Green's functions and linear differential equations: theory, applications, and computation
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press
2011
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| Series: | Chapman & Hall/CRC applied mathematics and nonlinear science series
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXV, 356 S. Ill., graph. Darst. |
| ISBN: | 9781439840085 |
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MARC
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| adam_text | Titel: Green s functions and linear differential equations
Autor: Kythe, Prem K
Jahr: 2011
Contents
Preface ............................................................... xi
Notations and Definitions......................................... xvii
1. Some Basic Results ............................................... 1
1.1. Euclidean Space .................................................. 1
1.1.1. Metrie Space ................................................ 2
1.1.2. Inner Product................................................2
1.2. Classes of Continuous Functions....................................3
1.3. Convergence......................................................3
1.3.1. Convergence of Sequences....................................3
1.3.2. Weak Convergence...........................................4
1.3.3. Metrie ...................................................... 5
1.3.4. Convergence of Infinite Series.................................5
1.3.5. Tests for Convergence of Positive Series........................6
1.4. Functionals.......................................................6
1.4.1. Examples of Linear Functionals ............................... 8
1.5. Linear Transformations............................................8
1.6. Cramer s Rule .................................................... 9
1.7. Green s Identities ................................................ 10
1.8. Differentiation and Integration.....................................13
1.8.1. Leibniz s Rules............................................. 13
1.8.2. Integration by Parts ......................................... 13
1.9. Inequalities...................................................... 14
1.9.1. Bessel s Inequality for Fourier Series..........................14
1.9.2. Bessel s Inequality for Square-Integrable Functions.............14
1.9.3. Schwarz s Inequality for Infinite Sequences.................... 15
1.10. Exercises ...................................................... 15
2. The Concept of Green s Functions ............................. 18
2.1. Generalized Functions............................................ 18
2.1.1. Heaviside Function..........................................26
2.1.2. Delta Function in Curvilinear Coordinates.....................27
2.2. Singular Distributions ............................................ 29
2.3. The Concept of Green s Functions ................................. 31
2.4. Linear Operators and Inverse Operators.............................34
vi CONTENTS
2.4.1. Linear Operators and Inverse Operators ....................... 34
2.4.2. Adjoint Operators...........................................35
2.5. Fundamental Solutions ...........................................41
2.6. Exercises........................................................44
3. Sturm-Liouville Systems ........................................ 47
3.1. Ordinary Differential Equations ................................... 47
3.1.1. Initial and Boundary Conditions..............................47
3.1.2. General Solution............................................48
3.1.3. Method of Variation of Parameters............................49
3.2. Initial Value Problems............................................51
3.2.1. One-Sided Green s Functions ................................ 51
3.2.2. Wronskian Method..........................................54
3.2.3. Systems of First-Order Differential Equations..................55
3.3. Boundary Value Problems.........................................55
3.3.1. Sturm-Liouville Boundary Value Problems .................... 56
3.3.2. Properties of Green s Functions...............................58
3.3.3. Green s Function Method....................................59
3.4. Eigenvalue Problem for Sturm-Liouville Systems....................64
3.4.1. Eigenpairs ................................................. 66
3.4.2. Orthonormal Systems ....................................... 67
3.4.3. Eigenfunction Expansion .................................... 69
3.4.4. Data for Eigenvalue Problems ................................72
3.5. Periodic Sturm-Liouville Systems..................................73
3.6. Singular Sturm-Liouville Systems ................................. 74
3.7. Exercises........................................................79
4. Bernoulli s Separation Method ................................. 84
4.1. Coordinate Systems .............................................. 84
4.2. Partial Differential Equations......................................85
4.3. Bernoulli s Separation Method .................................... 89
4.3.1. Laplace s Equation in a Cube.................................89
4.3.2. Laplace s Equation in a Cylinder ............................. 90
4.3.3. Laplace s Equation in a Sphere ...............................91
4.3.4. Helmholtz s Equation in Cartesian Coordinates.................92
4.3.5. Helmholtz s Equation in Spherical Coordinates.................93
4.3.6. Wave Equation ............................................. 94
4.4. Examples ....................................................... 95
4.5. Exercises ...................................................... 116
5. Integral Transforms ............................................ 121
5.1. Integral Transform Pairs ......................................... 121
5.2. Laplace Transform.............................................. 122
5.2.1. Definition of Dirac Delta Function........................... 125
5.3. Fourier Integral Theorems ....................................... 126
5.3.1. Properties of Fourier Transforms ............................ 127
5.3.2. Fourier Transforms of Derivatives of a Function...............127
CONTENTS vii
5.3.3. Convolution Theorems for Fourier Transform................. 127
5.4. Fourier Sine and Cosine Transforms .............................. 130
5.4.1. Properties of Fourier Sine and Cosine Transforms............. 130
5.4.2. Convolution Theorems for Fourier Sine and Cosine Transforms . 131
5.5. Finite Fourier Transforms........................................ 132
5.5.1. Properties................................................. 134
5.5.2. Periodic Extensions ........................................ 134
5.5.3. Convolution............................................... 135
5.6. Multiple Transforms ............................................ 136
5.7. Hankel Transforms.............................................. 137
5.8. Summary: Variables of Transforms ............................... 139
5.9. Exercises ...................................................... 139
6. Parabolic Equations ............................................ 143
6.1. 1-D Diffusion Equation..........................................144
6.1.1. Sturm-Liouville System for 1-D Diffusion Equation ........... 144
6.1.2. Green s Function for 1-D Diffusion Equation ................. 146
6.2. 2-D Diffusion Equation.......................................... 148
6.2.1. Dirichlet Problem for the General Parabolic Equation in a Square 149
6.3. 3-D Diffusion Equation.......................................... 151
6.3.1. Electrostatic Analog........................................151
6.4. Schrödinger Diffusion Operator .................................. 154
6.5. Min-Max Principle.............................................. 156
6.6. Diffusion Equation in a Finite Medium............................ 156
6.7. Axisymmetric Diffusion Equation ................................ 157
6.8. 1-D Heat Conduction Problem ................................... 158
6.9. Stefan Problem ................................................. 160
6.10. 1-D Fractional Diffusion Equation............................... 163
6.10.1. 1-D Fractional Diffusion Equation in Semi-Infinite Medium ... 165
6.11. 1-D Fractional Schrödinger Diffusion Equation ................... 166
6.12. Eigenpairs and Dirac Delta Function............................. 167
6.13. Exercises ..................................................... 170
7. Hyperbolic Equations .......................................... 175
7.1. 1-D Wave Equation ............................................. 175
7.1.1. Sturm-Liouville System for 1-D Wave Equation...............175
7.1.2. Vibrations of a Variable String .............................. 177
7.1.3. Green s Function for 1-D Wave Equation..................... 179
7.2. 2-D Wave Equation ............................................. 180
7.3. 3-D Wave Equation ............................................. 180
7.4. 2-D Axisymmetric Wave Equation................................ 182
7.5. Vibrations of a Circular Membrane ............................... 182
7.6. 3-D Wave Equation in a Cube .................................... 183
7.7. Schrödinger Wave Equation...................................... 186
7.8. Hydrogen Atom ................................................ 187
7.8.1. Harmonie Oscillator........................................190
viii CONTENTS
7.9. 1-D Fractional Nonhomogeneous Wave Equation...................190
7.10. Applications of the Wave Operator...............................193
7.10.1. Cauchy Problem for 2-D and 3-D Wave Equation ............ 193
7.10.2. d Alembert Solution of the Cauchy Problem for Wave Equation 194
7.10.3. Free Vibration of a Large Circular Membrane................196
7.10.4. Hyperbolic or Parabolic Equations in Terms of Green s Functions 196
7.11. Laplace Transform Method..................................... 198
7.12. Quasioptics and Diffraction.....................................201
7.12.1. Diffraction of Monochromatic Waves.......................201
(a) Fraunhofer Approximation..............................202
(b) Fresnel Approximation.................................204
7.13. Exercises ..................................................... 205
8. Elliptic Equations .............................................. 209
8.1. Green s Function for 2-D Laplace s Equation ......................209
8.2. 2-D Laplace s Equation in a Rectangle............................211
8.3. Green s Function for 3-D Laplace s Equation ......................212
8.3.1. Laplace s Equation in a Rectangular Parallelopiped............213
8.4. Harmonie Functions.............................................217
8.5. 2-D Helmholtz s Equation .......................................218
8.5.1. Closed-Form Green s Function for Helmholtz s Equation.......219
8.6. Green s Function for 3-D Helmholtz s Equation....................220
8.7. 2-D Poisson s Equation in a Circle................................221
8.8. Method for Green s Function in a Rectangle.......................226
8.9. Poisson s Equation in a Cube.....................................229
8.10. Laplace s Equation in a Sphere..................................231
8.11. Poisson s Equation and Green s Function in a Sphere..............235
8.12. Applications of Elliptic Equations ...............................237
8.12.1. Dirichlet Problem for Laplace s Equation....................237
8.12.2. Neumann Problem for Laplace s Equation...................237
8.12.3. Robin Problem for Laplace s Equation......................239
8.12.4. Dirichlet Problem for Helmholtz s Equation.................239
8.12.5. Dirichlet Problem for Laplace s Equation in the Half-Plane___240
8.12.6. Dirichlet Problem for Laplace s Equation in a Circle..........241
8.12.7. Dirichlet Problem for Laplace s Equation in the Quarter Plane . 241
8.12.8. Vibration Equation for the Unit Sphere......................243
8.13. Exercises ..................................................... 244
9. Spherical Harmonics ........................................... 251
9.1. Historical Sketch ............................................... 251
9.2. Laplace s Solid Spherical Harmonics..............................252
9.2.1. Orthonormalization ........................................ 254
9.2.2. Condon-Shortley Phase Factor ..............................256
9.2.3. Spherical Harmonics Expansion.............................257
9.2.4. Addition Theorem ......................................... 258
9.2.5. Laplace s Coefficients......................................259
CONTENTS ix
9.3. Surface Spherical Harmonics.....................................261
9.3.1. Poisson Integral Representation ............................. 266
9.3.2. Representation of a Function f(6, 4 ) .........................268
9.3.3. Addition Theorem for Spherical Harmonics...................269
9.3.4. Discrete Energy Spectrum .................................. 271
9.3.5. Further Developments......................................274
9.4. Exercises ...................................................... 276
10. Conformal Mapping Method ................................. 281
10.1. Definitions and Theorems.......................................281
10.1.1. Cauchy-Riemann Equations................................281
10.1.2. Conformal Mapping ...................................... 282
10.1.3. Symmetrie Points.........................................283
10.1.4. Cauchy s Integral Formula.................................284
10.1.5. Mean-Value Theorem ..................................... 284
10.2. Dirichlet Problem..............................................285
10.2.1. Dirichlet Problem for a Circle in the (x, */)-Plane............. 289
10.3. Neumann Problem.............................................290
10.4. Green s and Neumann s Functions...............................293
10.4.1. Laplacian................................................293
10.4.2. Green s Function for a Circle .............................. 295
10.4.3. Green s Function for an Ellipse.............................297
10.4.4. Green s Function for an Infinite Strip ....................... 299
10.4.5. Green s Function for an Annulus........................... 302
10.5. Computation of Green s Functions...............................303
10.5.1. Interpolation Method......................................304
10.6. Exercises ..................................................... 309
A. Adjoint Operators ............................................. 314
B. List of Fundamental Solutions ................................ 317
B.l. Linear Ordinary Differential Operator with Constant Coefficients___317
B.2. Fundamental Solutions for the Operators..........................317
B.3. Elliptic Operator ............................................... 317
B.4. Helmholtz Operator.............................................318
B.5. Fundamental Solution for the Cauchy-Riemann Operator........... 319
B.6. Fundamental Solution for the Diffusion Operator...................319
B.7. Schrödinger Operator........................................... 320
B.8. Fundamental Solution for the Wave Operator...................... 321
B.9. Fundamental Solution for the Fokker-Plank Operator...............321
B.10. Klein-Gordon Operator ........................................ 321
C. List of Spherical Harmonics................................... 322
C.l. Legendre s Equation............................................322
C.2. Associated Legendre s Equation..................................323
C.3. Relations with or without Condon-Shortley Phase Factor............324
C.4. Laguerre s Equation ............................................ 326
x CONTENTS
C.5. Associated Laguerre s Equation..................................327
D. Tables of Integral Transforms ................................ 329
D.I. Laplace Transform Pairs ........................................ 329
D.2. Fourier Cosine Transform Pairs..................................332
D.3. Fourier Sine Transform Pairs .................................... 333
D.4. Complex Fourier Transform Pairs................................334
D.S. Finite Sine Transform Pairs......................................335
D.6. Finite Cosine Transform Pairs ................................... 336
D.7. Zero-Order Hankel Transform Pairs ..............................337
E. Fractional Derivatives ......................................... 338
F. Systems of Ordinary Differential Equations ................. 341
Bibliography ....................................................... 345
Index ............................................................... 349
|
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| author | Kythe, Prem K. 1930- |
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| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV037390781 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T23:23:17Z |
| institution | BVB |
| isbn | 9781439840085 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022543673 |
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| owner_facet | DE-29T DE-19 DE-BY-UBM DE-824 DE-83 |
| physical | XXV, 356 S. Ill., graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Chapman & Hall/CRC applied mathematics and nonlinear science series A Chapman & Hall book |
| spelling | Kythe, Prem K. 1930- Verfasser (DE-588)12072362X aut Green's functions and linear differential equations theory, applications, and computation Prem K. Kythe Boca Raton, Fla. [u.a.] CRC Press 2011 XXV, 356 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC applied mathematics and nonlinear science series A Chapman & Hall book Differential equations, Partial Green's functions Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Green-Funktion (DE-588)4158123-4 gnd rswk-swf Lineare Differentialgleichung (DE-588)4206889-7 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Green-Funktion (DE-588)4158123-4 s DE-604 Lineare Differentialgleichung (DE-588)4206889-7 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022543673&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kythe, Prem K. 1930- Green's functions and linear differential equations theory, applications, and computation Differential equations, Partial Green's functions Partielle Differentialgleichung (DE-588)4044779-0 gnd Green-Funktion (DE-588)4158123-4 gnd Lineare Differentialgleichung (DE-588)4206889-7 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4158123-4 (DE-588)4206889-7 |
| title | Green's functions and linear differential equations theory, applications, and computation |
| title_auth | Green's functions and linear differential equations theory, applications, and computation |
| title_exact_search | Green's functions and linear differential equations theory, applications, and computation |
| title_full | Green's functions and linear differential equations theory, applications, and computation Prem K. Kythe |
| title_fullStr | Green's functions and linear differential equations theory, applications, and computation Prem K. Kythe |
| title_full_unstemmed | Green's functions and linear differential equations theory, applications, and computation Prem K. Kythe |
| title_short | Green's functions and linear differential equations |
| title_sort | green s functions and linear differential equations theory applications and computation |
| title_sub | theory, applications, and computation |
| topic | Differential equations, Partial Green's functions Partielle Differentialgleichung (DE-588)4044779-0 gnd Green-Funktion (DE-588)4158123-4 gnd Lineare Differentialgleichung (DE-588)4206889-7 gnd |
| topic_facet | Differential equations, Partial Green's functions Partielle Differentialgleichung Green-Funktion Lineare Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022543673&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kythepremk greensfunctionsandlineardifferentialequationstheoryapplicationsandcomputation |