In this post, we will see the two set volume of Partial Differential Equations Of Mathematical Physics by A. N. Tychonov; A. A. Samarski.

# About the books

This text reflects the authors’ unique approach to the study of the basic types of partial differential equations of mathematical physics. The systematic presentation of the material offers the reader a natural entree to the subject. Each of the basic types of equations which are to be studied is motivated by its physical origins. The derivation of an equation from the physics to its final mathematical structure is very instructive to the student.

The authors have gone to great length to make clear the meaning of a solution to an initial value or boundary-value problem. Various methods of solving such problems are treated in great detail, as are the questions of existence and uniqueness of solutions. Thus, the student gains an appreciation of the theoretical foundations of the subject and simultaneously acquires the manipulative skills for solving such problems.

The exercises which accompany each chapter have been selected to test the student’s ability both to formulate the correct mathematical statement of the problem and to apply the appropriate method for its solution. The applications treated by the authors are non-trivial and are completely worked out in detail.

The first volume covers the two dimensional class of partial differential equations of mathematical physics and is well suited as a basic text for both the undergraduate and graduate level at the university. The second volume will cover the three dimensional counterparts of the present volume and contain an additional chapter on the special functions which arise in mathematical physics.The second volume covers the three-dimensional aspects of the material contained in the first volume. There is an additional chapter on the special functions that arise in the solution of the problems treated; the basic properties and representations of these functions are derived in a simple and straight-forward manner. As in the first volume, the topics in each chapter are motivated by their physical origins and are supplemented by examples, as well as by exercises. A few additional but pertinent references have also been included.

The book were translated from the Russian by S. Radding. Volume 1 was published in 1964 and Volume 2 was published in 1967.

Credits to original uploader.

You can get Volume 1 here.

You can get Volume 2 here.

Follow us on The Internet Archive: https://archive.org/details/@mirtitles

Follow Us On Twitter: https://twitter.com/MirTitles

Write to us: mirtitles@gmail.com

Fork us at GitLab: https://gitlab.com/mirtitles/

Add new entries to the detailed book catalog here.

# Contents

# 1. CLASSIFICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

1-1. Differential equations of the second order with two independent variables 1

1-2. Differential equations of the second order with several independent variables 7

1-3. The canonical forms of linear equations with constant coefficients 9

# 2. HYPERBOLIC DIFFERENTIAL EQUATIONS

2-1. Simple problems which lead to hyperbolic differential equations and boundary-value problems 12

2-2. Wave-propagation method 36

2-3. Separation of variables 66

2-4. Problems with auxiliary conditions on the characteristics 101

2-5. Solutions of general linear hyperbolic differential equations 107

2-6. Applications to Chapter 2 117

# 3. PARABOLIC DIFFERENTIAL EQUATIONS

3-1. Simple problems which lead to parabolic differential equations 153

3-2. The method of separation of variables 171

3-3. Problems for the infinite straight line 188

3-4. Problems without initial conditions 210

3.5. Applications to Chapter 3 215

# 4. ELLIPTIC DIFFERENTIAL EQUATIONS

4-1. Problems which lead to Laplace’s differential equation 241

4-2. General properties of harmonic functions 252

4-3. Solutions of the boundary-value problems for simple regions by separation of variables 270

4-4, Green’s function (source function) 279

4-5. Potential theory 288

4-6. The difference method 325

4-7. Applications to Chapter 4 332

APPENDIX

Tables of error integrals and some cylindrical functions 363

Literature references of the editor 371

# 5. SPATIAL WAVE PROPAGATION 381

5-1. Initial Value Problems 381

1. The mean value method 381

2. The method of descent 383

3. Physical interpretation 385

4. The method of images 387

5-2. The Kirchhoff Formula 388

1. Derivation of the Kirchhoff formula 388

2. Consequences of the Kirchhoff wave formula 392

5-3. Vibrations of a Bounded Spatial Region 394

1. General scheme for the method of separation of variables.

Standing waves 394

2. Vibrations of a rectangular membrane 399

3. Vibrations of a circular membrane Problems 402

Problems 407

5-4. Applications of Chapter 5 408

1. Reduction of the equations of the theory of elasticity to wave equations 408

2. The equations of the electromagnetic field 410

1. The equations of electromagnetic fields and boundary conditions 410

2. Potential of an electromagnetic field 414

3. The electromagnetic field of an oscillator 416

# 6. SPATIAL HEAT PROPAGATION 422

6-1. Heat Propagation in an Unbounded Space 422

1. Green’s function 422

2. Heat propagation in an unbounded space 426

6-2. Heat Propagation in Bounded Regions 429

1. Scheme of the method of separation of variables

2. Cooling of a circular cylinder 429

3. Determination of the critical dimensions 432

6-3. Boundary-Valve Problems for Regions With Variable Boundaries 434

1. Green’s formula for the heat-conduction equation and

Green’s function 436

2. Solution of the boundary-value problem 441

3. Green’s function for an interval 443

6-4. Heat Potentials 445

1. Properties of heat potentials of single and double layers 445

2. Solution of boundary-value problems 446

Problems 449

6-5. Applications of Chapter 6 450

1. Diffusion of a cloud 450

2. The demagnetization of cylinders 453

3. The difference method for the heat-conduction equation 457

# 7. ELLIPTIC DIFFERENTIAL EQUATIONS (Continued) 465

7-1. Some Fundamental Problems which Lead to the Differential Equation 𝛥v+cv=0 465

1. Forced vibrations 465

2. Diffusion of gases with decomposition phenomena and with chain reactions 466

3. Diffusion in a moving medium 466

4. Formulation of the interior boundary-value problem for the equation 𝛥v+cv=0 467

7-2. Green’s Function 468

1. Green’s function 468

2. Integral representation of the solution 470

3. Potentials 473

7-3. Problems for an Unbounded Region: Radiation Principle 476

1. The equation Ju+cv=—f in an unbounded space 476

2. The principle of limiting absorption 477

3. The principle of limiting amplitude 478

4. Radiation conditions 480

7-4. Problems of the Mathematical Theory of Diffraction 484

1. Statement of the problem 484

2. Uniqueness of the solutions 485

3. Diffraction by a sphere 488

Problems 495

7-5. Applications of Chapter 7 497

1. Waves in conducting tubes (wave guides) 497

2. Electromagnetic vibrations in cavity resonators 507

1. The eigenvibrations of cylindrical endovibrators 507

2. Electromagnetic energy of eigenvibrations 511

3. Generation of vibrations in an endovibrator 513

3. Skin effect 515

4. The propagation of radio waves on the surface of the earth 519

# APPENDIX: SPECIAL FUNCTIONS 525

## A-1 Introduction 525

1. Differential equations of special functions 525

2. Formulation of the boundary value problems in the case k(a)=0 526

## A-2 Cylindrical Functions 526

1. The cylindrical functions 532

1. Power series 532

2. Recursion formulas 532

3. The Bessel function of the first kind of order n + 1/2 539

4. Asymptotic representation of cylindrical functions for

large x 540

2. Boundary-value problems for the Bessel differential equation 542

3. The different types of cylindrical functions 545

1. Hankel functions 545

2. Hankel and Neumann functions 546

3. Bessel functions with imaginary arguments 548

4. The function K_0(x) 549

4. Integral representations and asymptotic representations of the Bessel functions 553

1. Integral representations for the Bessel functions of

integral order 553

2. Asymptotic representations of Bessel functions of the

first kind 556

5. The Fourier-Bessel integral and some integrals which contain Bessel functions 559

1. Fourier-Bessel integral 559

2. Some integrals whose integrands contain Bessel functions 559

6. Representation of cylindrical functions by contour integrals 563

1. Representation of cylindrical functions by contour integrals 563

2. The saddle point method. Asymptotic representations 568

## A-3 Spherical Functions 571

1. Legendre polynomials 571

1. The generating function and Legendre polynomials 571

2. A recursion formula 573

3. The Legendre differential equation 573

4. The orthogonality of the Legendre polynomials 574

5. The norm of the Legendre polynomials 576

6. A differential relation for the Legendre polynomials 577

7. An integral formula. The boundedness of the Legendre polynomials

8. The associated Legendre polynomials 578

9. The closedness of systems of associated Legendre polynomials 580

2. Harmonic polynomials and spherical functions 584

1. Harmonic polynomials 584

2. Spherical functions 585

3. The orthogonality of the system of spherical functions 588

4. The completeness of systems of spherical functions 590

5. The expansion in spherical functions 591

3. Some examples of the application of spherical functions 594

1. The polarization of a sphere in a homogeneous field 594

2. The eigenvibrations of a sphere 597

3. The exterior boundary-value problem for the sphere 599

## A4 The Tschebyscheff-Hermite and the Tschebyscheff-Laguerre

Polynomials 601

1. The Tschebyscheff-Hermite polynomial 601

2. The Tschebyscheff-Laguerre polynomials 604

3. Simple problems for the Schrédinger equation 610

1. The Schrédinger equation 610

2. The harmonic oscillator 611

3. The eigenvalue problem of rotators 613

4. Motion of electrons in a Coulomb field 614

Index 619

Pingback: Partial Differential Equations Of Mathematical Physics (Vols. 1 and 2) – Tychonov, Samarski — Mir Books | Chet Aero Marine