Exact Solutions First-Order Equations with Two Independent VariablesEquations of the Form f(x,y)?w/?x + g(x,y)?w/?y = 0Equations 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)wEquations 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 VariablesEquations of the Form f(x,y,z)?w/?x + g(x,y,z)?w/?y + h(x,y,z)?w/?z = 0Equations 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 VariableConstant Coefficient EquationsHeat Equation with Axial or Central Symmetry and Related EquationsEquations Containing Power Functions and Arbitrary ParametersEquations Containing Exponential Functions and Arbitrary ParametersEquations Containing Hyperbolic Functions and Arbitrary ParametersEquations Containing Logarithmic Functions and Arbitrary ParametersEquations Containing Trigonometric Functions and Arbitrary ParametersEquations Containing Arbitrary FunctionsEquations of Special Form Second-Order Parabolic Equations with Two Space VariablesHeat Equation ?w/?t = a?2wHeat Equation with a Source ?w/?t = a?2w + ?(x,y,t)Other Equations Second-Order Parabolic Equations with Three or More Space VariablesHeat Equation ?w/?t = a?3wHeat Equation with Source ?w/?t = a?3w + ?(x,y,z,t)Other Equations with Three Space VariablesEquations with n Space Variables Second-Order Hyperbolic Equations with One Space VariableConstant Coefficient EquationsWave Equation with Axial or Central SymmetryEquations Containing Power Functions and Arbitrary ParametersEquations Containing the First Time DerivativeEquations Containing Arbitrary Functions Second-Order Hyperbolic Equations with Two Space VariablesWave Equation ?2w/?t2 = a2?2wNonhomogeneous 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 VariablesWave Equation ?2w/?t2 = a2?3wNonhomogeneous 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 VariablesEquations with n Space Variables Second-Order Elliptic Equations with Two Space VariablesLaplace Equation ?2w = 0Poisson Equation ?2w = - ?(x)Helmholtz Equation ?2w + ?w = - ?(x)Other Equations Second-Order Elliptic Equations with Three or More Space VariablesLaplace Equation ?3w = 0Poisson Equation ?3w = - ?(x)Helmholtz Equation ?3w + ?w = - ?(x)Other Equations with Three Space VariablesEquations with n Space Variables Higher-Order Partial Differential EquationsThird-Order Partial Differential EquationsFourth-Order One-Dimensional Nonstationary EquationsTwo-Dimensional Nonstationary Fourth-Order EquationsThree- and n-Dimensional Nonstationary Fourth-Order EquationsFourth-Order Stationary EquationsHigher-Order Linear Equations with Constant CoefficientsHigher-Order Linear Equations with Variable Coefficients Systems of Linear Partial Differential EquationsPreliminary Remarks. Some Notation and Helpful RelationsSystems of Two First-Order EquationsSystems of Two Second-Order EquationsSystems of Two Higher-Order EquationsSimplest Systems Containing Vector Functions and Operators div and curlElasticity EquationsStokes Equations for Viscous Incompressible FluidsOseen Equations for Viscous Incompressible FluidsMaxwell Equations for Viscoelastic Incompressible FluidsEquations of Viscoelastic Incompressible Fluids (General Model)Linearized Equations for Inviscid Compressible Barotropic FluidsStokes Equations for Viscous Compressible Barotropic FluidsOseen Equations for Viscous Compressible Barotropic FluidsEquations of ThermoelasticityNondissipative Thermoelasticity Equations (the Green–Naghdi Model)Viscoelasticity EquationsMaxwell Equations (Electromagnetic Field Equations)Vector Equations of General FormGeneral Systems Involving Vector and Scalar Equations: Part IGeneral Systems Involving Vector and Scalar Equations: Part II Analytical Methods Methods for First-Order Linear PDEsLinear PDEs with Two Independent VariablesFirst-Order Linear PDEs with Three or More Independent Variables Second-Order Linear PDEs: Classification, Problems, Particular SolutionsClassification of Second-Order Linear Partial Differential EquationsBasic Problems of Mathematical PhysicsProperties and Particular Solutions of Linear Equations Separation of Variables and Integral Transform MethodsSeparation of Variables (Fourier Method)Integral Transform Method Cauchy Problem. Fundamental SolutionsDirac Delta Function. Fundamental SolutionsRepresentation of the Solution of the Cauchy Problem via the Fundamental Solution Boundary Value Problems. Green’s FunctionBoundary Value Problems for Parabolic Equations with One Space Variable. Green’s FunctionBoundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s Function. Goursat ProblemBoundary Value Problems for Elliptic Equations with Two Space VariablesBoundary Value Problems with Many Space Variables. Green’s FunctionConstruction of the Green’s Functions. General Formulas and Relations Duhamel’s Principles. Some TransformationsDuhamel’s Principles in Nonstationary ProblemsTransformations Simplifying Initial and Boundary Conditions Systems of Linear Coupled PDEs. Decomposition MethodsAsymmetric and Symmetric DecompositionsFirst-Order Decompositions. ExamplesHigher-Order Decompositions Some Asymptotic Results and FormulasRegular Perturbation Theory Formulas for the EigenvaluesSingular Perturbation Theory Elements of Theory of Generalized FunctionsGeneralized Functions of One VariableGeneralized Functions of Several Variables Symbolic and Numerical Solutions with Maple, Mathematica, and MATLAB® Linear Partial Differential Equations with MapleIntroductionAnalytical Solutions and Their VisualizationsAnalytical Solutions of Mathematical ProblemsNumerical Solutions and Their Visualizations Linear Partial Differential Equations with MathematicaIntroductionAnalytical Solutions and Their VisualizationsAnalytical Solutions of Mathematical ProblemsNumerical Solutions and Their Visualizations Linear Partial Differential Equations with MATLAB®IntroductionNumerical Solutions of Linear PDEsConstructing Finite-Difference ApproximationsNumerical Solutions of Systems of Linear PDEs Tables and Supplements Elementary Functions and Their PropertiesPower, Exponential, and Logarithmic FunctionsTrigonometric FunctionsInverse Trigonometric FunctionsHyperbolic FunctionsInverse Hyperbolic Functions Finite Sums and Infinite SeriesFinite Numerical SumsFinite Functional SumsInfinite Numerical SeriesInfinite Functional Series Indefinite and Definite IntegralsIndefinite IntegralsDefinite Integrals Integral TransformsTables of Laplace TransformsTables of Inverse Laplace TransformsTables of Fourier Cosine TransformsTables of Fourier Sine Transforms Curvilinear Coordinates, Vectors, Operators, and Differential RelationsArbitrary Curvilinear Coordinate SystemsCartesian, Cylindrical, and Spherical Coordinate SystemsOther Special Orthogonal Coordinates Special Functions and Their PropertiesSome Coefficients, Symbols, and NumbersError Functions. Exponential and Logarithmic IntegralsSine Integral and Cosine Integral. Fresnel IntegralsGamma Function, Psi Function, and Beta FunctionIncomplete Gamma and Beta FunctionsBessel Functions (Cylindrical Functions)Modified Bessel FunctionsAiry FunctionsDegenerate Hypergeometric Functions (Kummer Functions)Hypergeometric FunctionsLegendre Polynomials, Legendre Functions, and Associated Legendre FunctionsParabolic Cylinder FunctionsElliptic IntegralsElliptic FunctionsJacobi Theta FunctionsMathieu Functions a