# Handbook of Linear Partial Differential Equations for Engineers and Scientists, Second Edition

## Preface

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 (R)

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.

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## About the Author

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.

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## Table of Contents

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 = h1(x,y)w + h0(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 f1 w/ x + f2 w/ y + f3 w/ z = f4, fn = fn(x,y,z)

Equations of the Form f1 w/ x + f2 w/ y + f3 w/ z = f4w, fn = fn(x,y,z)

Equations of the Form f1 w/ x + f2 w/ y + f3 w/ z = f4w + f5, fn = fn(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 2w

Heat Equation with a Source w/ t = a 2w + (x,y,t)

Other Equations

Second-Order Parabolic Equations with Three or More Space Variables

Heat Equation w/ t = a 3w

Heat Equation with Source w/ t = a 3w + (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 2w/ t2 = a2 2w

Nonhomogeneous Wave Equation 2w/ t2 = a2 2w + (x,y,t)

Equations of the Form 2w/ t2 = a2 2w bw + (x,y,t)

Telegraph Equation 2w/ t2 + k( w/ t) = a2 2w bw + (x,y,t)

Other Equations with Two Space Variables

Second-Order Hyperbolic Equations with Three or More Space Variables

Wave Equation 2w/ t2 = a2 3w

Nonhomogeneous Wave Equation 2w/ t2 = a2 3+ (x,y,z,t)Equations of the Form 2w/ t2 = a2 3w bw + (x,y,z,t)

Telegraph Equation 2w/ t2 + k( w/ t) = a2 3w 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 2w = 0

Poisson Equation 2w = (x)

Helmholtz Equation 2w + w = (x)

Other Equations

Second-Order Elliptic Equations with Three or More Space Variables

Laplace Equation 3w = 0

Poisson Equation 3w = (x)

Helmholtz Equation 3w + w = (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 (R)

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 (R)

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

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