# Handbook of Linear Partial Differential Equations for Engineers and Scientists

## Preview

## Book Description

- Includes nearly 4,000 linear partial differential equations (PDEs) with solutions
- Presents solutions of numerous problems relevant to heat and mass transfer, wave theory, hydrodynamics, aerodynamics, elasticity, acoustics, electrodynamics, diffraction theory, quantum mechanics, chemical engineering sciences, electrical engineering, and other fields
- Outlines basic methods for solving various problems in science and engineering
- Contains much more linear equations, problems, and solutions than any other book currently available
- Provides a database of test problems for numerical and approximate analytical methods for solving linear PDEs and systems of coupled PDEs

New to the Second Edition

- More than 700 pages with 1,500+ new first-, second-, third-, fourth-, and higher-order linear equations with solutions
- Systems of coupled PDEs with solutions
- Some analytical methods, including decomposition methods and their applications
- Symbolic and numerical methods for solving linear PDEs with Maple, Mathematica, and MATLAB
^{®} - Many new problems, illustrative examples, tables, and figures

To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the material in increasing order of complexity.

## Table of Contents

**Exact Solutions**

**First-Order Equations with Two Independent Variables**Equations of the Form

*f*(

*x,y*)

*∂w/∂x + g*(

*x,y*)

*∂w/∂y =*0

**Equations of the Form**

*f*(

*x,y*)

*∂w/∂x + g*(

*x,y*)

*∂w/∂y = h*(

*x,y*)

**Equations of the Form**

*f*(

*x,y*)

*∂w/∂x + g*(

*x,y*)

*∂w/∂y = h*(

*x,y*)

*w*

**Equations of the Form**

*f*(

*x,y*)

*∂w/∂x + g*(

*x,y*)

*∂w/∂y = h*

_{1}(

*x,y*)

*w + h*

_{0}(

*x,y*)

**First-Order Equations with Three or More Independent Variables**Equations of the Form

*f*(

*x,y,z*)

*∂w/∂x + g*(

*x,y,z*)

*∂w/∂y + h*(

*x,y,z*)

*∂w/∂z =*0

**Equations of the Form**

*f*

_{1}

*∂w/∂x + f*

_{2}

*∂w/∂y + f*

_{3}

*∂w/∂z = f*

_{4}

*, f*(

_{n}= f_{n}*x,y,z*)

**Equations of the Form**

*f*

_{1}

*∂w/∂x + f*

_{2}

*∂w/∂y + f*

_{3}

*∂w/∂z = f*

_{4}

*w, f*(

_{n}= f_{n}*x,y,z*)

**Equations of the Form**

*f*

_{1}

*∂w/∂x + f*

_{2}

*∂w/∂y + f*

_{3}

*∂w/∂z = f*

_{4}

*w + f*

_{5}

*, f*(

_{n}= f_{n}*x,y,z*)

**Second-Order Parabolic Equations with One Space Variable**Constant Coefficient Equations

**Heat Equation with Axial or Central Symmetry and Related Equations**

**Equations Containing Power Functions and Arbitrary Parameters**

**Equations Containing Exponential Functions and Arbitrary Parameters**

**Equations Containing Hyperbolic Functions and Arbitrary Parameters**

**Equations Containing Logarithmic Functions and Arbitrary Parameters**

**Equations Containing Trigonometric Functions and Arbitrary Parameters**

**Equations Containing Arbitrary Functions**

**Equations of Special Form**

**Second-Order Parabolic Equations with Two Space Variables**Heat Equation

*∂w/∂t = a∆*

_{2}w**Heat Equation with a Source**

*∂w/∂t = a∆*Փ(

_{2}w +*x,y,t*)

**Other Equations**

**Second-Order Parabolic Equations with Three or More Space Variables**Heat Equation

*∂w/∂t = a∆*

_{3}w**Heat Equation with Source**

*∂w/∂t = a∆*Փ(

_{3}w +*x,y,z,t*)

**Other Equations with Three Space Variables**

**Equations with**

*n*Space Variables

**Second-Order Hyperbolic Equations with One Space Variable**Constant Coefficient Equations

**Wave Equation with Axial or Central Symmetry**

**Equations Containing Power Functions and Arbitrary Parameters**

**Equations Containing the First Time Derivative**

**Equations Containing Arbitrary Functions**

**Second-Order Hyperbolic Equations with Two Space Variables**Wave Equation

*∂*

^{2}w/∂t^{2}= a^{2}∆_{2}w**Nonhomogeneous Wave Equation**

*∂*Փ(

^{2}w/∂t^{2}= a^{2}∆_{2}w +*x,y,t*)

**Equations of the Form**

*∂*Փ(

^{2}w/∂t^{2}= a^{2}∆_{2}w − bw +*x,y,t*)

**Telegraph Equation**

*∂*Փ(

^{2}w/∂t^{2}+ k(∂w/∂t) = a^{2}∆_{2}w − bw +*x,y,t*)

**Other Equations with Two Space Variables**

**Second-Order Hyperbolic Equations with Three or More Space Variables**Wave Equation

*∂*

^{2}w/∂t^{2}= a^{2}∆_{3}w**Nonhomogeneous Wave Equation**

*∂*Փ(

^{2}w/∂t^{2}= a^{2}∆_{3}+*x,y,z,t*)Equations of the Form

*∂*Փ(

^{2}w/∂t^{2}= a^{2}∆_{3}w − bw +*x,y,z,t*)

**Telegraph Equation**

*∂*Փ(

^{2}w/∂t^{2}+ k(∂w/∂t) = a^{2}∆_{3}w − bw +*x,y,z,t*)

*)*

**Other Equations with Three Space Variables**

**Equations with**

*n*Space Variables

**Second-Order Elliptic Equations with Two Space Variables**Laplace Equation

*∆*0

_{2}w =**Poisson Equation**

*∆*Փ(

_{2}w = −**x**)

**Helmholtz Equation**

*∆*Փ(

_{2}w + λw = −**x**)

**Other Equations**

**Second-Order Elliptic Equations with Three or More Space Variables**Laplace Equation

*∆*0

_{3}w =**Poisson Equation**

*∆*Փ(

_{3}w = −**x**)

**Helmholtz Equation**

*∆*Փ(

_{3}w + λw = −**x**)

**Other Equations with Three Space Variables**

**Equations with**

*n*Space Variables

**Higher-Order Partial Differential Equations**Third-Order Partial Differential Equations

**Fourth-Order One-Dimensional Nonstationary Equations**

**Two-Dimensional Nonstationary Fourth-Order Equations**

**Three- and**

*n*-Dimensional Nonstationary Fourth-Order Equations

**Fourth-Order Stationary Equations**

**Higher-Order Linear Equations with Constant Coefficients**

**Higher-Order Linear Equations with Variable Coefficients**

**Systems of Linear Partial Differential Equations**Preliminary Remarks. Some Notation and Helpful Relations

**Systems of Two First-Order Equations**

**Systems of Two Second-Order Equations**

**Systems of Two Higher-Order Equations**

**Simplest Systems Containing Vector Functions and Operators div and curl**

**Elasticity Equations**

**Stokes Equations for Viscous Incompressible Fluids**

**Oseen Equations for Viscous Incompressible Fluids**

**Maxwell Equations for Viscoelastic Incompressible Fluids**

**Equations of Viscoelastic Incompressible Fluids (General Model)**

**Linearized Equations for Inviscid Compressible Barotropic Fluids**

**Stokes Equations for Viscous Compressible Barotropic Fluids**

**Oseen Equations for Viscous Compressible Barotropic Fluids**

**Equations of Thermoelasticity**

**Nondissipative Thermoelasticity Equations (the Green–Naghdi Model)**

**Viscoelasticity Equations**

**Maxwell Equations (Electromagnetic Field Equations)**

**Vector Equations of General Form**

**General Systems Involving Vector and Scalar Equations: Part I**

**General Systems Involving Vector and Scalar Equations: Part II**

Analytical Methods

**Methods for First-Order Linear PDEs**Linear PDEs with Two Independent Variables

**First-Order Linear PDEs with Three or More Independent Variables**

**Second-Order Linear PDEs: Classification, Problems, Particular Solutions**Classification of Second-Order Linear Partial Differential Equations

**Basic Problems of Mathematical Physics**

**Properties and Particular Solutions of Linear Equations**

**Separation of Variables and Integral Transform Methods**Separation of Variables (Fourier Method)

**Integral Transform Method**

**Cauchy Problem. Fundamental Solutions**Dirac Delta Function. Fundamental Solutions

**Representation of the Solution of the Cauchy Problem via the Fundamental Solution**

**Boundary Value Problems. Green’s Function**Boundary Value Problems for Parabolic Equations with One Space Variable. Green’s Function

**Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s Function. Goursat Problem**

**Boundary Value Problems for Elliptic Equations with Two Space Variables**

**Boundary Value Problems with Many Space Variables. Green’s Function**

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

**Duhamel’s Principles. Some Transformations**Duhamel’s Principles in Nonstationary Problems

**Transformations Simplifying Initial and Boundary Conditions**

**Systems of Linear Coupled PDEs. Decomposition Methods**Asymmetric and Symmetric Decompositions

**First-Order Decompositions. Examples**

**Higher-Order Decompositions**

**Some Asymptotic Results and Formulas**Regular Perturbation Theory Formulas for the Eigenvalues

**Singular Perturbation Theory**

**Elements of Theory of Generalized Functions**Generalized Functions of One Variable

**Generalized Functions of Several Variables**

Symbolic and Numerical Solutions with Maple, Mathematica, and MATLAB^{® }

**Linear Partial Differential Equations with Maple**Introduction

**Analytical Solutions and Their Visualizations**

**Analytical Solutions of Mathematical Problems**

**Numerical Solutions and Their Visualizations**

**Linear Partial Differential Equations with Mathematica**Introduction

**Analytical Solutions and Their Visualizations**

**Analytical Solutions of Mathematical Problems**

**Numerical Solutions and Their Visualizations**

**Linear Partial Differential Equations with MATLAB ^{®}**Introduction

**Numerical Solutions of Linear PDEs**

**Constructing Finite-Difference Approximations**

**Numerical Solutions of Systems of Linear PDEs**

Tables and Supplements

**Elementary Functions and Their Properties**Power, Exponential, and Logarithmic Functions

**Trigonometric Functions**

**Inverse Trigonometric Functions**

**Hyperbolic Functions**

**Inverse Hyperbolic Functions**

**Finite Sums and Infinite Series**Finite Numerical Sums

**Finite Functional Sums**

**Infinite Numerical Series**

**Infinite Functional Series**

**Indefinite and Definite Integrals**Indefinite Integrals

**Definite Integrals**

**Integral Transforms**Tables of Laplace Transforms

**Tables of Inverse Laplace Transforms**

Tables of Fourier Cosine Transforms

**Tables of Fourier Sine Transforms**

**Curvilinear Coordinates, Vectors, Operators, and Differential Relations**Arbitrary Curvilinear Coordinate Systems

**Cartesian, Cylindrical, and Spherical Coordinate Systems**

**Other Special Orthogonal Coordinates**

**Special Functions and Their Properties**Some Coefficients, Symbols, and Numbers

**Error Functions. Exponential and Logarithmic Integrals**

**Sine Integral and Cosine Integral. Fresnel Integrals**

**Gamma Function, Psi Function, and Beta Function**

**Incomplete Gamma and Beta Functions**

**Bessel Functions (Cylindrical Functions)**

**Modified Bessel Functions**

**Airy Functions**

**Degenerate Hypergeometric Functions (Kummer Functions)**

**Hypergeometric Functions**

**Legendre Polynomials, Legendre Functions, and Associated Legendre Functions**

**Parabolic Cylinder Functions**

**Elliptic Integrals**

**Elliptic Functions**

**Jacobi Theta Functions**

**Mathieu Functions and Modified Mathieu Functions**

**Orthogonal Polynomials**

**Nonorthogonal Polynomials**

References

Index

## Author(s)

### Biography

**Andrei D. Polyanin**, D.Sc., is an internationally renowned scientist of broad interests and is active in various areas of mathematics, mechanics, and chemical engineering sciences. He is one of the most prominent authors in the field of reference literature on mathematics. Professor Polyanin graduated with honors from the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University in 1974. He received his Ph.D. in 1981 and D.Sc. in 1986 at the Institute for Problems in Mechanics of the Russian Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences. He is also professor of applied mathematics at Bauman Moscow State Technical University and at National Research Nuclear University MEPhI. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. Professor Polyanin has authored more than 30 books in English, Russian, German, and Bulgarian as well as more than 170 research papers, three patents, and a number of fundamental handbooks. Professor Polyanin is editor-in-chief of the website *EqWorld—The World of Mathematical Equations*, editor of the book series *Differential and Integral Equations and Their Applications*, and a member of the editorial board of the journals *Theoretical Foundations of Chemical Engineering*, *Mathematical Modeling and Computational Methods*, and *Bulletin of the National Research Nuclear University MEPhI*. In 1991, Professor Polyanin was awarded the Chaplygin Prize of the Russian Academy of Sciences for his research in mechanics. In 2001, he received an award from the Ministry of Education of the Russian Federation.

**Vladimir E. Nazaikinskii**, D.Sc., is an actively working mathematician specializing in partial differential equations, mathematical physics, and noncommutative analysis. He was born in 1955 in Moscow, graduated from the Moscow Institute of Electronic Engineering in 1977, defended his Ph.D. in 1980 and D.Sc. in 2014, and worked at the Institute for Automated Control Systems, Moscow Institute of Electronic Engineering, Potsdam University, and Moscow State University. Currently he is a senior researcher at the Institute for Problems in Mechanics, Russian Academy of Sciences. He is the author of seven monographs and more than 90 papers on various aspects of noncommutative analysis, asymptotic problems, and elliptic theory.

### Featured Author Profiles

## Reviews

Praise for the Previous Edition"… one-stop shopping for scientists and engineers who need a cookbook solution for partial differential equations. The logical organization—by type of equation … and number of variables—makes finding entries easy. … This very useful book has no competitors."

—CHOICE, October 2002

"… a good example of a reference information resource named 'Handbook.' It is an information tool: comprehensive, condensed, descriptive in 'Contents,' authoritative, and practical. … In one volume it contains over 2,000 solutions to linear partial differential equations. … It is not a solution manual to accompany a textbook, but an information resource of advanced level for professionals. … a great addition for research and academic collections."

—E-Streams, Vol. 6, No. 2

"… I have been reading the Polyanin booksHandbook of Linear Partial Differential Equations for Engineers andScientistsandHandbook of Exact Solutions for Ordinary Differential Equations. I think these books are extraordinary, and are destined to become classics. … CRC Press has provided an invaluable service to science and engineering by publishing these books."

—William Schiesser, Lehigh University, Bethlehem, Pennsylvania, USA