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2nd Edition

# Handbook of Linear Partial Differential Equations for Engineers and Scientists

By

,

## Vladimir E. Nazaikinskii

ISBN 9781466581456
Published January 15, 2016 by Chapman and Hall/CRC
1643 Pages 28 B/W Illustrations

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## 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.

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 = 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 + Փ(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 + Փ(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 = a22w
Nonhomogeneous Wave Equation 2w/∂t2 = a22w + Փ(x,y,t)
Equations of the Form 2w/∂t2 = a22w − bw + Փ(x,y,t)
Telegraph Equation 2w/∂t2 + k(∂w/∂t) = a22w − bw + Փ(x,y,t)
Other Equations with Two Space Variables

Second-Order Hyperbolic Equations with Three or More Space Variables
Wave Equation 2w/∂t2 = a23w
Nonhomogeneous Wave Equation 2w/∂t2 = a23+ Փ(x,y,z,t)Equations of the Form 2w/∂t2 = a23w − bw + Փ(x,y,z,t)
Telegraph Equation 2w/∂t2 + k(∂w/∂t) = a23w − bw + Փ(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 = − Փ(x)
Helmholtz Equation 2w + λw = − Փ(x)
Other Equations

Second-Order Elliptic Equations with Three or More Space Variables
Laplace Equation 3w = 0
Poisson Equation 3w = − Փ(x)
Helmholtz Equation 3w + λw = − Փ(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.

## 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 books Handbook of Linear Partial Differential Equations for Engineers and Scientists and Handbook 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
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