2nd Edition

Handbook of Linear Partial Differential Equations for Engineers and Scientists

    1644 Pages 28 B/W Illustrations
    by Chapman & Hall

    • 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

    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.

    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