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- M. M. Smirnov
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*Table of contents : Front CoverTitle PagePREFACECONTENTSINTRODUCTIONPART I DIFFERENTIAL EQUATIONSOF THE HYPERBOLIC TYPE 1. METHODS OF FINDING THE GENERAL SOLUTION TO EQUATIONS OF THE HYPERBOLIC TYPE 2. THE CAUCHY PROBLEM ON A PLANE 3. THE APPLICATION OF THE METHOD OF CHARACTERISTICS TO THE STUDY OF LOW-AMPLITUDE VIBRATIONS OF A STRING 4. LONGITUDINAL VIBRATIONS OF A ROD 5. APPLICATION OF THE METHOD OF CHARACTERISTICS TO THE STUDY OF ELECTRICAL VIBRATIONS IN CONDUCTORS 6. THE WAVE EQUATION 7. FUNCTIONALLY INVARIANT SOLUTIONS 7. APPLICATION OF THE FOURIER METHOD TO THE STUDY OF FREE VIBRATIONS OF STRINGS AND RODS 9. FORCED VIBRATIONS OF STRINGS AND RODS 10. TORSIONAL VIBRATIONS OF A HOMOGENEOUS ROD 11. ELECTRIC OSCILLATIONS IN LINES 12. BESSEL FUNCTIONS 13. SMALL-AMPLITUDE VIBRATIONS OF A THREAD SUSPENDED FROM ONE END 15. LEGENDRE POLYNOMIALS 16. THE APPLICATION OF THE FOURIER METHOD TO THE STUDY OF SMALL - AMPLITUDE VIBRATIONS OF RECTANGULAR AND CIRCULAR MEMBRANES 17. INTEGRAL FORMULAE THAT ARE APPLICABLE TO THE THEORY OF DIFFERENTIAL EQUATIONS OF THE ELLIPTIC TYPEPART II DIFFERENTIAL EQUATIONSOF THE ELLIPTIC TYPE 18. LAPLACE AND POISSON EQUATIONS 19. POTENTIAL THEORY 20 .ELEMENTS OF THE THEORY OF LOGARITHMIC POTENTIAL 21. SPHERICAL FUNCTIONS 22. SEVERAL QUESTIONS ON GRAVIMETRY AND THE THEORY OF THE SHAPE OF THE EARTH 23. APPLICATION OF THE THEORY OF SPHERICAL FUNCTIONS TO THE SOLUTION OF PROBLEMS IN MATHEMATICAL PHYSICS 24. GRAVITY WAVES ON THE SURFACE OF A LIQUID 25. THE HELMHOLTZ EQUATION 26. THE EMISSION AND SCATTERING OF SOUND 27. COMMENTS ON EQUATIONS OF THE ELLIPTIC TYPE IN THE GENERAL FORMPART III EQUATIONS OF THE PARABOLIC TYPE 28. THE SIMPLEST PROBLEMS LEADING TO THE HEAT-FLOW EQUATION. SOME GENERAL THEOREMS 29. HEAT-FLOW IN AN INFINITE ROD 30. THE APPLICATION OF THE FOURIER METHOD TO THE SOLUTION OF BOUNDARY- VALUE PROBLEMSPART IV SUPPLEMENTARY MATERIAL 31. THE USE OF INTEGRAL OPERATORS IN SOLVING PROBLEMS IN MATHEMATICAL PHYSICS 32. EXAMPLES OF THE APPLICATION OF FINITE INTEGRAL TRANSFORMATIONS 33. EXAMPLES OF THE APPLICATION OF INTEGRAL TRANSFORMATIONS WITH INFINITE LIMITS 34. MAXWELL'S EQUATIONS 35. EMISSION OF ELECTROMAGNETIC WAVES 36. DIRECTED ELECTROMAGNETIC WAVES 37. ELECTROMAGNETIC HORNS AND RESONATORS 38. MOTION OF A VISCOUS FLUID 38. GENERALIZED FUNCTIONS +REFERENCES*

DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS

N. S. KOSHLYAKOVt M. M. SMIRNOV Leningrad S ta te U niversity, Leningrad

E. B. GLINER A . F . Io ffe P hysical Technical In stitu te o f the Academ y o f Sciences o f the U S S R , Leningrad

Translated by SCRIPTA TECHNICA, INC. T ranslation E ditor

HERBERT J. EAGLE D epartm ent o f M athem atics, B row n U niversity, Providence, R .I.

1964 NORTH-HOLLAND PUBLISHING COMPANY—AMSTERDAM

© 1964

N orth-H olland Publishing Company

No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY, AMSTERDAM SOLE DISTRIBUTORS FOR U .S.A .: INTERSCIENCE PUBLISHERS, A DIVISION OF

JOHN WILEY &SONS, INC., NEW YORK

Printed in The Netherlands

PREFACE The rapid development of contemporary technology requires ever more extensive mathematical preparation for engineers. This has resulted in a demand for a more complete exposition of the applications of the fundamen tal mathematical disciplines for engineers, technicians, and students in technological institutions. The present book examines a number of physical and technical prob lems which involve second-order partial differential equations. Consider able attention is also given to the theory of such equations. In addition, the text includes several chapters and sections of a general nature (indicated by an asterisk). The material in these sections does not as yet have direct application; nonetheless, it is important for an understanding of contem porary scientific literature on mathematical physics. Among the applications studied are the vibrations of strings, mem branes, and shafts; electric oscillations in lines; the electrostatic prob lem; the basic gravimetric problem; the emission of electromagnetic waves and their distribution along wave guides and in horns; the emission and dispersion of sound; gravity waves on the surface of a liquid; heat flow in a solid body, and so forth. Solutions are given to both very simple and more complicated problems, making it possible for the reader to master the methods considered in the book and also the physics of the phenomena in question. In almost every chapter, there are problems whose basic pur pose is to develop the reader's technical skill. Approximate methods for solving problems in mathematical physics are not discussed, since their exposition would require a considerable in crease in the size of the book. Also excluded are certain specialized prob lems (for example, those associated with the physics of atomic reactors) that have arisen only in the last few years. The preparation of the book was carried out under the guidance and with the cooperation of Member-Correspondent of the Academy of Sciences of the USSR, Professor Nikolai Sergeevich Koshlyakov, whose untimely death occurred before publication of the book. A noted specialist in the field of analytic number theory and higher transcendental functions, Prof. Koshlyakov published a number of works in the field of mathematical phys ics. In the course of his career, which included more than 30 years of scientific and pedagogical activity, as well as 15 years of research in ap plied problems, Prof. Koshlyakov always devoted a great deal of attention to the mathematical education of engineers. An excellent lecturer and teacher, he enjoyed the constant respect and devotion of his listeners and students. His textbook Osnovnye Differentsial'nye Uravneniya Matematicheskoi Fiziki (Basic Differential Equations of Mathematical Physics),

viii

PREFACE

several chapters of which are used in the present book, has seen four edi tions (the latest in 1936). The authors of the Introduction and Parts I and III are N. S. Koshlyakov and M. M. Smirnov; the authors of parts II and IV are N. S. Koshlyakov and E. B. Gliner. We take this occasion to express our deep gratitude to I. M. Gel'fand, G.I. Z el'tser and G. P. Samosyuk for graciously reading the individual sec tions of the book, to G. Yu. Dzhanelidze and S. I. Amosov for making a tho rough review of the manuscript, and especially to Scientific Instructor G.P. Akilov. All of these made a number of valuable comments leading to an im provement in the text and to the correction of a number of errors. Leningrad, September 5, 1951

E. B. Gliner M. M. Smirnov

CONTENT S Introduction

1 Part I. Differential equations of the hyperbolic type

Chapter I. Methods of finding the general solution to equations of the hyperbolic type 1. General remarks. Examples 2. The Euler-Darboux equation Chapter II. The Cauchy problem on a plane

19 19 24 31

1. The Cauchy problem and its solution by the Riemann method 2. Examples of applications of Riemann's method

31 35

Chapter III. The application of the method of characteristics to the study of low-amplitude vibrations of a string

42

1. 2. 3. 4. 5. 6.

Derivation of the equation for the vibrations of a string Vibrations of a homogeneous infinite string Vibrations of a string fixed at both ends A property of the characteristics Wave reflection in a fastened string The concept of generalized solutions

Chapter IV. Longitudinal vibrations of a rod

42 45 50 53 54 55 59

1. The differential equation for longitudinal vibrations of a ho mogeneous rod of constant cross section. The initial and boundary conditions 2. The vibrations of a rod with one end fixed 3. Axial Impact on a rod

59 61 64

Chapter V. Application of the method of characteristics to the study of electrical vibrations in conductors

70

1. 2. 3. 4. 5. 6.

Differential equations for free electrical oscillations The telegraph equation Integration of the telegraph equation by the Riemann method Electrical oscillations in an infinite conductor Oscillations in a line that is free of distortion Boundary conditions for a conductor of finite length

70 71 72 74 76 78

X

CONTENTS

Chapter VI. The wave equation 1. The differential equation for transverse vibrations of a membrane 2. The hydrodvnamic equations and the propagation of sound waves 3. Poisson's formula 4. The propagation of sound waves in space 5. Cylindrical waves 6. Plane waves 7. Spherical waves 8. The inhomogeneous wave equation 9. A uniqueness theorem Chapter VII. Functionally invariant solutions

80 80 82 87 90 92 93 95 100 103 106

1. Functionally invariant solutions to equations of the hyper bolic type with two independent variables 2. Functionally invariant solutions to the wave equation 3. The problem of reflection of plane elastic waves

106 111 113

Chapter VUI. 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 2. The vibration of a plucked string 3. The vibrations of a struck string 4. Longitudinal vibrations of a rod 5. The general plan of the Fourier method

117 122 123 123 126

Chapter IX. Forced vibrations of strings and rods 1. Forced vibrations of a string that is fixed at the ends 2. Forced vibrations of a string under the action of a concen trated force 3. Forced vibrations of a heavy rod 4. Forced vibrations of a string with moving ends 5. The uniqueness of the solution to a mixed problem Chapter X. Torsional vibrations of a homogeneous rod 1. The differential equation for torsional vibrations of a cylind rical rod 2. The vibrations of a rod with fastened disk Chapter XI. Electric oscillations in lines 1. Transient phenomena in electric lines 2. Steady-state processes following the application of a voltage

134 134 137 139 141 144 147 147 149 156 156 156

CONTENTS

Chapter XII. Bessel functions 1. Bessel's equation 2. Certain particular cases of Bessel functions 3. Hie orthogonality of the Bessel functions and the roots of these functions 4. The expansion of an arbitrary function in a series of Bessel functions 5. Some integral representations of the Bessel functions 6. Hankel's functions 7. Bessel's functions with imaginary argument Chapter Xm. Small-amplitude vibrations of a thread suspended from one end 1. The free vibrations of a suspended thread 2. Forced vibrations of a suspended thread Chapter XIV. Small-amplitude radial vibrations of a gas 1. Radial vibrations of a gas in a sphere 2. The radial vibrations of a gas in an infinite cylindrical tube Chapter XV. Legendre polynomials 1. Legendre's differential equation 2. The orthogonality of the Legendre polynomials and their norm 3. Certain properties of Legendre polynomials 4. Integral representations of Legendre polynomials 5. Hie generating function 6. Recursion formulae relating the Legendre polynomials and their derivatives 7. Legendre functions of the second kind 8. Small-amplitude vibrations of a rotating string Chapter XVI. The application of the Fourier method to the study of small-amplitude vibrations of rectangular and circular mem branes 1. Free vibrations of a rectangular membrane 2. Free vibrations of a circular membrane

Xi

161 161 165 166 170 172 175 176 180 180 183 190 190 195 201 201 203 205 206 208 209 210 210

216 216 220

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 2. The Ostrogradskli-Gauss formulaand the Green theorem 3* Transformation of Green's theorem 4. Levy's functions 5. The Green-Stoke6 theorem

231 233 237 238 242

xii

CONTENTS

6* TTie Green-Stokes theorem for two dimensions 7. Representation of certain differential expressions in ortho gonal coordinate systems Chapter XVIII. Laplace and Poisson equations 1. Laplace and Poisson equations. Examples of problems lead ing to the Laplace equations 2. Boundary-value problems 3. Harmonic functions 4. Uniqueness of the solutionsto boundary-value problems 5. Fundamental solutions to Laplace's equation. The basic for mula in the theory of harmonic functions 6. Poisson’s formula. The solution to Dirichlet's problem for a sphere 7. Green's function 8. Harmonic functions in the plane Chapter XDC. Potential theory 1. 2. 3. 4.

Newtonian potential Potentials of different orders Multipoles Analysis of a potential in term s of multipoles. Spherical functions 5. The potentials of single and double layers 6. Lyapunov surfaces 7. The convergence and the continuous dependence of improper integrals on parameters 8. The behaviour of a single-layer potential and of its normal derivatives upon crossing the layer 9* The tangential derivatives of the single-layer potential and its derivatives in an arbitrary direction 10. The behaviour of the double-layer potential when the layer is crossed Chapter XX. Elements of the theory of logarithmic potential 1. Logarithmic potential 2. The double-layer logarithmic potential 3. Discontinuity in the normal derivative of the logarithmic po tential on a curve 4. The logarithmic potential of masses distributed over an area Chapter XXI. Spherical functions 1. The construction of a system of linearly independent spheri cal functions 2. The orthogonality of spherical functions 3. Expansions in spherical functions 4. The use of spherical functions for solvingboundaryproblems 5. Green's function of the Dirichlet problem fora sphere 6. Green's function for the Neumann problem fora sphere

246 247 256 256 262 264 269 274 280 283 288 294 294 296 299 302 306 307 310 313 317 323 325 325 327 330 331 333 333 337 339 343 345 348

CONTENTS

Chapter XXII. Several questions on gravimetry and the theory of the shape of the earth 1. 2. 3. 4. 5.

Equipotential distributions The energy of a gravitational field. Gauss' problem Gravitational fields. Stokes' theorem The basic gravimetric problem The solution of the basic problem of gravimetry by Green's method

Chapter XXIII. Application of the theory of spherical functions to the solution of problems in mathematical physics 1. The electrostatic potential of a conducting sphere divided into two hemispheres by a dielectric layer 2. The problem erf steady-state temperature ina sphere 3. The problem of charge distribution on an inductively charged sphere 4. The flow of an incompressible liquid arounda sphere Chapter XXIV Gravity waves on the surface of a liquid 1. 2. 3. 4.

Statement of the problem Two-dimensional waves in a basin of finitedepth Annular waves Stationary phase method

Chapter XXV. The Helmholtz equation 1. The connection between the Helmholtz equation and certain hyperbolic and parabolic operations 2. Spherical symmetrical solutions to the Helmholtz equation in a bounded region 3. Eigenvalues and eigenfunctions of a general boundary-value problem. Expansions in eigenfunctions 4. The separation of variables in the Helmholtz equation in cylindrical and spherical coordinates 5. Spherically symmetric solutions of the Helmholtz equation in an infinite region 6. Integral formulae 7. Series expansions in particular solutions of the Helmholtz equation in an infinite region 8 Questions concerning the uniqueness of solutions to the ex ternal boundary-value problems for the Helmholtz equation Chapter XXVI. The emission and scattering of sound 1. 2. 3. 4. 5.

The fundamental relationships for soundfields The acoustic field of a vibrating cylinder The acoustic field of a pulsating sphere. Point sources Emission from an opening in a plane wall The acoustic field due to arbitrary oscillation of the surface of a sphere

x iii

352 352 355 358 362 365 369 369 371 373 377 381 381 384 390 393 398 398 400 406 411 416 422 428 431 435 435 436 439 441 443

xiv

CONTENTS

6. Investigation of the field of a sphere with arbitrary vibration of its surface. Acoustic or vibrational multipoles 7. The scattering of sound

447 452

Chapter XXVII*. Comments on questions of the elliptic type in the general form

456

1. 2. 3. 4. 5. 6.

The general form of equations of the elliptic type The basic boundary-value problem Conjugate boundary-value problems Fundamental solutions. Green's function Uniqueness theorem Conditions of solubility of boundary-value problems

456 457 458 459 462 464

Part III. Equations of the parabolic type Chapter XXVIII. The simplest problems leading to the heat-flow equation. Some general theorems 1. The heat-flow equation in an isotropic body. Initial and boun dary conditions 2. The diffusion equation 3. The heat-flow equation in a torus 4. An extreme-value theorem. The uniqueness of the solution to the first boundary-value problem 5. The uniqueness of the solution to the Cauchy problem

471 471 474 475 477 479

Chapter XXIX. Heat-flow in an infinite rod

480

1. Heat-flow in an infinite rod 2. Heat-flow in a semi-infinite rod

480 487

Chapter XXX. The application of the Fourier method to the solution of boundary-value problems 1. 2. 3. 4. 5. 6.

Heat-flow in a finite rod The inhomogeneous heat-flow equation Heat-flow in an infinite cylinder Heat-flow in a cylinder of finite dimensions Heat-flow in a homogeneous sphere Heat-flow in a rectangular plate

Part IV. Supplementary material Chapter XXXI. The use of integral operators in solving problems in mathematical physics 1. 2. 3. 4. 5.

Basic definitions. Method of application of integral operators Conditions allowing the use of integral operators Finite integral transformations Integral transformations ininfinite intervals Summary of the results

492 492 498 501 507 509 515

521 522 522 525 530 537

CONTENTS

Chapter XXXII. Examples of the application of finite integral tran s formations 1. 2. 3. 4. 5. 6.

Vibrations of a heavy thread Vibrations of a membrane Heat-flow in acylindrical rod Heat-flow in a circular tube Heat-flow in asphere Steady-state heat-flow in a parallelepiped

Chapter XXXIII. Examples of the application of integral transform a tions with infinite limits 1. The problem of the vibrations of aninfinitely long 6trlng 2. Linear heat-flow in a semi-infinite rod 3. The distribution of heat in a cylindrical rod whose surface is kept at two different temperatures 4. The steady thermal state of an infinite wedge Chapter XXXIV. Maxwell's equations 1. 2. 3. 4.

The system of Maxwell's equations Electromagnetic field potentials Boundary conditions Representation of an electromagnetic field by means of two scalar functions 5* A uniqueness theorem

Chapter XXXV. Emission of electromagnetic waves 1. General remarks 2. A vertical emitter in a homogeneous medium over an ideally conducting plane 3. A vertical emitter in a homogeneous medium over a sphere of finite conductivity 4. A magnetic antenna over a medium of finite conductivity 5. The field of an arbitrary system of emitters 6. A horizontal emitter over a medium of finite conductivity Chapter XXXVI. Directed electromagnetic waves 1. Transverse electric, transverse magnetic, and transverse electromagnetic waves 2. Waves between ideally conducting planes separated by a di electric 3. Further examination of directed waves 4. TM wave in a waveguide of circular cross section 5. TE waves in a waveguide of circular cross section 6. Waves in a coaxial cable 7. Waves in a dielectric rod

xv

542 542 545 548 553 555 559 563 563 565 567 570 574 574 578 581 588 591 596 596 598 603 605 612 614 621 621 622 627 635 637 638 640

xvl

CONTENTS

Chapter XXXVII. Electromagnetic horns and resonators

647

1. Sectorial horns and resonators 2. Spherical resonators

647 651

Chapter XXXVIII. Motion of a viscous fluid

653

1. Equations of motion of a viscous fluid 2. Motion of a viscous fluid in the space over a rotating disk of infinite radius 3. Motion of a viscous fluid in a plane diffuser Chapter XXXIX*. Generalized functions 1. Introduction 2. Generalized functions 3. Properties of fundamental and generalized functions. The most important operations on generalized functions 4. Differentiation of generalized functions. The concept of gen eralized solutions of differential equations 5. The Dirac delta function 6. Convolutions of generalized functions 7. The concept of fundamental solutions 8. The concept of a generalized Fourier transform References

653 658 660 665 665 666 669 675 679 681 686 692 700

I NT RODUCTI ON 1. The fundamental differential equations of mathematical physics Many problems in mechanics and physics involve the study of secondorder partial differential equations. The following are some examples: (1) The study of various types of waves - elastic, acoustic, and electro magnetic - and of other oscillational phenomena leads to the wave equation Z2u

3 12

2 /3 ^ u

_

C

'b2u

3

' 3x2 + 3y2 + 3z 2' ’

(1)

where c is the velocity of propagation of the wave in the given medium. (2) The processes of heat flow in a homogeneous isotropic body and other diffusion phenomena are described by the heat-flow equation: 3m_ 2

3 ^m 32u\ ^ 3 r2

dy2

dz2^

2

( )

(3) Study of a steady thermal state in a homogeneous isotropic body leads to Poisson's equation: 32u + 32m (3) Ax. y, z) 3*2 3y2 0Z2 In the absence of internal heat sources, eq. (3) becomes Laplace's equation 32m 32m 32m (4) 3x2

3^2

gz 2

The potentials of a gravitational or of a constant electric field in which there are no masses or electric charges also satisfy Laplace's equation. Equations (1) - (4) are often called the fundamental equations of mathe matical physics. A detailed study of these equations makes possible the theoretical treatment of a large number of physical phenomena and the so lution of many physical and technical problems. Each of eqs. (1) - (4) has an infinite number of particular solutions. In solving a specific physical problem, it is necessary to choose from among these solutions the one that satisfies certain additional conditions imposed by the physical situation. Thus, problems in mathematical physics reduce to finding solutions to partial differential equations that satisfy certain ad ditional conditions. The most common of these additional conditions are the so-called boundary conditions (conditions that must be satisfied at the boun dary of the medium in question) and initial conditions (which must be satis fied at the particular instant of time at which consideration of a physical process begins). Let us note one very important point. A problem in mathematical phys-

2

INTRODUCTION

ics is considered to be stated correctly i f the problem has exactly one sta ble solution satisfying all the conditions. By "stable", we mean that small changes in any of the given conditions of the problem must cause a corre spondingly small change in the solution. The existence and uniqueness re quirement means that among the given conditions there are none that are incompatible and that these conditions are sufficient to determine a unique solution. The stability requirement is necessary for the following reason. In the given conditions for a specific problem, especially if they are ob tained from experiment, there is always some error, and it is necessary that a small error in the given conditions causes only a small inaccuracy in the solution. This requirement expresses the physical determinacy of the stated problem. Determining whether a problem in mathematical physics is stated cor rectly is a very important and at the same time extremely difficult question in the theory of partial differential equations. We shall not make a complete study of this question in the present book. The following three sections are devoted to the classification of sec ond-order equations of the form i j * ! Aij(x V - > xn> d x jjx j*

=0 >

and, in the case of two independent variables, to their reduction to canoni cal form. 2. The reduction of second-order equations to canonical form Let us examine a second-order equation with two independent variables that is linear with respect to the second-order derivatives: a2^ „ 3%u ( 3u 3u\ + 2B (5) Sx 3y + C ^ 2 + F \X’y ’U’Tx’3 j ) = ° ’ where A, B, and C are functions of x and y that have continuous derivatives up to the second order. Let us replace x and y by the new independent variables £ and rj. Sup pose that £ =

We must examine separately the cases f?2 - AC > 0 , f?2 - AC< 0, and f?2 - AC = 0 throughout the entire region. The case in which the expression f?2 - AC changes sign in the region will be examined later. CASE I: £2 . AC greater than zero. In this case, eq. (5) is said to be of the hyperbolic type. We may assume that either A * 0 or C * 0. We shall examine separately the case when A = C = 0. Without loss of generality, we may assume that A * 0 everywhere. Then, eq. (12) may be written in the form

INTRODUCTION

4

\a % ♦ (B +

f»j (a | | ♦ ( i . y p m c ) H ) . 0 .

This equation can be separated into two equations: A | | + (B+VB2^AC)|e=0,

(12a)

A | | + (B - ./B2-AC) |S = 0 .

(12b)

Consequently, the solutions of each of eqs. (12a) and (12b) will be solutions of eq. (12). To integrate eqs. (12a) and (12b), we set up corresponding systems of differential equations *) d x _____ dy d x ______ dy A ~ B +J b 2 -AC ’ A ~ B - J b 2 -AC ' A dy - (B +Jb 2 - AC) dx = 0 , „ ____ A dy - (B - Jb2 - AC) dx = 0 .

(13)

We note that eqs. (13) may be written in the form of a single equation A dy2 - 2B dx dy + C dx2 = 0 .

(13a)

Suppose that y) = constant, 0), the solutions (14) will be real and distinct. Here, we have two distinct families of real char acteristics. In eq. (6), let us set €= where y) are solutions of eq. (12). Then, on the basis of (10), A = € = 0 in eq. (9). The coefficient E is everywhere different from zero in the region in question - a consequence of (7) and (11). Dividing eq. (9) by the coefficient 2B * 0, we reduce it to the form d^u / 3u 3u\ . 3T3 ^ = M ^ ’“ ’34’3 ^ • (15> This is the canonical form of an equation of the hyperbolic type. If A = C = 0, eq. (5) is of the hyperbolic type and is already in canoni cal form.

INTRODUCTION

5

If eq. (5) is linear with respect to the first-order derivatives and to the function w itself, the transformed equation will also be linear: 3 3h +

(16)

+ c(£,v) “

«(€>n)

Setting i=H +v , V -v , we reduce eq. (15) to the form B^u B^u / 3u 3u\ 3^2 " 9^2 yi ,v ,u , 8i/’^ i) ■ This is the second canonical form of an equation of the hyperbolic type. CASE II: Suppose that B% - AC = 0 throughout the region in question. In this case, eq. (5) is of the parabolic type. We shall assume that throughout the region the coefficients in eq. (5) do not vanish simultaneously. The con dition that B2 - AC = 0 implies that at every point of this region one of the coefficients A and C is different from zero. Without loss of generality, we may assume that A is everywhere different from zero. Then, eq6. (12a) and (12b) are identical and take the form A% ^ =0. 3x e + BBy

(17)

It is easy to see that every solution of eq. (17), where satisfies the equation Bx

- AC = 0, also (18)

By

We note that for an equation of the parabolic type the solutions (14) co incide, and we have only one family of real characteristics oi(x, y) = const. Let us set i =0, y > 0 ) . 3xl 3y2

( 21)

( 22)

(23)

In the reduction of equations of the ellip tic type to canonical form , we shall confine o u rselv es to analytic coefficients A , B, and C. T hus, we shall be able to find the solution to eqs. (12a) and (12b) in the form of an analytic function.

7

INTRODUCTION

This equation is of the hyperbolic type because fj2 . AC = x2y 2 > 0 . We set up the equation of the characteristics (13a) according to the general theory: x 2 dy2 _ y2 (^2 = o or x dy + y dx = 0 ,

x dy - y dx = 0 .

Integrating these equations, we obtain xy = C i ,

y / x = C2 .

Consequently, we need to introduce new variables 4 and t), defined by the form ulae 4 = xy ,

ti= y / x .

Then, from (8), we obtain 3%u _ 2 ^ 2u 3x2 ^ 34^

„ y2 x2 3£ 3r;

y 2 d^u *4 dij2

y 3u *3 g^ ’

3 o S2** o S2" 1 9^“ ~ 9 =X 9 +2 + ~n—S • 3y2

342

3i3v

x 2 3r,2

Substituting the values of the second derivatives in eq. (23), we reduce it to the canonical form: 3^91

1 9m

.

34 3r; ' 24 3i) “ °

...

„

(4 > °. V > 0) .

3. The reduction of mixed-type second-order equations to canonical form Let us again consider the equation 32m 32u ( 3u 3u\ 92m (24) A ^ 2 + 2B 3 F ^ + C g^2 + F \x ,y ,u ’3x’3y) = 0 ' If we consider eq. (24) not in the neighbourhood of a point but in the en tire region D, the classification of second-order equations into three types that was made in section 2 is not exhaustive, since the expression B2 - AC may not, in the general case, retain its sign throughout the entire region. Therefore, the characteristics may be partially real and partially imagin ary. Thus, if B2 - AC changes sign in the region D, eq. (24) is said to be an equation of mixed type. The curve y, determined by the equation B2 - AC =0, is said to be the parabolic curve of eq. (19), and the parts into which the region D is divided by the curve are said to be the elliptic and hyperbolic parts of the region, depending upon whether B2 - AC is less than or greater than zero (respectively) in that part.

INTRODUCTION

8

Just as in section 2, we introduce, instead of x and y, the new inde pendent variables £ and rj: €=4(*.3>), rt=r\(x,y). Then, eq. (24), in the new independent variables 4 and rj, becomes *) _ H S n _ H d v d(x, y) ~ Sx dy dy dx f/„aH dH\ dH / . 3H „dH \dH \ = p l(B 3^ + c T y ) 3 ^ + (A l i + B d j ) 17) f . /3 B \2

„_3B3B

- / d ^ 2] ^ .

=p|A^ +2B^ 37+cW r ° along y. Because of the continuity of the functions 4 , B, C, and H, the Jacobian will be different from zero in some neighbourhood of the curve y. There fore, in that neighbourhood, we may take €=?(*.3>). V = tl(x,y) = H(x,y) in eqs. (25). It then follows from (27) and (33) that B = 0 in the left member of eq. (26). The coefficient C is different from zero in a neighbourhood of the curve y. Dividing eq. (26) by it, we obtain A d^u d2u „ / f 3u 3u\ C 3 4 2 + 3ij2

or, taking into consideration (27), (30), (32), and (34), we finally obtain n v it \ s2u 32m V K i(4,n) + g^2

e- I t 3u 3u\ F i \ ^ ,T}’u ’ d i ’d7 ) / ’

,ns.\ ^

where K i (£,ti) is different from zero in some neighbourhood of the curve y. CASE II. The parabolic curve y is the characteristic or the envelope of the family of characteristics of eq. (24); that is, at all points of the curve y, A

=0 .

dx dy

(36)

Let us assume that A > 0 and C > 0 along y; then, since B2 - AC = 0, eq. (36) can be written in the form

+

=

(37)

where « = sgn B. If B = 0, then either A = 0 or C = 0 along y , and from (37), we have either Hy = 0 or Hx = 0. For the function v = r)(x, y), we take a solution of the equation »(*. V)

* m(x, y) ^ = 0 ,

(38)

where the functions n(x, y) and m(x, y) satisfy the condition Am2 + 2Bmn + Cn2 * 0

(39)

INTRODUCTION

10

in some neighbourhood of y. For example, if either A or C is different from zero in the region D, we may take v = x for m = 1 and n = 0 or tj = y for m = 0 and n = 1. For the function £ = £{x, y), we take a solution of the equation (40) where v(x,y) is a solution of eq. (38). With this choice of the functions £(x, y) and ri(x,y), eqs. (28) and (29) are satisfied. We note that it is always possible to choose a solution of eq. (40) such that £(x, y) = 0 at the point of intersection of the curves v(x, y) = 0 and y. Just as in Case I, it is easy to show that the Jacobian 3(£,T))/3{x,y) is different from zero along the curve y. Then, because of the continuity of the functions A, B, C, m, and w, this Jacobian will also be different from zero in some neighbourhood of the curve y. Let us now set H(x, y) = H(£,r))-, then, we obtain 3ff _3_Hdx dH dy _ H^ y j ^ f > x 3£ ~ 3x d£ + dy d£ £xr/y - £yrjx ’

(41) =3 H U 9y _ 3q " dx 9tj + dy dr/ ~

Hx i y - Hy i x £xqy - £ynx

Along y, 3H/3£ will be different from zero because the curve r? = constant is nowhere tangential to the curve y. From eq. (37), it follows that along y, ^

=-a tJ C ,

g~ = V ’JA

[a (x,y)* Q ], (42)

since along y eq. (40) is reduced to the form

which follows from the condition that - AC = 0. From the fact that H(£,r/) = 0, we have 3£/3t, = -H v/ff£ , and from (42), it follows that £ = constant along y. But £ = 0 at the point of intersection with the curve y and, consequently, £ =0 along y. Therefore, B(0,rj) = 0 and we may write U(£,V) = £H(_m,v)£,v) =£N{£,n) ,

(43)

where 0 < 6(£,v) < 1 and *0. In the transformed equation (26), E = 0 and C * 0 in the neighbourhood

INTRODUCTION

11

of the curve_y, which follows from (28) and (29). Dividing eq. (26) by the coefficient C * 0, we obtain

but

= cftl^ix.y) M(x,y) = £np2l{nM = i nK 2(i ,v) , and we finally have (44) where 78 different from zero in a neighbourhood of the curve y. Example. Let us examine the equation

This equation is of mixed type since AC - B2 = x2 - y2 - 1 = H(x, y) . In the region 1 - x2 + y2 > 0, the equation belongs to the hyperbolic type and in the region 1 - x2 + y2 < 0, it belongs to the elliptic type. The curve x? - y2 = 1 is the parabolic curve. Since 3H3H 3x 3y +

= 4x2(1-x^) + 8x2y2 _ 4 y 2 (i+;y2) = 4 {y 2 - x 2)(x 2 - y 2 _ 1) = 0

along the curve y, we have Case n. According to the general theory, we take for the functions Z(x,y) and n(x, y) solutions of eqs. (38) and (40). For example, we take n = 1 + x and m = -y. Then, eq. (38) takes the form: (1 + x)

=0.

Its partial solution is then

Substituting t\(x,y) into eq. (40), we obtain y(l+x) | | + (1+x +y2) | ^ = 0 .

INTRODUCTION

12

T his equation h as th e p a rtia l solution *2 _ v 2

(1+x)2 T hus, we need to introduce the new v aria b les 4 and form ulae x2 - y2 - 1 4(1 +x)2 Then, fro m form ulae (8), we have

3m _ 1 + x + y2 3u 3x 32m

y

2 (l+ x )3 34

(1 + x + y2)2 32m

3x2 ~ 4(1+x)6

3u

y2

32m

3y2 ~4(1+x)4 342 3 2m _ Sx3y~

y

y

3m

1 3m

3 y ~ ~ 2(1 +x)2 3£ + 1 + x 3q ’

y ( l+ x + y2) 3^m

3{2 '

(i+*)5

3497,

y2 32m (1 + x)4 3tj2

32m _

according to the

V =1 +X ' 3m _

( l + r)^ 3n ’

tj

y

l+ x + 3 y 2 3m 2(1 +x)4 34

32m

(1+ x)3 34 3r)

1

2y 3m (1 + x)3 3?) ’

3^m

( l+ x ) 2 3q2

1

^

3m

2 (l+ x )2 34 ’

y (l + x + y2) 3 2m 1 + x + 2y2 32m 4 (l + x)5 342 + 2 (l+ x )4 34 3q

y 3 2u y 3m 1 3m (1 + x)3 3tj2 (1+x)3 34 (1+ x)2 3tj Substituting th e expression (46) into eq. (45), we red u ce it to canonical form : 3 ^ 32m 1 3m 0. 342 + 37,2 + 2 3« 4. The classification of second-order equations with many independent

variables C onsider th e second-order lin e ar equation

n

2 i,j=1

„ n a^M y D du + Z, B; -z— + Cm + F = 0 dxi 3Xj i=1 * 3x,-

(47)

w here the Aij, the Bj, C, and F a r e r e a l functions of th e independent v a ria b le s x i , x 2 ) . . . , x n. Instead of Xf , X2, . . . , x n , let us introduce new independent v aria b les

tl>€2>£3>- •' >£»■

Suppose that the

INTRODUCTION

13

tk = Zk(xl , x2>--- >xrd (* = 1,2,. .. ,») (48) are twice continuously differentiable functions and that the Jacobian of the transformation is different from zero throughout the region in question. Then, 3u _ y 3u 3*k (49) 3xi = k= l3* k 3i ’ d*u y __=

a2u

ili + y

a y /50n

3H 3xj k,l=1 3^ k 3^l 3xi 3xj + k=l 3%k 3xi 3xj ' Substituting the values of the derivatives in (49) and (50) into eq. (47), we obtain 3 2w

2

y

k % \ ‘±kl 3Zk 3U + k=l

3u

j;

k 3 Lk

+ Cm + F = 0

(51)

where

~r _ y

,

B _

y

.

y D 3t k

Akl if= i ij 3xi 3xj ’ Ek if= i ij 3xi 3xj + i=i * 3xi Let us consider some particular point M(*^, . . . , x%). Suppose that at this point, 3i k / 3xi ~ uki • The transformation formulae n ~Akl= . 2

i ,]=1

A ija k ittlj

(52)

(53)

coincide with the formulae for the transformation of the coefficients of the quadratic form n 2 Aij Pi Pj (54) i j =1 if we make the change of variables n p i = 2 a-ki 9k > k=l reducing it to the form

(55)

n 2 %kl 9k 91 • (56) k,l =1 Consequently, the coefficients Afj in eq. (47) are transformed at the given point (xj, Xjj,. . . , x^) in exactly the same way as are the coefficients of the quadratic form (54) by the linear transformation (55). The coeffi cients Aij of the form (54) are assumed to be constant and equal to the val ues of the coefficients A ^ x j , . . . , % ) in eq. (47) at the point (x^, x?, . . . , *$). In algebra, it is proved that a non-singular transformation (o5) exists

INTRODUCTION

14

that reduces the quadratic form (54) with real coefficients A^j to the form m Z ±qj (m « m) , (57) such that the number of term s with positive and negative signs in the form (57) is determined exclusively by the form (54) and does not depend on the choice of the transformation (55). This is the law of invariance of quadratic forms 1). Let us now suppose that we have found a transformation (55) that re duces the form (54) to the form (57). Then, the transformation (48) satis fying the condition (52) reduces eq. (47) to the form (51): n a*u + Cu + F = 0 A *X=iAkl s ik + fe?iBk n k where

Z

A kl{x\ , . . . , x®) = ± 1

if fe = Z « m

Aki(x®,. . . , x$J) = 0

if k * I and k = l> m.

This form of eq. (47) is called its canonical form at the point ( * ? , * § , . . . , x°).

Thus, for every point (*f,-*2 >• • • >•*>$)» it is possible to find a nonsingular transformation (48) of the independent variables that reduces eq. (47) to canonical form at that point. We call eq. (47) at the point (x®, * £ , . . . , x^) an equation of elliptic type if, in eq. (51), all « coefficients *8, ■• • >■*«) are different from zero and have the same sign. It is of hyperbolic type if n - 1 of the coefficients A ^ x 1» Xjj) have the same sign and the other coefficient has the op posite sign. It is of ultrahyperbolic type if there is more than one positive coefficient among the x^,. . . , xfy and more than one negative coeffi cient and m = n. It is of parabolic type in the broad sense if at least one of the coefficients A ^ x ^ , x§,. . . , x^) is equal to zero. It is of parabolic type in the narrow sense (or simply it is of parabolic type) if exactly one of the coefficients is equal to zero and all other coefficients have the same sign. Eq. (47) is said to be of the elliptic (hyperbolic, and so on) type in the region D if it is of the elliptic (hyperbolic, and so on) type at every point of the region D. We note that it is generally impossible to obtain a canonical form of eq. (47) by means of the same transformation of the Independent variables for the entire region in which the differential equation is of the type in question. For if we attempted, by means of transformation (48), to reduce eq. (47) to canonical form throughout a certain region, it would be neces sary to Impose on the n functions £j(xi,. . . , x^) the £w(«-l) conditions —

v , H k Hi Aj j ~ a —0 (k * Z) • i,j=1 A i i x~. dXj Bxj If n exceeds 3, this system is not soluble in the general case, because Akl - .4-

INTRODUCTION

15

the number of equations exceeds the number of functions 4i(*i,.. .,%} to be determined. For n = 3, a solution does exist for the system, but, in con trast with the case of n = 2, it is not possible in the general case to impose further conditions on the coefficients of the derivatives 3 ^ / 4^ We note also that if the coefficients Aij in eq. (47) are constants, the equation may be reduced to canonical form simultaneously at all points of the region by means of a single linear transformation of the independent variables. Problems 1. Reduce the following equations to canonical form: d^u „ . 3^m „ - 2 sin x' 3x , 3y ,

a)

3X2

y2

b)

=o

3 2 m

3u

„

= 0 ,

(x > 0, y > 0) , o

3 2 m II

A

„

3y2

X

o

3 ^m

x2

^ - COS

3y*

O

o

c)

+

3x2

n 3^u

- C O S '2*

Answers: a) b)

c)

3 2 “ 3 4 3 rj

3 2 m ^

2

=

n 0

i-

t ^ x +y - c o s x ,

’

3 2 m +

^ 2

+

1

3 m

24

3 4

+

=

o

^ 3 7 ,2

1

3m

2n

3 t)

’

V = x - y + cos x i=y2 ,

"

°

r\ = x 2

’

t=l I x ’

n =y ■

2. Show that the equation d_[(' £ \ 2 3m1 _ _L (i 3x (V " h) 3xj ~ a2 V ' h) g,2 can be reduced to the form 3 2u _ 32v 3x2 a2 3 3^u 3du 3u n (; - * ) r ) T ) T = 0

3x2 can be reduced to the form

dy^

3^w 3( 9tj

3x

1

w

4 (£ +tj)2

/n ^ ,v (0 < X < 1)

=0 .

Method of solution: Set £ = .Jl - x + {y ,

?) = V I-x - iy ,

m /4 + h

5. Reduce to canonical form: 3x2

3y2

3x

(°< * < * )

Answer: 3^w 34

1 w 4 s in 2 ( £ - Tj ) ’

where 4 =i ( j ' + “ )>

»J=-i(3' - u ) ,

a) = arc cos j , 1’

w =-p= -/sin (£ -n)

PART I

DIFFERENTIAL EQUATIONS OF THE HYPERBOLIC TYPE

Chapter I METHODS OF FI NDI NG THE GE NE RAL S OLUTI ON TO EQUATI ONS OF THE H Y P E R B O L I C T YP E 1. General remarks. Examples For an ordinary differential equation yW = f ( x , y , y ' , . . . J n -V) the solution , c n),

y

containing n arbitrary constants, is called the general solution in the do main D of the variables x , y , y ' , . . . ,y(w-l), if, for a suitable choice of con stants C-[,C2 , . . . ,CW, it is possible to solve an arbitrary Cauchy problem inD. For partial differential equations, the matter becomes more compli cated. However, even here, we may seek the "general solution", now con taining arbitrary functions, the number of which, generally, is equal to the order of the differential equation. For a second-order hyperbolic equation with two independent variables, we shall call a solution containing two a r bitrary functions the general solution if, for a suitable choice of arbitrary functions, it is possible to solve the Cauchy problem with arbitrary initial conditions on a non-characteristic curve; that is, to find the solution to the second-order equation satisfying on the given curve the initial conditions Su Sn I = * , where d/dn Indicates differentiation with respect to the normal to the curve I. Knowing the general solution makes it possible, in certain cases, to obtain solutions in closed form for some problems in mathematical physics. Example 1. Let us consider the wave equation u\ i -

0 such that if we re place f(x) and F(x) by f\{x) and Fi(x) in such a way that If(x) - f\(x) | < tj ,

| F(x) - Fx(x) | < 7? ,

the difference between the new and the original solution will be less in ab solute value than e. This follows directly from eq. (8). Example 2. Find the solution to the equation 2 d^u

o d%u

„

(9)

satisfying the initial conditions 3u

=F(x) . ( 10) y=l As was shown above (see the Introduction), eq. (9) can, by means of a change of variables M| y = l = / W -

dy

22

GENERAL SOLUTION TO EQUATIONS OF HYPERBOLIC TYPE

4 = xy , v = y/ x , be reduced to the canonical form d^u Jl 3m _.

[Ch. I ( 11)

( 12)

34 3b ' 24 3q “ ° 1

Setting (13)

w = 3w /3b ,

we reduce eq. (12) to the linear equation 3w 1 w =0 , 94 24 the general solution of which is of the form w=-/4 0o(b). Substituting (14) into (13), we obtain the equation

(14)

*ofo) , which can be integrated in quadrature: a = -Ji I 0o(b) db +

[Ch. n

THE CAUCHY PROBLEM IN A PLANE

40

dv,

_ dG 9a. = 1 dl'n^-z " 4 (40 -t,0)S2

/ dGx \da/T,=-£ ’

dv, _ dG 9a. = 1 (£ -Vq) (j -€0) /dG\ 3” n=-£ =dff a»I n=-C 4 («0 - t)0)C2 W „ =_£ ' it follows that £f>+Vr d_v_ 9w. - m H + 3v[r,=-i 2(t o-Vo )i K~doJT)=-£ ' ^ To use eq. (40), we still need to find the values of the function w on the line rj = -£ and at the points P and Q. It is easy to see that tfll _ > = w ( i , - i ) =-f 2i u(x, 0) = 42T / U 2) .

(44)

From this, we easily obtain w(P) = w «0 »*«o) Remembering that

,

m^ q.^ o)

u>(Q)

= w(-rtQ,rt0) =4-2T)0 AVq) .

(45)

M£ o, t)q) 42 'lxq

we find from eqs. (40) and (42) - (45) that u(x0 , y 0)

2v'X-,

+---^ -h0 tp+% 4(6o ■*6) Vxp

M *)% .

Returning now to the original variables x and y, and omitting the subscripts from these letters, we obtain the solution of the Cauchy problem for eq. (25): u^x

i

*/4x + j y A x + 4 x y + } y 2) + 4 4 x - j y f i x - 4 x y + } y 2)

2ifx x Jx+zy +—

* ( x , y , z ) dz ,

& Tx-b where $(x,y, z) =Jz F(z2) c ( 7y2 - (z-V*)2\ . yAz2) r , ( \y2 - {z-Jx)2\ ' 4z j x ' K 4z 4 f >

THE CAUCHY PROBLEM IN A PLANE

C h. II]

41

Problems 1. Find the Riemann function for the Euler-Darboux equation 3^u 9 du a 3u q dx 3y x - y d x x - y 3 y ~ Answer: v(x,y;x0, y0)

=

(yQ-x)~P

( y -*o)~a

(y-x)a+P F(a, 0,

l;o ) ,

where _ (* - x0) (y - y0) a ~ (x - y 0) ( y - x 0) ‘ 2.

Integrate, by the Riemann method, the equation

9\ 3%u „ 3u , „ 11 X^ ^ 2 - ^ y 2 - 2 x V x - V, = 0 with the conditions “ ly=0 = ^ ) >

(0 0, for x + at < 0 .

u2 ~ 0

Method of solution: integrate the equation 32m o 3x2 ’ 3/2 and use the conditions 32mo 32u± 9uj 9 i M = M — =+ Tr 3/2 x =0 9 : 9x x=0 3

Pd'P, F ig. 16.

If the rod is somewhat stretched or compressed along its longitudinal axis and then released, it will start vibrating. Let us direct the x-axis along the axis of the rod, and let us assume that, in a state of rest, the ends of the rod are at the points x = 0 and x = I. Suppose that x is the ab scissa of an arbitrary cross section of the rod when at rest. Let us denote by u(x, t) the displacement of this section at the time t. Then, the displace ment of the section whose abscissa is x + dx will be equal to

Thus, It is clear that the relative lengthening of the rod at the cross section whose abscissa is x is given by the derivative du(x, t)/dx . Recalling that the rod undergoes only small oscillations, we can compute the tension T in this cross section. From Hooke's law, we have (1)

60

LONGITUDINAL VIBRATIONS OF A ROD

[C h.IV

where E is the modulus of elasticity of the material of which the rod is composed, and S is the cross sectional area. Let us consider the element of the rod included between two cross sections whose abscissas, when the rod is at rest, are x and x + dx. This element is acted on by the forces of tension Tx and Tx+ E S — dx ( 2) x+dx dx