Differential equations of mathematical physics:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Amsterdam
North-Holland Publ. Co.
1964
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 701 S. Ill., graph. Darst. |
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| 100 | 1 | |a Košljakov, Nikolaj S. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Differencial'nye uravnenija matematičeskoj fiziki |
| 245 | 1 | 0 | |a Differential equations of mathematical physics |c Nikolaj S. Košljakov ; Modest M. Smirnov ; E. B. Gliner. Transl. by Scripta Technica, Inc. Trans. editor Herbert J. Eagle |
| 264 | 1 | |a Amsterdam |b North-Holland Publ. Co. |c 1964 | |
| 300 | |a XVI, 701 S. |b Ill., graph. Darst. | ||
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| 700 | 1 | |a Eagle, Herbert J. |e Sonstige |4 oth | |
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Record in the Search Index
| _version_ | 1804116376511578112 |
|---|---|
| adam_text | CONTENTS
Introduction 1
Part I. Differential equations of the hyperbolic type
Chapter I. Methods of finding the general solution to equations of the
hyperbolic type 19
1. General remarks. Examples 19
2. The Euler Darboux equation 24
Chapter II. The Cauchy problem on a plane 31
1. The Cauchy problem and its solution by the Riemann method 31
2. Examples of applications of Riemann s method 35
Chapter III. The application of the method of characteristics to the
study of low amplitude vibrations of a string 42
1. Derivation of the equation for the vibrations of a string 42
2. Vibrations of a homogeneous infinite string 45
3. Vibrations of a string fixed at both ends 50
4. A property of the characteristics 53
5. Wave reflection in a fastened string 54
6. The concept of generalized solutions 55
Chapter IV. Longitudinal vibrations of a rod 59
1. The differential equation for longitudinal vibrations of a ho¬
mogeneous rod of constant cross section. The initial and
boundary conditions 59
2. The vibrations of a rod with one end fixed 61
3. Axial impact on a rod 64
Chapter V. Application of the method of characteristics to the study
of electrical vibrations in conductors 70
1. Differential equations for free electrical oscillations 70
2. The telegraph equation 71
3. Integration of the telegraph equation by the Riemann method 72
4. Electrical oscillations in an infinite conductor 74
5. Oscillations in a line that is free of distortion 76
6. Boundary conditions for a conductor of finite length 78
x CONTENTS
Chapter VI. The wave equation 80
1. The differential equation for transverse vibrations of a
membrane 80
2. The hydrodynamic equations and the propagation of sound
waves 82
3. Poisson s formula 87
4. The propagation of sound waves in space 90
5. Cylindrical waves 92
6. Plane waves 93
7. Spherical waves 95
8. The inhomogeneous wave equation 100
9. A uniqueness theorem 103
Chapter VII. Functionally invariant solutions 106
1. Functionally invariant solutions to equations of the hyper¬
bolic type with two independent variables 106
2. Functionally invariant solutions to the wave equation 111
3. The problem of reflection of plane elastic waves 113
Chapter VIII. Application of the Fourier method to the study of free
vibrations of strings and rods 117
1. The Fourier method for the equation of free vibrations of a
string 117
2. The vibration of a plucked string 122
3. The vibrations of a struck string 123
4. Longitudinal vibrations of a rod 123
5. The general plan of the Fourier method 126
Chapter K. Forced vibrations of strings and rods 134
1. Forced vibrations of a string that is fixed at the ends 134
2. Forced vibrations of a string under the action of a concen¬
trated force 137
3. Forced vibrations of a heavy rod 139
4. Forced vibrations of a string with moving ends 141
5. The uniqueness of the solution to a mixed problem 144
Chapter X. Torsional vibrations of a homogeneous rod 147
1. The differential equation for torsional vibrations of a cylind¬
rical rod 147
2. The vibrations of a rod with fastened disk 149
Chapter XI. Electric oscillations in lines 156
1. Transient phenomena in electric lines 15$
2. Steady state processes following the application of a voltage 156
CONTENTS xi
Chapter XII. Bessel functions 161
1. Bessel s equation 161
2. Certain particular cases of Bessel functions 165
3. The orthogonality of the Bessel functions and the roots of
these functions 166
4. The expansion of an arbitrary function in a series of Bessel
functions 170
5. Some integral representations of the Bessel functions 172
6. Hankel s functions 175
7. Bessel s functions with imaginary argument 176
Chapter XIII. Small amplitude vibrations of a thread suspended from
one end 180
1. The free vibrations of a suspended thread 180
2. Forced vibrations of a suspended thread 183
Chapter XIV. Small amplitude radial vibrations of a gas 190
1. Radial vibrations of a gas in a sphere 190
2. The radial vibrations of a gas in an infinite cylindrical tube 195
Chapter XV. Legendre polynomials 201
1. Legendre s differential equation 201
2. The ^orthogonality of the Legendre polynomials and their
norm 203
3. Certain properties of Legendre polynomials 205
4. Integral representations of Legendre polynomials 206
5. The generating function 208
6. Recursion formulae relating the Legendre polynomials and
their derivatives 209
7. Legendre functions of the second kind 210
8. Small amplitude vibrations of a rotating string 210
Chapter XVI. The application of the Fourier method to the study of
small amplitude vibrations of rectangular and circular mem¬
branes 216
1. Free vibrations of a rectangular membrane 216
2. Free vibrations of a circular membrane 220
Part II. Differential equations of the elliptic type
Chapter XVII. Integral formulae that are applicable to the theory of
differential equations of the elliptic type 231
1. Definitions and notations 231
2. The Ostrogradskii Gauss formula and the Green theorem 233
3* Transformation of Green s theorem 237
4. Levy s functions 238
5. The Green Stokes theorem 242
xii CONTENTS
6* The Green Stokes theorem for two dimensions 246
7. Representation of certain differential expressions in ortho¬
gonal coordinate systems 247
Chapter XVIII. Laplace and Poisson equations 256
1. Laplace and Poisson equations. Examples of problems lead¬
ing to the Laplace equations 256
2. Boundary value problems 262
3. Harmonic functions 264
4. Uniqueness of the solutions to boundary value problems 269
5. Fundamental solutions to Laplace s equation. The basic for¬
mula in the theory of harmonic functions 274
6. Poisson s formula. The solution to Dirichlet s problem for a
sphere 280
7. Green s function 283
8. Harmonic functions in the plane 288
Chapter XIX. Potential theory 294
1. Newtonian potential 294
2. Potentials of different orders 296
3. Multipoles 299
4. Analysis of/a potential in terms of multipoles. Spherical
functions 302
5. The potentials of single and double layers 306
6. Lyapunov surfaces 307
7. The convergence and the continuous dependence of improper
integrals on parameters 310
8. The behaviour of a single layer potential and of its normal
derivatives upon crossing the layer 313
9* The tangential derivatives of the single layer potential and
its derivatives in an arbitrary direction 317
10. The behaviour of the double layer potential when the layer is
crossed 323
Chapter XX. Elements of the theory of logarithmic potential 325
1. Logarithmic potential 325
2. The double layer logarithmic potential 327
3. Discontinuity in the normal derivative of the logarithmic po¬
tential on a curve 330
4. The logarithmic potential of masses distributed over an area 331
Chapter XXI. Spherical functions 333
1. The construction of a system of linearly independent spheri¬
cal functions 333
2. The orthogonality of spherical functions 337
3. Expansions in spherical functions 339
4. The use of spherical functions for solving boundary problems 343
5. Green s function of the Dirichlet problem for a sphere 345
6. Green s function for the Neumann problem for a sphere 348
CONTENTS xiii
Chapter XXII. Several questions on gravimetry and the theory of the
shape of the earth 352
1. Equipotential distributions 352
2. The energy of a gravitational field. Gauss problem 355
3. Gravitational fields. Stokes theorem 358
4. The basic gravimetric problem 362
5. The solution of the basic problem of gravimetry by Green s
method 365
Chapter XXIII. Application of the theory of spherical functions to the
solution of problems in mathematical physics 369
1. The electrostatic potential of a conducting sphere divided
into two hemispheres by a dielectric layer 369
2. The problem of steady state temperature in a sphere 371
3. The problem of charge distribution on an inductively charged
sphere 373
4. The flow of an incompressible liquid around a sphere 377
Chapter XXIV. Gravity waves on the surface of a liquid 381
1. Statement of the problem 381
2. Two dimensional waves in aibasin of finite depth 384
3. Annular waves 390
4. Stationary phase method 393
Chapter XXV. The Helmholtz equation 398
1. The connection between the Helmholtz equation and certain
hyperbolic and parabolic operations 398
2. Spherical symmetrical solutions to the Helmholtz equation in
a bounded region 400
3. Eigenvalues and eigenfunctions of a general boundary value
problem. Expansions in eigenfunctions 406
4. The separation of variables in the Helmholtz equation in
cylindrical and spherical coordinates 411
5. Spherically symmetric solutions of the Helmholtz equation in
an infinite region 416
6. Integral formulae 422
7. Series expansions in particular solutions of the Helmholtz
equation in an infinite region 428
8* Questions concerning the uniqueness of solutions to the ex¬
ternal boundary value problems for the Helmholtz equation 431
Chapter XXVI. The emission and scattering of sound 435
1. The fundamental relationships for sound fields 435
2. The acoustic field of a vibrating cylinder 436
3. The acoustic field of a pulsating sphere. Point sources 439
4. Emission from an opening in a plane wall 441
5. The acoustic field due to arbitrary oscillation of the surface
of a sphere 443
xiv CONTENTS
6. Investigation of the field of a sphere with arbitrary vibration
of its surface. Acoustic or vibrational multipoles 447
7. The scattering of sound 452
Chapter XXVII*. Comments on questions of the elliptic type in the
general form 456
1. The general form of equations of the elliptic type 456
2. The basic boundary value problem 457
3. Conjugate boundary value problems 458
4. Fundamental solutions. Green s function 459
5. Uniqueness theorem 462
6. Conditions of solubility of boundary value problems 464
Part HI. Equations of the parabolic type
Chapter XXVIII. The simplest problems leading to the heat flow
equation. Some general theorems 471
1. The heat flow equation in an isotropic body. Initial and boun¬
dary conditions 471
2. The diffusion equation 474
3. The heat flow equation in a torus 475
4. An extreme value theorem. The uniqueness of the solution to
the first boundary value problem 477
5. .The uniqueness of the solution to the Cauchy problem 479
Chapter XXEX. Heat flow in an infinite rod 480
1. Heat flow in an infinite rod 480
2. Heat flow in a semi infinite rod 487
Chapter XXX. The application of the Fourier method to the solution
of boundary value problems 492
1. Heat flow in a finite rod 492
2. The inhomogeneous heat flow equation 498
3. Heat flow in an infinite cylinder 501
4. Heat flow in a cylinder of finite dimensions 507
5. Heat flow in a homogeneous sphere 509
6. Heat flow in a rectangular plate 515
Part IV. Supplementary material
Chapter XXXI. The use of integral operators in solving problems in
mathematical physics 522
1. Basic definitions. Method of application of integral operators 522
2. Conditions allowing the use of integral operators 522
3. Finite integral transformations 525
4. Integral transformations in infinite intervals 530
5. Summary of the results 537
CONTENTS xv
Chapter XXXII. Examples of the application of finite integral trans¬
formations 542
1. Vibrations of a heavy thread 542
2. Vibrations of a membrane 545
3. Heat flow in a cylindrical rod 548
4. Heat flow in a circular tube 553
5. Heat flow in a sphere 555
6. Steady state heat flow in a parallelepiped 559
Chapter XXXIII. Examples of the application of integral transforma¬
tions with infinite limits 563
1. The problem of the vibrations of an infinitely long string 563
2. Linear heat flow in a semi infinite rod 565
3. The distribution of heat in a cylindrical rod whose surface is
kept at two different temperatures 567
4. The steady thermal state of an infinite wedge 570
Chapter XXXIV. Maxwell s equations 574
1. The system of Maxwell s equations 574
2. Electromagnetic field potentials 578
3. Boundary conditions 581
4. Representation of an electromagnetic field by means of two
scalar functions 588
5* A uniqueness theorem 591
Chapter XXXV. Emission of electromagnetic waves 596
1. General remarks 596
2. A vertical emitter in a homogeneous medium over an ideally
conducting plane 598
3. A vertical emitter in a homogeneous medium over a sphere
of finite conductivity 603
4. A magnetic antenna over a medium of finite conductivity 605
5. The field of an arbitrary system of emitters 612
6. A horizontal emitter over a medium of finite conductivity 614
Chapter XXXVI. Directed electromagnetic waves 621
1. Transverse electric, transverse magnetic, and transverse
electromagnetic waves 621
2. Waves between ideally conducting planes separated by a di¬
electric 622
3. Further examination of directed waves 627
4. TM wave in a waveguide of circular cross section 635
5. TE waves in a waveguide of circular cross section 637
6. Waves in a coaxial cable 638
7. Waves in a dielectric rod 640
xvi CONTENTS
Chapter XXXVII. Electromagnetic horns and resonators 647
1. Sectorial horns and resonators 647
2. Spherical resonators 651
Chapter XXXVIII. Motion of a viscous fluid 653
1. Equations of motion of a viscous fluid 653
2. Motion of a viscous fluid in the space over a rotating disk of
infinite radius 658
3. Motion of a viscous fluid in a plane diffuser 660
Chapter XXXIX*. Generalized functions 665
1. Introduction 665
2. Generalized functions 666
3. Properties of fundamental and generalized functions. The
most important operations on generalized functions 669
4. Differentiation of generalized functions. The concept of gen¬
eralized solutions of differential equations 675
5. The Dirac delta function 679
6. Convolutions of generalized functions 681
7. The concept of fundamental solutions 686
8. The concept of a generalized Fourier transform 692
References 700
|
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| building | Verbundindex |
| bvnumber | BV001936587 |
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| indexdate | 2024-07-09T15:37:32Z |
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| language | English |
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| publisher | North-Holland Publ. Co. |
| record_format | marc |
| spelling | Košljakov, Nikolaj S. Verfasser aut Differencial'nye uravnenija matematičeskoj fiziki Differential equations of mathematical physics Nikolaj S. Košljakov ; Modest M. Smirnov ; E. B. Gliner. Transl. by Scripta Technica, Inc. Trans. editor Herbert J. Eagle Amsterdam North-Holland Publ. Co. 1964 XVI, 701 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Physik - Partielle Differentialgleichung - Ordnung 2 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Spezielle Funktion (DE-588)4182213-4 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Mathematische Physik (DE-588)4037952-8 s DE-604 Potenzialtheorie (DE-588)4046939-6 s Spezielle Funktion (DE-588)4182213-4 s Partielle Differentialgleichung (DE-588)4044779-0 s Smirnov, Modest M. Verfasser aut Gliner, Ėrast B. Verfasser aut Eagle, Herbert J. Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001262214&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Košljakov, Nikolaj S. Smirnov, Modest M. Gliner, Ėrast B. Differential equations of mathematical physics Mathematische Physik - Partielle Differentialgleichung - Ordnung 2 Partielle Differentialgleichung (DE-588)4044779-0 gnd Mathematische Physik (DE-588)4037952-8 gnd Spezielle Funktion (DE-588)4182213-4 gnd Differentialgleichung (DE-588)4012249-9 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4037952-8 (DE-588)4182213-4 (DE-588)4012249-9 (DE-588)4046939-6 |
| title | Differential equations of mathematical physics |
| title_alt | Differencial'nye uravnenija matematičeskoj fiziki |
| title_auth | Differential equations of mathematical physics |
| title_exact_search | Differential equations of mathematical physics |
| title_full | Differential equations of mathematical physics Nikolaj S. Košljakov ; Modest M. Smirnov ; E. B. Gliner. Transl. by Scripta Technica, Inc. Trans. editor Herbert J. Eagle |
| title_fullStr | Differential equations of mathematical physics Nikolaj S. Košljakov ; Modest M. Smirnov ; E. B. Gliner. Transl. by Scripta Technica, Inc. Trans. editor Herbert J. Eagle |
| title_full_unstemmed | Differential equations of mathematical physics Nikolaj S. Košljakov ; Modest M. Smirnov ; E. B. Gliner. Transl. by Scripta Technica, Inc. Trans. editor Herbert J. Eagle |
| title_short | Differential equations of mathematical physics |
| title_sort | differential equations of mathematical physics |
| topic | Mathematische Physik - Partielle Differentialgleichung - Ordnung 2 Partielle Differentialgleichung (DE-588)4044779-0 gnd Mathematische Physik (DE-588)4037952-8 gnd Spezielle Funktion (DE-588)4182213-4 gnd Differentialgleichung (DE-588)4012249-9 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| topic_facet | Mathematische Physik - Partielle Differentialgleichung - Ordnung 2 Partielle Differentialgleichung Mathematische Physik Spezielle Funktion Differentialgleichung Potenzialtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001262214&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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