LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS

HANDBOOK OF

LINEAR PARTIAL

DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS

Andrei D. Polyanin CHAPMAN & HALL/CRC

A CRC Press Company

Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data

Polianin, A. D. (Andrei Dmitrievich) Handbook of linear partial differential equations for engineers and scientists / by Andrei D. Polyanin p. cm. Includes bibliographical references and index. ISBN 1-58488-299-9

1. Differential equations, Linear--Numerical solution--Handbooks, manuals, etc. I. Title.

QA377 .P568 2001 515 ′.354—dc21

2001052427 CIP

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FOREWORD

Linear partial differential equations arise in various fields of science and numerous applications, e.g., heat and mass transfer theory, wave theory, hydrodynamics, aerodynamics, elasticity, acous- tics, electrostatics, electrodynamics, electrical engineering, diffraction theory, quantum mechanics, control theory, chemical engineering sciences, and biomechanics.

This book presents brief statements and exact solutions of more than 2000 linear equations and problems of mathematical physics. Nonstationary and stationary equations with constant and variable coefficients of parabolic, hyperbolic, and elliptic types are considered. A number of new solutions to linear equations and boundary value problems are described. Special attention is paid to equations and problems of general form that depend on arbitrary functions. Formulas for the effective construction of solutions to nonhomogeneous boundary value problems of various types are given. We consider second-order and higher-order equations as well as the corresponding boundary value problems. All in all, the handbook presents more equations and problems of mathematical physics than any other book currently available.

For the reader’s convenience, the introduction outlines some definitions and basic equations, problems, and methods of mathematical physics. It also gives useful formulas that enable one to express solutions to stationary and nonstationary boundary value problems of general form in terms of the Green’s function.

Two supplements are given at the end of the book. Supplement A lists properties of the most common special functions (the gamma function, Bessel functions, degenerate hypergeometric func- tions, Mathieu functions, etc.). Supplement B describes the methods of generalized and functional separation of variables for nonlinear partial differential equations. We give specific examples and an overview application of these methods to construct exact solutions for various classes of second-, third-, fourth-, and higher-order equations (in total, about 150 nonlinear equations with solutions are described). Special attention is paid to equations of heat and mass transfer theory, wave theory, and hydrodynamics as well as to mathematical physics equations of general form that involve arbitrary functions.

The equations in all chapters are in ascending order of complexity. Many sections can be read independently, which facilitates working with the material. An extended table of contents will help the reader find the desired equations and boundary value problems. We refer to specific equations using notation like “1.8.5.2,” which means “Equation 2 in Subsection 1.8.5.”

To extend the range of potential readers with diverse mathematical backgrounds, the author strove to avoid the use of special terminology wherever possible. For this reason, some results are presented schematically, in a simplified manner (without details), which is however quite sufficient in most applications.

Separate sections of the book can serve as a basis for practical courses and lectures on equations of mathematical physics. The author thanks Alexei Zhurov for useful remarks on the manuscript. The author hopes that the handbook will be useful for a wide range of scientists, university

teachers, engineers, and students in various areas of mathematics, physics, mechanics, control, and engineering sciences.

Andrei D. Polyanin

BASIC NOTATION

Latin Characters

fundamental solution Im[ ✁ ✂ ] imaginary part of a complex quantity ✁

Green’s function ✆

< ✞ ✟ < ✝ ; ✠ = 1, ✡☛✡☛✡ , } Re[ ✁ ] real part of a complex quantity ✁

-dimensional Euclidean space, ✆ = {−

, ☞ ✌ , ✍ cylindrical coordinates, = ✎ ✞ 2 + ☞ ✏ 2 and ✞ = cos ✌ , ✏ = sin ✌

, ☞ ✑ , ✌ spherical coordinates, = ✎ ✞ 2 + ✏ 2 + ✍ 2 and ✞

= ☞ sin ✑ cos ✌ , ✏ = sin ✑ sin ✌ , ✍ = ☞ cos ✑

time ( ≥ 0) unknown function (dependent variable)

, ✏ , ✍ space (Cartesian) coordinates ✆

1 , ✡☛✡☛✡ , ✞ ☎ Cartesian coordinates in ✆ -dimensional space

-dimensional vector, x = { ✞ 1 , ✡☛✡☛✡ , ✞ ☎ }

| ✆ x| magnitude (length) of -dimensional vector, |x| =

-dimensional vector, y = { ✏ 1 , ✡☛✡☛✡ , ✏ ☎ }

Greek Characters

Laplace operator

2 two-dimensional Laplace operator, 2 = ✖ 2 + ✖ 2

3 three-dimensional Laplace operator, 3 = ✖ 2 + ✖ ☎ 2 ✖ ✗ ✖ ✘ + ✖ 2

-dimensional Laplace operator, ✕ ☎

) Dirac delta function; ✢ ✣ ( ✏ ) ( ✞ − ✏ ) ✥ ✏ = ( ✞ ), where ( ✞ ) is any continuous function,

Kronecker delta, ✩ ☎ ✧ = ★ 1 if ✆ = ,

0 if

) Heaviside unit step function, ( ✞ )= ★ 1 if

0 if ✞ <0

Brief Notation for Derivatives

= ✫ 2 (partial derivatives)

(derivatives for = ( ✞ ))

Special Functions (See Also Supplement A)

Airy function; Ai( ✞ )= ✵ ✶ 3 ✞ ✷ ✸ ✲ 2 ✞ 3 ✸ 2

2 + Ce ✹

2 ✟ + ✹ cosh[(2 ✠ + ✻ ) ✞ ] even modified Mathieu functions, where ✻ = 0, 1;

Ce 2 ☎ + ✹ ( ✞ , ✺ ) = ce 2 ☎ + ✹ ( ✼✽✞ , ✺ ) Ce 2 ☎ + ✹ ( ✞ , ✺ ) = ce 2 ☎ + ✹ ( ✼✽✞ , ✺ )

2 ✟ cos 2 ✠ ✞ even -periodic Mathieu functions; these satisfy the

equation ✦ ✏ +( −2 ✺ cos 2 ✦ ✞ ) ✏ = 0, where = 2 ☎ ( ✺ )

are eigenvalues

ce ✚ 2 ☎ +1 ☎ (

2 ✟ +1 cos[(2 ✠ +1) ✞ ] even 2 -periodic Mathieu functions; these satisfy the

cos 2 ✦ ✞ ) ✏ = 0, where = 2 ☎ +1 ( ✺ )

are eigenvalues

= ( ✞ ) parabolic cylinder function (see Paragraph 7.3.4-1); it

+ ✲✽❀ + 1 1 2 2 − 4 ✞ ✏ =0 erf ✳

satisfies the equation ✳

= ✗ exp ✲ − ❂ 2 ✥ ❂

error function

erfc =

exp ✲ − ❂ 2 ✥ ❂

complementary error function

Hermite polynomial

)= ❅ ( ✞ )+ ❃ ✼❇❆ ✿ ✿ ( ✞ ) Hankel function of first kind, ✼ 2 = −1

Hankel function of second kind, = −1

hypergeometric function, ( ) = ( + 1) ✡☛✡☛✡ ( + − 1)

modified Bessel function of first kind

Bessel function of first kind

2 ✰ sin( ❀ modified Bessel function of second kind

generalized Laguerre polynomial

Legendre polynomial

associated Legendre functions

2 ☎ + Se ✹

2 ✟ + ✹ sinh[(2 ✠ + ✟ ✻ ) ✞ ] odd modified Mathieu functions, where ✻ = 0, 1;

2 sin 2 odd -periodic Mathieu functions; these satisfy the

equation ✦ ✏ +( −2 ✺ cos 2 ✞ ) ✏ = 0, where ✦ = ❉ 2 ☎ ( ✺ )

are eigenvalues

2 ✟ +1 sin[(2 ✠ +1) ] odd 2 -periodic Mathieu functions; these satisfy the

equation ✦ +( −2 cos 2 ) = 0, where = ❉ 2 ☎ +1 ( ✺ )

are eigenvalues

( ✰ ✞ ) cos( ❀ )− ❅ − ( ✞

( )= sin( ✰ ❀ Bessel function of second kind )

incomplete gamma function

( P )= ✢ ✱

gamma function

degenerate hypergeometric function,

AUTHOR

Andrei D. Polyanin, D.Sc., Ph.D., is a noted scientist of broad interests, who works in various areas of mathematics, mechanics, and chemical engineering sciences.

A. D. Polyanin graduated from the Department of Mechanics and Mathematics of the Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, A. D. Polyanin has been a member of the staff of the Institute for Problems in Me- chanics of the Russian Academy of Sciences.

Professor Polyanin has made important contributions to de- veloping new exact and approximate analytical methods of the theory of differential equations, mathematical physics, integral equations, engineering mathematics, nonlinear mechanics, theory of heat and mass transfer, and chemical hydrodynamics. He ob-

tained exact solutions for several thousand ordinary differential, partial differential, mathematical physics, and integral equations.

Professor Polyanin is an author of 27 books in English, Russian, German, and Bulgarian, as well as over 120 research papers and three patents. He has written a number of fundamental handbooks, including A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations , CRC Press, 1995; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; and A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations , Gordon and Breach, 2001.

In 1991, A. D. Polyanin was awarded a Chaplygin Prize of the USSR Academy of Sciences for his research in mechanics. Address: Institute for Problems in Mechanics, RAS, 101 Vernadsky Avenue, Building 1, 117526 Moscow, Russia

E-mail: polyanin@ipmnet.ru

CONTENTS

Foreword Basic Notation and Remarks Author Introduction. Some Definitions, Formulas, Methods, and Solutions

0.1. Classification of Second-Order Partial Differential Equations

0.1.1. Equations with Two Independent Variables

0.1.2. Equations with Many Independent Variables

0.2. Basic Problems of Mathematical Physics

0.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems

0.2.2. First, Second, Third, and Mixed Boundary Value Problems

0.3. Properties and Particular Solutions of Linear Equations

0.3.1. Homogeneous Linear Equations

0.3.2. Nonhomogeneous Linear Equations

0.4. Separation of Variables Method

0.4.1. General Description of the Separation of Variables Method

0.4.2. Solution of Boundary Value Problems for Parabolic and Hyperbolic Equations

0.5. Integral Transforms Method

0.5.1. Main Integral Transforms

0.5.2. Laplace Transform and Its Application in Mathematical Physics

0.5.3. Fourier Transform and Its Application in Mathematical Physics

0.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution

0.6.1. Cauchy Problem for Parabolic Equations

0.6.2. Cauchy Problem for Hyperbolic Equations

0.7. Nonhomogeneohus Boundary Value Problems with One Space Variable. Representation of Solutions via the Green’s Function

0.7.1. Problems for Parabolic Equations

0.7.2. Problems for Hyperbolic Equations

0.8. Nonhomogeneous Boundary Value Problems with Many Space Variables. Representa- tion of Solutions via the Green’s Function

0.8.1. Problems for Parabolic Equations

0.8.2. Problems for Hyperbolic Equations

0.8.3. Problems for Elliptic Equations

0.8.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types

0.9. Construction of the Green’s Functions. General Formulas and Relations

0.9.1. Green’s Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains

0.9.2. Green’s Functions Admitting Incomplete Separation of Variables

0.9.3. Construction of Green’s Functions via Fundamental Solutions

0.10. Duhamel’s Principles in Nonstationary Problems

0.10.1. Problems for Homogeneous Linear Equations

0.10.2. Problems for Nonhomogeneous Linear Equations

0.11. Transformations Simplifying Initial and Boundary Conditions

0.11.1. Transformations That Lead to Homogeneous Boundary Conditions

0.11.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions

1. Parabolic Equations with One Space Variable

1.1. Constant Coefficient Equations

1.1.1. 2 Heat Equation

1.1.2. ✒ Equation of the Form

1.1.3. ✒ Equation of the Form

1.1.4. ✒ Equation of the Form =

1.1.5. ✒ Equation of the Form

1.2. Heat Equation with Axial or Central Symmetry and Related Equations

2 1.2.1. ✳ Equation of the Form

1.2.2. ✒ Equation of the Form

1.2.3. ✳ Equation of the Form

1.2.4. ✒ Equation of the Form

1−2 ❯ ❱ ✖ 1.2.5. ❯ Equation of the Form ✖ ❚

1.2.6. ✒ Equation of the Form

1.3. Equations Containing Power Functions and Arbitrary Parameters

1.3.1. ✒ Equations of the Form

1.3.2. ✒ Equations of the Form

1.3.3. ✒ Equations of the Form

1.3.4. ✓ Equations of the Form ✖ ❚ =( ✞ + ❉ ) ✖ ❚

1.3.5. ✒ Equations of the Form =(

1.3.6. ✒ Equations of the Form

1.3.7. ✒ Equations of the Form

1.3.8. 2 Liquid-Film Mass Transfer Equation (1 −

1.3.9. ✘ Equations of the Form (

1.4. Equations Containing Exponential Functions and Arbitrary Parameters

1.4.1. ✒ Equations of the Form =

1.4.2. ✒ Equations of the Form

1.4.3. ✒ Equations of the Form

1.4.4. ✒ Equations of the Form

Equations of the Form ✓ ✗ ✖ ❚ 2 ✞

1.4.6. Other Equations

1.5. Equations Containing Hyperbolic Functions and Arbitrary Parameters

1.5.1. Equations Containing a Hyperbolic Cosine

1.5.2. Equations Containing a Hyperbolic Sine

1.5.3. Equations Containing a Hyperbolic Tangent

1.5.4. Equations Containing a Hyperbolic Cotangent

1.6. Equations Containing Logarithmic Functions and Arbitrary Parameters

Equations of the Form ✓ ✞

1.6.2. ✒ Equations of the Form

1.7. Equations Containing Trigonometric Functions and Arbitrary Parameters

1.7.1. Equations Containing a Cosine

1.7.2. Equations Containing a Sine

1.7.3. Equations Containing a Tangent

1.7.4. Equations Containing a Cotangent

1.8. Equations Containing Arbitrary Functions

1.8.1. ✒ Equations of the Form

1.8.2. ✒ Equations of the Form

1.8.3. ✒ Equations of the Form

1.8.4. ✒ Equations of the Form

1.8.5. ✒ Equations of the Form

1.8.6. ✓ Equations of the Form = ( ✞ ) ✖ ❚ 2

1.8.7. ✒ Equations of the Form

Equations of the Form ✒

1.8.9. ✒ Equations of the Form ( ) =

1.9. ❭ Equations of Special Form

1.9.1. Equations of the Diffusion (Thermal) Boundary Layer ❵ 2 2

1.9.2. One-Dimensional Schr ¨odinger Equation ✼❫ ❪ ❳ ✖ ❚ ✬ =− 2 ❴ ✧ ✖ ❚ 2 + ❛

2. Parabolic Equations with Two Space Variables

2.1. Heat Equation ✖ ❚ ✬

2.1.1. Boundary Value Problems in Cartesian Coordinates

2.1.2. Problems in Polar Coordinates

2.1.3. Axisymmetric Problems

2.2. ✒ Heat Equation with a Source

2.2.1. Problems in Cartesian Coordinates

2.2.2. Problems in Polar Coordinates

2.2.3. Axisymmetric Problems

2.3. Other Equations

2.3.1. Equations Containing Arbitrary Parameters

2.3.2. Equations Containing Arbitrary Functions

3. Parabolic Equations with Three or More Space Variables ✕

3.1. Heat Equation ✖ ❚ ✬

3.1.1. Problems in Cartesian Coordinates

3.1.2. Problems in Cylindrical Coordinates

3.1.3. Problems in Spherical Coordinates

3.2. Heat Equation with Source ✖ ❚ ✬

3.2.1. Problems in Cartesian Coordinates

3.2.2. Problems in Cylindrical Coordinates

3.2.3. Problems in Spherical Coordinates

3.3. Other Equations with Three Space Variables

3.3.1. Equations Containing Arbitrary Parameters

3.3.2. Equations Containing Arbitrary Functions

3.3.3. ✒ Equations of the Form

( ✞ , ✏ , ✍ ) ✖ = div[ ( ✞ , ✏ , ✍ )∇ ]− ✺ ( ✞ , ✏ , ✍ ) + ( ✞ , ✏ , ✍ , )

3.4. ✆ Equations with Space Variables

3.4.1. ✒ Equations of the Form ☎ ❙

3.4.2. Other Equations Containing Arbitrary Parameters

3.4.3. Equations Containing Arbitrary Functions

4. Hyperbolic Equations with One Space Variable

4.1. Constant Coefficient Equations

4.1.1. 2 Wave Equation

Equations of the Form ✒

4.1.3. ✦ Equation of the Form ✖ ❚ 2 = 2 ✖ 2 ✗

4.1.4. ✒ Equation of the Form

4.1.5. ✒ Equation of the Form

4.2. Wave Equation with Axial or Central Symmetry 2

2 4.2.1. ✳ Equations of the Form

4.2.2. ✒ Equation of the Form

4.2.3. Equation of the Form ✖ ❚ 2 = 2 ✲ ✳ ✖ ❚ 2 + 2 ✖ ✖ ❚

4.2.4. ✒ Equation of the Form

4.2.5. ✒ Equation of the Form

4.2.6. ✒ Equation of the Form 2

4.3. Equations Containing Power Functions and Arbitrary Parameters

4.3.1. ✒ Equations of the Form

4.3.2. ✒ Equations of the Form

4.3.3. Other Equations

4.4. Equations Containing the First Time Derivative 2

4.4.1. ✒ Equations of the Form

4.4.2. ✒ Equations of the Form

4.4.3. ✗ Other Equations

4.5. Equations Containing Arbitrary Functions 2

4.5.1. ✒ Equations of the Form

4.5.2. ✒ Equations of the Form

4.5.3. ✖ Other Equations

5. Hyperbolic Equations with Two Space Variables

5.1. ✕ Wave Equation

5.1.1. Problems in Cartesian Coordinates

5.1.2. Problems in Polar Coordinates

5.1.3. Axisymmetric Problems

Nonhomogeneous Wave Equation ✒

5.2.1. Problems in Cartesian Coordinates

5.2.2. Problems in Polar Coordinates

5.2.3. Axisymmetric Problems

5.3. ✒ Equations of the Form

5.3.1. Problems in Cartesian Coordinates

5.3.2. Problems in Polar Coordinates

5.3.3. Axisymmetric Problems

5.4. ✒ Telegraph Equation

5.4.1. ✖ Problems in Cartesian Coordinates

5.4.2. Problems in Polar Coordinates

5.4.3. Axisymmetric Problems

5.5. Other Equations with Two Space Variables

6. Hyperbolic Equations with Three or More Space Variables

2 6.1. ✕ Wave Equation

6.1.1. Problems in Cartesian Coordinates

6.1.2. Problems in Cylindrical Coordinates

6.1.3. Problems in Spherical Coordinates

6.2. ✒ Nonhomogeneous Wave Equation

6.2.1. Problems in Cartesian Coordinates

6.2.2. Problems in Cylindrical Coordinates

6.2.3. Problems in Spherical Coordinates

6.3. ✒ Equations of the Form 2

6.3.1. Problems in Cartesian Coordinates

6.3.2. Problems in Cylindrical Coordinates

6.3.3. Problems in Spherical Coordinates

6.4. ✒ Telegraph Equation

6.4.1. Problems in Cartesian Coordinates

6.4.2. Problems in Cylindrical Coordinates

6.4.3. Problems in Spherical Coordinates

6.5. Other Equations with Three Space Variables

6.5.1. Equations Containing Arbitrary Parameters 2

6.5.2. ✒ Equation of the Form

) ✓ ✖ 2 = div ( ✞ , ✏ , ✍ )∇ − ✺ ( ✞ ✏

6.6. ✆ Equations with Space Variables

6.6.1. ✓ Wave Equation ✖ ❚ 2

6.6.2. ✒ Nonhomogeneous Wave Equation

6.6.3. ✒ Equations of the Form = 2 ✕ ☎ ✓ ❙ −

6.6.4. ✖ Equations Containing the First Time Derivative

7. Elliptic Equations with Two Space Variables

7.1. ✓ Laplace Equation 2 =0

7.1.1. Problems in Cartesian Coordinate System

7.1.2. Problems in Polar Coordinate System

7.1.3. Other Coordinate Systems. Conformal Mappings Method

7.2. ❙ Poisson Equation 2 =− (x)

7.2.1. Preliminary Remarks. Solution Structure

7.2.2. Problems in Cartesian Coordinate System

7.2.3. Problems in Polar Coordinate System

7.2.4. Arbitrary Shape Domain. Conformal Mappings Method

7.3. ❙ Helmholtz Equation 2 +

=− (x)

7.3.1. General Remarks, Results, and Formulas

7.3.2. Problems in Cartesian Coordinate System

7.3.3. Problems in Polar Coordinate System

7.3.4. Other Orthogonal Coordinate Systems. Elliptic Domain

7.4. Other Equations

7.4.1. ✤ Stationary Schr ¨odinger Equation 2 = (

7.4.2. Convective Heat and Mass Transfer Equations

7.4.3. Equations of Heat and Mass Transfer in Anisotropic Media

7.4.4. Other Equations Arising in Applications

7.4.5. ❙ Equations of the Form (

8. Elliptic Equations with Three or More Space Variables ✕

8.1. ✓ Laplace Equation 3 =0

8.1.1. Problems in Cartesian Coordinates

8.1.2. Problems in Cylindrical Coordinates

8.1.3. Problems in Spherical Coordinates

8.1.4. Other Orthogonal Curvilinear Systems of Coordinates ✕

8.2. ❙ Poisson Equation 3 + (x) = 0

8.2.1. Preliminary Remarks. Solution Structure

8.2.2. Problems in Cartesian Coordinates

8.2.3. Problems in Cylindrical Coordinates

8.2.4. Problems in Spherical Coordinates ✕

8.3. ❙ Helmholtz Equation 3 + =− (x)

8.3.1. General Remarks, Results, and Formulas

8.3.2. Problems in Cartesian Coordinates

8.3.3. Problems in Cylindrical Coordinates

8.3.4. Problems in Spherical Coordinates

8.3.5. Other Orthogonal Curvilinear Coordinates

8.4. Other Equations with Three Space Variables

8.4.1. Equations Containing Arbitrary Functions ✦

8.4.2. ❙ Equations of the Form div [ (

8.5. Equations with Space Variables ✕

8.5.1. Laplace Equation ☎ ✓ =0

8.5.2. Other Equations

9. Higher-Order Partial Differential Equations

9.1. Third-Order Partial Differential Equations

9.2. Fourth-Order One-Dimensional Nonstationary Equations

9.2.1. ✒ Equations of the Form

2 ✗ 9.2.2. 4 Equations of the Form ✖ ❚ 2 + ✖ ❚ 4

9.2.3. ✒ Equations of the Form

9.2.4. ✦ Equations of the Form ✖ ❚ 2 + 2 4 ✖ ❙ ❚ 4

9.2.5. Other Equations

9.3. Two-Dimensional Nonstationary Fourth-Order Equations

9.3.1. 4 Equations of the Form + 2 4 ✳ ❙

9.3.2. ✕ Two-Dimensional Equations of the Form

9.3.3. ✓ Three- and -Dimensional Equations of the Form 2

9.3.4. ✒ Equations of the Form

9.3.5. ✒ Equations of the Form

9.4. Fourth-Order Stationary Equations ✕ ✕

9.4.1. ✓ Biharmonic Equation

9.4.2. ❙ Equations of the Form = (

9.4.3. ❙ Equations of the Form

9.4.4. Equations of the Form ✖ ❚ 4 ✖ ❚ 4 ✞ ✖ ✏ ✗ + ✖ ✘ = ( , )

9.4.5. ❙ Equations of the Form

9.4.6. Stokes Equation (Axisymmetric Flows of Viscous Fluids)

9.5. Higher-Order Linear Equations with Constant Coefficients

9.5.1. Fundamental Solutions. Cauchy Problem

9.5.2. Elliptic Equations

9.5.3. Hyperbolic Equations

9.5.4. Regular Equations. Number of Initial Conditions in the Cauchy Problem

9.5.5. Some Special-Type Equations

9.6. Higher-Order Linear Equations with Variable Coefficients

9.6.1. Equations Containing the First Time Derivative

9.6.2. Equations Containing the Second Time Derivative

9.6.3. Nonstationary Problems with Many Space Variables

9.6.4. Some Special-Type Equations

Supplement A. Special Functions and Their Properties

A.1. Some Symbols and Coefficients

A.1.1. Factorials

A.1.2. Binomial Coefficients

A.1.3. Pochhammer Symbol

A.1.4. Bernoulli Numbers

A.2. Error Functions and Exponential Integral

A.2.1. Error Function and Complementary Error Function

A.2.2. Exponential Integral

A.2.3. Logarithmic Integral

A.3. Sine Integral and Cosine Integral. Fresnel Integrals

A.3.1. Sine Integral

A.3.2. Cosine Integral

A.3.3. Fresnel Integrals

A.4. Gamma and Beta Functions

A.4.1. Gamma Function

A.4.2. Beta Function

A.5. Incomplete Gamma and Beta Functions

A.5.1. Incomplete Gamma Function

A.5.2. Incomplete Beta Function

A.6. Bessel Functions

A.6.1. Definitions and Basic Formulas

A.6.2. Integral Representations and Asymptotic Expansions

A.6.3. Zeros and Orthogonality Properties of Bessel Functions

A.6.4. Hankel Functions (Bessel Functions of the Third Kind)

A.7. Modified Bessel Functions

A.7.1. Definitions. Basic Formulas

A.7.2. Integral Representations and Asymptotic Expansions

A.8. Airy Functions

A.8.1. Definition and Basic Formulas

A.8.2. Power Series and Asymptotic Expansions

A.9. Degenerate Hypergeometric Functions

A.9.1. Definitions and Basic Formulas

A.9.2. Integral Representations and Asymptotic Expansions

A.10. Hypergeometric Functions

A.10.1. Definition and Some Formulas

A.10.2. Basic Properties and Integral Representations

A.11. Whittaker Functions

A.12. Legendre Polynomials and Legendre Functions

A.12.1. Definitions. Basic Formulas

A.12.2. Zeros of Legendre Polynomials and the Generating Function

A.12.3. Associated Legendre Functions

A.13. Parabolic Cylinder Functions

A.13.1. Definitions. Basic Formulas

A.13.2. Integral Representations and Asymptotic Expansions

A.14. Mathieu Functions

A.14.1. Definitions and Basic Formulas

A.15. Modified Mathieu Functions

A.16. Orthogonal Polynomials

A.16.1. Laguerre Polynomials and Generalized Laguerre Polynomials

A.16.2. Chebyshev Polynomials and Functions

A.16.3. Hermite Polynomial

A.16.4. Jacobi Polynomials

Supplement B. Methods of Generalized and Functional Separation of Variables in Nonlinear Equations of Mathematical Physics

B.1. Introduction

B.1.1. Preliminary Remarks

B.1.2. Simple Cases of Variable Separation in Nonlinear Equations

B.1.3. Examples of Nontrivial Variable Separation in Nonlinear Equations

B.2. Methods of Generalized Separation of Variables

B.2.1. Structure of Generalized Separable Solutions

B.2.2. Solution of Functional Differential Equations by Differentiation

B.2.3. Solution of Functional Differential Equations by Splitting

B.2.4. Simplified Scheme for Constructing Exact Solutions of Equations with Quadratic Nonlinearities

B.3. Methods of Functional Separation of Variables

B.3.1. Structure of Functional Separable Solutions

B.3.2. Special Functional Separable Solutions

B.3.3. Differentiation Method

B.3.4. Splitting Method. Reduction to a Functional Equation with Two Variables

B.3.5. Some Functional Equations and Their Solutions. Exact Solutions of Heat and Wave Equations

B.4. First-Order Nonlinear Equations

B.4.1. Preliminary Remarks

B.4.2. Individual Equations

B.5. Second-Order Nonlinear Equations

B.5.1. Parabolic Equations

B.5.2. Hyperbolic Equations

B.5.3. Elliptic Equations

B.5.4. Equations Containing Mixed Derivatives

B.5.5. General Form Equations

B.6. Third-Order Nonlinear Equations

B.6.1. Stationary Hydrodynamic Boundary Layer Equations

B.6.2. Nonstationary Hydrodynamic Boundary Layer Equations

B.7. Fourth-Order Nonlinear Equations

B.7.1. Stationary Hydrodynamic Equations (Navier–Stokes Equations)

B.7.2. Nonstationary Hydrodynamic Equations

B.8. Higher-Order Nonlinear Equations ❈

B.8.1. ✳ Equations of the Form

= ✲▲✞

, ✓ , , ✖ ❚ , ✡☛✡☛✡ , ✖ ❣ ❚

B.8.2. ✳ Equations of the Form

2 = ✲▲✞ , , , ✖ ❚ ✡☛✡☛✡ ✖ ❣ ❚

B.8.3. ❣ Other Equations

References

Introduction

Some Definitions, Formulas, Methods, and Solutions

0.1. Classification of Second-Order Partial Differential

Equations

0.1.1. Equations with Two Independent Variables

0.1.1-1. Examples of equations encountered in applications. Three basic types of partial differential equations are distinguished—parabolic, hyperbolic, and

elliptic . The solutions of the equations pertaining to each of the types have their own characteristic qualitative differences.

The simplest example of a parabolic equation is the heat equation

where the variables ✄ and play the role of time and the spatial coordinate, respectively. Note that equation (1) contains only one highest derivative term.

The simplest example of a hyperbolic equation is the wave equation

where the variables ✄ and play the role of time and the spatial coordinate, respectively. Note that the highest derivative terms in equation (2) differ in sign.

The simplest example of an elliptic equation is the Laplace equation

where ☎ and play the role of the spatial coordinates. Note that the highest derivative terms in equation (3) have like signs.

Any linear partial differential equation of the second-order with two independent variables can

be reduced, by appropriate manipulations, to a simpler equation which has one of the three highest derivative combinations specified above in examples (1), (2), and (3).

0.1.1-2. Types of equations. Characteristic equations. Consider a second-order partial differential equation with two independent variables which has the

general form

, ✞ are some functions of and that have continuous derivatives up to the second-order inclusive.*

Given a point ( ☎ , ), equation (4) is said to be

parabolic 2 if ✝ ✆ − ✞ = 0,

2 hyperbolic ✆ if − ✞ > 0, elliptic

if ✝ 2 − ✆ ✞ <0

at this point. In order to reduce equation (4) to a canonical form, one should first write out the characteristic equation

which splits into two equations

(5) and

(6) and find their general integrals.

0.1.1-3. Canonical form of parabolic equations (case ✝ 2 − ✆ ✞ = 0).

In this case, equations (5) and (6) coincide and have a common general integral,

By passing from ☎ , to new independent variables

, ✓ in accordance with the relations

= ✓ ( , ) is any twice differentiable function that satisfies the condition of nondegeneracy

( ✕ , of the Jacobian ✖ )

( ✗ , ✘ ) in the given domain, we reduce equation (4) to the canonical form

, one can take ✓ = or ✓ = . It is apparent that, just as the heat equation (1), the transformed equation (7) has only one highest-derivative term.

As ☎

1 does not depend on the derivative ✕ , equation (7) is an ordinary differential equation for the variable ✓ , in which ✒ serves as a parameter.

In the degenerate case where the function ✁

2 0.1.1-4. Canonical form of hyperbolic equations (case ✆ ✝ − ✞ > 0). The general integrals

of equations (5) and (6) are real and different. These integrals determine two different families of real characteristics.

* The right-hand side of equation (4) may be nonlinear. The classification and the procedure of reducing such equations to a canonical form are only determined by the left-hand side of the equation.

By passing from ☎ , to new independent variables

, ✓ in accordance with the relations

we reduce equation (4) to

This is the so-called first canonical form of a hyperbolic equation. The transformation

brings the above equation to another canonical form,

where ✟ 3 =4 ✟ 2 . This is the so-called second canonical form of a hyperbolic equation. Apart from notation, the left-hand side of the last equation coincides with that of the wave equation (2).

2 0.1.1-5. Canonical form of elliptic equations (case ✆ ✝ − ✞ < 0). In this case the general integrals of equations (5) and (6) are complex conjugate; these determine

two families of complex characteristics. Let the general integral of equation (5) have the form

where ☎ ( , ) and

( ✄ , ☎ ) are real-valued functions. By passing from , to new independent variables ✒ , ✓ in accordance with the relations

we reduce equation (4) to the canonical form

Apart from notation, the left-hand side of the last equation coincides with that of the Laplace equation (3).

0.1.2. Equations with Many Independent Variables

Consider a second-order partial differential equation with ✭

independent variables 1 , ✫✬✫✬✫ , that has the form ✭

where ✯ ✰ are some functions that have continuous derivatives with respect to all variables to the

, }. [The right-hand side of equation (8) may be nonlinear. The left-hand side only is required for the classification of this equation.]

second-order inclusive, and x = { ✭ 1 ,

At a point x = x 0 , the following quadratic form is assigned to equation (8): ✭

(x 0 ) ✒ ✯ ✒ ✰ .

TABLE 1 Classification of equations with many independent variables

Type of equation (8) at a point x = x 0 Coefficients of the canonical form (11) Parabolic (in the broad sense)

At least one coefficient of the ✞ ✯ is zero Hyperbolic (in the broad sense)

All ✞ ✯ are nonzero and some ✞ ✯ differ in sign Elliptic

All ✞ ✯ are nonzero and have like signs

By an appropriate linear nondegenerate transformation ✭ ✲ ✲

the quadratic form (9) can be reduced to the canonical form ✭

where the coefficients ✞ ✯ assume the values 1, −1, and 0. The number of negative and zero coefficients in (11) does not depend on the way in which the quadratic form is reduced to the canonical form.

Table 1 presents the basic criteria according to which the equations with many independent variables are classified. Suppose all coefficients of the highest derivatives in (8) are constant, ✆ ✯ ✰ = const. By introducing ✭ ✲ ✲

the new independent variables ✲

in accordance with the formulas ✯

are the coefficients of the linear transformation (10), we reduce equation (8) to the canonical ✴ form ✴

Here, the coefficients ✭

are the same as in the quadratic form (11), and y = { 1 , ✫✬✫✬✫ , }. ✲

Among the parabolic equations, it is conventional to distinguish the parabolic equations in the narrow sense, i.e., the equations for which only one of the coefficients, ✲ ✞ , is zero, while the other ✞ ✯ is the same, and in this case the right-hand side of equation (12) must contain the ☎

first-order partial derivative with respect to .

In turn, the hyperbolic equations are divided into normal hyperbolic equations— for which all ✞ ✯ but one have like signs—and ultrahyperbolic equations—for which there are two or more positive ✞ ✯ and two or more negative ✞ ✯ .

Specific equations of parabolic, elliptic, and hyperbolic types will be discussed further in Subsection 0.2. ✸✺✹

References for Section 0.1: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), S. J. Farlow (1982), D. Colton (1988), E. Zauderer (1989), A. N. Tikhonov and A. A. Samarskii (1990), I. G. Petrovsky (1991), W. A. Strauss (1992), R. B. Guenther and J. W. Lee (1996), D. Zwillinger (1998).

0.2. Basic Problems of Mathematical Physics

0.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems

Every equation of mathematical physics governs infinitely many qualitatively similar phenomena or processes. This follows from the fact that differential equations have infinitely many particular Every equation of mathematical physics governs infinitely many qualitatively similar phenomena or processes. This follows from the fact that differential equations have infinitely many particular

Throughout this section, we consider linear equations in the ✭ ✻ ✪ ✻ -dimensional Euclidean space

or in an open domain ✼ ✽ (exclusive of the boundary) with a sufficiently smooth boundary = ✼ .

0.2.1-1. Parabolic equations. Initial and boundary conditions. In general, a linear second-order partial differential equation of the parabolic type with ✪ independent

variables can be written as

Parabolic equations govern unsteady thermal, diffusion, and other phenomena dependent on time ✂

. Equation (1) is called homogeneous if ✂ ✻ ✭ ❁ (x, ) ≡ 0. ✁

). Find a function that satisfies equation (1) for > 0 and the initial condition

Cauchy problem ✂ ( ≥ 0, x

). Find a function that satisfies equation (1) for > 0, the initial condition (3), and the boundary condition

Boundary value problem ✂ *( ≥ 0, x

(4) In general, ❄ x,

is a first-order linear differential operator in the space variables x with coefficient de- ✂

pendent on x and . The basic types of boundary conditions are described below in Subsection 0.2.2. The initial condition (3) is called homogeneous if ✂ ❃ (x) ≡ 0. The boundary condition (4) is called homogeneous if ❅ (x, ) ≡ 0.

0.2.1-2. Hyperbolic equations. Initial and boundary conditions. Consider a second-order linear partial differential equation of the hyperbolic type with ✪ independent

variables of the general form

(5) where the linear differential operator ✿ ✂ x, ❀ is defined by (2). Hyperbolic equations govern unsteady

2 ✏ + (x, ) ✂ − ✿ x,

[ ]= ❁ (x, ),

wave processes, which depend on time . Equation (5) is said to be homogeneous if ✂

(x, ) ≡ 0. ✁

). Find a function that satisfies equation (5) for > 0 and the initial conditions

Cauchy problem ✂ ( ≥ 0, x

* Boundary value problems for parabolic and hyperbolic equations are sometimes called mixed or initial-boundary value problems .

). Find a function that satisfies equation (5) for > 0, the initial conditions (6), and boundary condition (4). The initial conditions (6) are called homogeneous if ❃ 0 (x) ≡ 0 and ❃ 1 (x) ≡ 0. Goursat problem. On the characteristics of a hyperbolic equation with two independent variables, ✁

Boundary value problem ✂ ( ≥ 0, x

the values of the unknown function are prescribed.

0.2.1-3. Elliptic equations. Boundary conditions. In general, a second-order linear partial differential equation of elliptic type with ✪ independent

variables can be written as

(7) where

x [ ]= ❁ (x),

Elliptic equations govern steady-state thermal, diffusion, and other phenomena independent of time ✂ . Equation (7) is said to be homogeneous if ✁ ❁ (x) ≡ 0.

Boundary value problem . Find a function that satisfies equation (7) and the boundary condition

(9) In general, ❄ x is a first-order linear differential operator in the space variables x. The basic types of boundary conditions are described below in Subsection 0.2.2.

x ✾ [ ]=

(x) at x ✽

The boundary condition (9) is called homogeneous if ❅ (x) ≡ 0. The boundary value problem (7)–(9) is said to be homogeneous if ❁ ≡ 0 and ❅ ≡ 0.

0.2.2. First, Second, Third, and Mixed Boundary Value Problems

For any (parabolic, hyperbolic, and elliptic) second-order partial differential equations, it is con- ventional to distinguish four basic types of boundary value problems, depending on the form of the boundary conditions (4) [see also the analogous equation (9)]. For simplicity, here we confine ✆

ourselves to the case where the coefficients ✯ ✰ of equations (1) and (5) have the special form

This situation is rather frequent in applications; such coefficients are used to describe various phenomena (processes) in isotropic media.

First boundary value problem. ✾ The function (x, ) takes prescribed values at the boundary of the domain,

(10) Second boundary value problem. ✾ The derivative along the (outward) normal is prescribed at the

(x, ✾ )=

1 (x, ) for x ✽

boundary of the domain,

(11) In heat transfer problems, where ✁ is temperature, the left-hand side of the boundary condition (11)

= ❅ 2 (x, ) for x ✽

is proportional to the heat flux per unit area of the surface ✾ .

Third boundary value problem.

A linear relationship between the unknown function and its ✾

normal derivative is prescribed at the boundary of the domain,

+ ❊ (x, ) = ❅ 3 (x, ) for x ✽

Usually, it is assumed that ✁

(x, ) = const. In mass transfer problems, where is concentration, the boundary condition (12) with ❅ 3 ≡ 0 describes a surface chemical reaction of the first order. Mixed boundary value problems. ✾ Conditions of various types, listed above, are set at different portions of the boundary . If ❅ 1 ≡ 0, ❅ 2 ≡ 0, or ❅ 3 ≡ 0, the respective boundary conditions (10), (11), (12) are said to be homogeneous. ✸✺✹

References for Section 0.2: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), M. A. Pinsky (1984), R. Leis (1986), R. Haberman (1987), A. A. Dezin (1987), A. G. Mackie (1989), A. N. Tikhonov and A. A. Samarskii (1990), I. Stakgold (2000).

0.3. Properties and Particular Solutions of Linear Equations

0.3.1. Homogeneous Linear Equations

0.3.1-1. Preliminary remarks. For brevity, in this paragraph a homogeneous linear partial differential equation will be written as

For second-order linear parabolic and hyperbolic equations, the linear differential operator ✁ [ ] is defined by the left-hand side of equations (1) and (5) from Subsection 0.2.1, respectively. It is

assumed that equation (1) is an arbitrary homogeneous linear partial differential equation of any ✂ ✄ ✄ ✭

order in the variables ❋ , 1 , ✫✬✫✬✫ ,

with sufficiently smooth coefficients.

A linear operator possesses the properties

An arbitrary homogeneous linear equation (1) has a trivial solution, ✁

A function is called a classical solution of equation (1) if ✁ , when substituted into (1), turns the equation into an identity and if all partial derivatives of that occur in (1) are continuous; the notion of a classical solution is directly linked to the range of the independent variables. In what follows, we usually write “solution” instead of “classical solution” for brevity.

0.3.1-2. Usage of particular solutions for the construction of other particular solutions.

Below are some properties of particular solutions of homogeneous linear equations. ✲

(x, ) be any particular solutions of the homogeneous equation (1). Then the linear combination

. Let 1 = 1 (x, ), 2 = 2 (x, ), ✫✬✫✬✫ ,

with arbitrary constants ● 1 , ● 2 , ✫✬✫✬✫ , ●

is also a solution of equation (1); in physics, this property

is known as the principle of linear superposition. ✲

Suppose { ✁ } is an infinite sequence of solutions of equation (1). Then the series ✲

irrespective of its convergence, is called a formal solution of (1). If the solutions are classical, the series is uniformly convergent, and the sum of the series has all the necessary particular derivatives, then the sum of the series is a classical solution of equation (1).

. Let the coefficients of the differential operator ✁ ✁ ✂

be independent of time ✁ . If equation (1) has

a particular solution ❑ = ❑ (x, ), then the partial derivatives of ❑ with respect to time,*

are also solutions of equation (1)

3 ❍ . Let the coefficients of the differential operator ✄ ✭ ✁

be independent of the space variables ✂ ✁

1 , ✫✬✫✬✫ , . If equation (1) has a particular solution ❑ = ❑ (x, ), then the partial derivatives of ✲ ❑

with respect to the space coordinates,

are also solutions of equation (1) ❋

are independent of only one space coordinate, say ✁ ✂ ✲ 1 , and equation (1) has a particular solution ❑ = ❑ (x, ), then the partial derivatives

If the coefficients of ✄

are also solutions of equation (1).

be constant and let equation (1) have a particular solution ✁ ❑ = ❑ (x, ). Then any particular derivatives of ❑ with respect to time and the space coordinates (inclusive mixed ✲

. Let the coefficients of

are solutions of equation (1).

. Suppose equation (1) has a particular solution dependent on a parameter ❋ ◆ , ❑ = ❑ (x, ; ◆ ), and the coefficients of ✁ are independent of ◆ (but can depend on time and the space coordinates). Then, ✲

by differentiating ❑ with respect to ◆ , one obtains other solutions of equation (1),

Let some constants ◆ 1 , ✫✬✫✬✫ , ◆ belong to the range of the parameter ✲ ✲ ◆ . Then the sum

are arbitrary constants, is also a solution of the homogeneous linear equation (1). The number of terms in sum (3) can be both finite and infinite.

6 ✁ ❍ . Another effective way of constructing solutions involves the following. The particular solution ✂ ❋

(x, ; ◆ ), which depends on the parameter ◆ (as before, it is assumed that the coefficients of are independent of ✏ ◆ ), is first multiplied by an arbitrary function ( ◆ ). Then the resulting expression is

integrated with respect to ◆ over some interval [ ❖ , ]. Thus, one obtains a new function,

(x, ; ◆

which is also a solution of the original homogeneous linear equation. The properties listed in Items 1 ❍ –6 ❍ enable one to use known particular solutions to construct

other particular solutions of homogeneous linear equations of mathematical physics.

* Here and in what follows, it is assumed that the particular solution ❙ ❚ is differentiable sufficiently many times with respect to ❯ and ❱ 1 , ❲▼❲▼❲ , ❱ ❳ (or the parameters).

TABLE 2 Homogeneous linear partial differential equations that admit separable solutions

No Form of equation (1) Form of particular solutions

Equation coefficients are ✭ (x, )=

exp( ❨ + 1 1 + ■✬■✬■ + ), constant

are related by an algebraic equation ✴ ✴

Equation coefficients are ✂

(x, ❀ )= ❩ ✧ (x),

independent of time ✭

is an arbitrary constant, x = { 1 , ✫✬✫✬✫ , }

Equation coefficients are independent ✂

(x, ) = exp( ✭ 1 1 + ■✬■✬■ + ) ✧ ( ),

of the coordinates 1 , ✫✬✫✬✫ ,

are arbitrary constants ✴ ✲ ✲ ✴ ✲

Equation coefficients are independent ✭

(x, ) = exp( 1 1 + ■✬■✬■ +

of the coordinates 1 , ✫✬✫✬✫ ,

are arbitrary constants ✴

(x, )= ( ✧ (x),

depends on only , ( ) satisfies the equation ❀ [ ]+ = 0, operator ✿ x depends on only x

5 ✏ operator

(x) satisfies the equation ✿ x [ ✧ ]− ❨ ✧ =0

1 (x, )= ( ) 1 ( 1 ) ( ), [ ✲ ]+ [ ]+ ■✬■✬■ + ✿ ✂ ✲ [ ],

( ) satisfies the equation ✿ ❀ [ ]+ ❨ ✏ = 0,

6 operator ❀ depends on only ✄ ,

( ) satisfies the equation ✿ ✭ [ ✧ ]+ ✧ = 0, operator

depends on only

]+ ✲ = 0, ✭ ❊ = 2, ✫✬✫✬✫ , ✪ , operator ✿ ❀ depends on only ✄ ,

operator ✵

depends on only

0.3.1-3. Separable solutions. Many homogeneous linear partial differential equations have solutions that can be represented as the

product of functions depending on different arguments. Such solutions are referred to as separable solutions.

Table 2 presents the most commonly encountered types of homogeneous linear differential equa- tions with many independent variables that admit exact separable solutions. Linear combinations of ✭

particular solutions that correspond to different values of the separation parameters, ❨ , 1 , ✫✬✫✬✫ , , are also solutions of the equations in question. For brevity, the word “operator” is used to denote ✴ ✴

“linear differential operator.” For a constant coefficient equation (see the first row in Table 2 ), the separation parameters must satisfy the algebraic equation

which results from substituting the solution into the equation (1). In physical applications, equa- ✴ tion (4) is usually referred to as a variance equation. Any ✪ of the ✪ + 1 separation parameters in (4)

can be treated as arbitrary.

Note that constant coefficient equations also admit more sophisticated solutions; see the second and third rows, the last column. The eighth row of Table 2 presents the case of incomplete separation of variables where the ✭

solution is separated with respect to the space variables ❫ 1 , ✫✬✫✬✫ , ❫ but is not separated with respect to time ✙ ✚✜✛ ❜ . ✢ ✣✥✤ ✦

For stationary equations, which do not depend on ❜ , one should set ❨ = 0, ✿ ❀ [ ❵ ] ≡ 0, and ( ❜ ) ≡ 1 in rows 1, 6, and 7 of Table 2 .

0.3.1-4. Solutions in the form of infinite series in ❜ .

1 ❍ . The equation

where ❜ ❣ is an arbitrary linear differential operator of the second (or any) order that only depends

on the space variables, has the formal series solution ✲

where ❃ (x) is an arbitrary infinitely differentiable function. This solution satisfies the initial condition

(x, 0) = ❃ (x).

2 ❍ . The equation

where ❜ ❣ ❛ is a linear differential operator, just as in Item 1 ❍

, has a formal solution represented by the ✲ ✲

sum of two series as

where ❃ (x) and ❅ (x) are arbitrary infinitely differentiable functions. This solution satisfies the initial conditions ❵ (x, 0) = ❃ (x) and ❀ ❵ ❛ (x, 0) = ❅ (x).

0.3.2. Nonhomogeneous Linear Equations

0.3.2-1. Simplest properties of nonhomogeneous linear equations. For brevity, we write a nonhomogeneous linear partial differential equation in the form

(5) where the linear differential operator ❋ is defined above, see the beginning of Paragraph 0.3.1-1.

[ ❵ ]= ❁ (x, ❜ ),

Below are the simplest properties of particular solutions of the nonhomogeneous equation (5).

) is a particular solution of the nonhomogeneous equation (5) and 0 (x, ❜ ) is a particular solution of the corresponding homogeneous equation (1), then the sum

where ● is an arbitrary constant, is also a solution of the nonhomogeneous equation (5). The following, more general statement holds: The general solution of the nonhomogeneous equation (5) is the sum of the general solution of the corresponding homogeneous equation (1) and any particular solution of the nonhomogeneous equation (5).

2 ❍ . Suppose ❵ 1 and ❵ 2 are solutions of nonhomogeneous linear equations with the same left-hand side and different right-hand sides, i.e.,

1 ]= ❁ 1 (x, ),

[ ❜ ❵ 2 ]= ❁ 2 (x, ❜ ).

Then the function ❵ = ❵ 1 + ❵ 2 is a solution of the equation

[ ❵ ]= ❁ 1 (x, ❜ )+ ❁ 2 (x, ❜ ).

0.3.2-2. Fundamental and particular solutions of stationary equations. Consider the second-order linear stationary (time-independent) nonhomogeneous equation

(6) Here, ✿ x is a linear differential operator of the second (or any) order of general form whose ✭ ✻

x [ ❵ ]= ❁ (x).

coefficients are dependent on x, where x ✽

A distribution ❦ ❦ = ❦ ❦ (x, y) that satisfies the equation with a special right-hand side

(7) is called a fundamental solution corresponding to the operator ✿ x . In (7), ❆ (x) is an ❧ -dimensional

x [ ❦ ❦ ]= ❆ (x − y)

Dirac delta function and the vector quantity y = { ♠ 1 , ♥✬♥✬♥ , ♠ ✻ ♦ ♦ } appears in equation (7) as an ❧ -dimensional free parameter. It is assumed that y ✽

The ❧ -dimensional Dirac delta function possesses the following basic properties:

1. ❆ (x) = ❆ ( ❫ 1 ) ❆ ( ❫ 2 ) ♥✬♥✬♥♣❆ ( ❫ P ♦ q r ),

2. ❁ (y) ❆ (x − y) s y= ✲ ❁ (x), where ❆ ( ❫ ) is the one-dimensional Dirac delta function, ❁ (x) is an arbitrary continuous function,

and s y= s ♠ 1 ♥✬♥✬♥✥s ♠ ♦ . For constant coefficient equations, a fundamental solution always exists; it can be found by means of the ❧ -dimensional Fourier transform (see Paragraph 0.5.3-2). The fundamental solution ❦ ❦ = ❦ ❦ (x, y) can be used to construct a particular solution of the linear stationary nonhomogeneous equation (6) for arbitrary continuous ❁ (x); this particular solution is expressed as follows: