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 = 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 = a2∆2w
Nonhomogeneous Wave Equation ∂2w/∂t2 = a2∆2w + Փ(x,y,t)
Equations of the Form ∂2w/∂t2 = a2∆2w − bw + Փ(x,y,t)
Telegraph Equation ∂2w/∂t2 + k(∂w/∂t) = a2∆2w − 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 = a2∆3w
Nonhomogeneous Wave Equation ∂2w/∂t2 = a2∆3+ Փ(x,y,z,t)Equations of the Form ∂2w/∂t2 = a2∆3w − bw + Փ(x,y,z,t)
Telegraph Equation ∂2w/∂t2 + k(∂w/∂t) = a2∆3w − 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