall/CRC 2008 XXXIII, 1108 S. Ill. txt rdacontent n rdamedia nc rdacarrier Handbooks of mathematical equations Includes bibliographical references and index Integral equations Handbooks, manuals, etc Integralgleichung (DE-588)4027229-1 gnd rswk-swf Exakte Lösung (DE-588)4348289-2 gnd rswk-swf (DE-588)4155008-0 Formelsammlung gnd-content (DE-588)4173536-5 Patentschrift gnd-content Integralgleichung (DE-588)4027229-1 s Exakte Lösung (DE-588)4348289-2 s 1\p DE-604 Manžirov, Aleksandr V. Verfasser (DE-588)120620758 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016138406&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016138406&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Poljanin, Andrej D. 1951- Manžirov, Aleksandr V. Handbook of integral equations Integral equations Handbooks, manuals, etc Integralgleichung (DE-588)4027229-1 gnd Exakte Lösung (DE-588)4348289-2 gnd |
| subject_GND | (DE-588)4027229-1 (DE-588)4348289-2 (DE-588)4155008-0 (DE-588)4173536-5 |
| title | Handbook of integral equations |
| title_auth | Handbook of integral equations |
| title_exact_search | Handbook of integral equations |
| title_exact_search_txtP | Handbook of integral equations |
| title_full | Handbook of integral equations Andrei D. Polyanin ; Alexander V. Manzhirov |
| title_fullStr | Handbook of integral equations Andrei D. Polyanin ; Alexander V. Manzhirov |
| title_full_unstemmed | Handbook of integral equations Andrei D. Polyanin ; Alexander V. Manzhirov |
| title_short | Handbook of integral equations |
| title_sort | handbook of integral equations |
| topic | Integral equations Handbooks, manuals, etc Integralgleichung (DE-588)4027229-1 gnd Exakte Lösung (DE-588)4348289-2 gnd |
| topic_facet | Integral equations Handbooks, manuals, etc Integralgleichung Exakte Lösung Formelsammlung Patentschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016138406&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016138406&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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