Partial Differential Equations

Victor Henner is a professor at the Department of Physics and Astronomy at the University of Louisville. He has Ph.Ds from the Novosibirsk Institute of Mathematics in Russia and Moscow State University. He co-wrote with Tatyana Belozerova Ordinary and Partial Differential Equations. Tatyana Belozerova is a professor at Perm State University in Russia. Along with Ordinary and Partial Differential Equations, she co-wrote with Victor Henner Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions.Alexander Nepomnyashchy is a mathematics professor at Northwestern University and hails from the Faculty of Mathematics at Technion-Israel Institute of Technology. His research interests include non-linear stability theory and pattern formation.

• IntroductionBasic definitionsExamplesFirst-order equations Linear first-order equationsGeneral solutionInitial conditionQuasilinear first-order equationsCharacteristic curvesExamplesSecond-order equationsClassification of second-order equationsCanonical formsHyperbolic equationsElliptic equationsParabolic equationsThe Sturm-Liouville ProblemGeneral considerationExamples of Sturm-Liouville ProblemsOne-Dimensional Hyperbolic EquationsWave EquationBoundary and Initial ConditionsLongitudinal Vibrations of a Rod and Electrical OscillationsRod oscillations: Equations and boundary conditionsElectrical Oscillations in a CircuitTraveling Waves: D'Alembert MethodCauchy problem for nonhomogeneous wave equationD'Alembert's formulaThe Green's functionWell-posedness of the Cauchy problemFinite intervals: The Fourier Method for Homogeneous EquationsThe Fourier Method for Nonhomogeneous EquationsThe Laplace Transform Method: simple cases Equations with Nonhomogeneous Boundary ConditionsThe Consistency Conditions and Generalized SolutionsEnergy in the HarmonicsDispersion of wavesCauchy problem in an infinite regionPropagation of a wave trainOne-Dimensional Parabolic EquationsHeat Conduction and Diffusion: Boundary Value ProblemsHeat conductionDiffusion equationOne-dimensional parabolic equations and initial and boundary conditionsThe Fourier Method for Homogeneous EquationsNonhomogeneous EquationsThe Green's function and Duhamel's principleThe Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary ConditionsLarge time behavior of solutionsMaximum principleThe heat equation in an infinite regionElliptic equationsElliptic differential equations and related physical problemsHarmonic functionsBoundary conditionsExample of an ill-posed problemWell-posed boundary value problems Maximum principle and its consequencesLaplace equation in polar coordinatesLaplace equation and interior BVP for circular domainLaplace equation and exterior BVP for circular domainPoisson equation: general notes and a simple casePoisson IntegralApplication of Bessel functions for the solution of Poisson equations in a circleThree-dimensional Laplace equation for a cylinderThree-dimensional Laplace equation for a ballAxisymmetric caseNon-axisymmetric caseBVP for Laplace Equation in a Rectangular DomainThe Poisson Equation with Homogeneous Boundary ConditionsGreen's function for Poisson equationsHomogeneous boundary conditionsNonhomogeneous boundary conditionsSome other important equationsHelmholtz equationSchrӧdinger equationTwo Dimensional Hyperbolic EquationsDerivation of the Equations of MotionBoundary and Initial ConditionsOscillations of a Rectangular MembraneThe Fourier Method for Homogeneous Equations with Homogeneous Boundary ConditionsThe Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary ConditionsThe Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary ConditionsSmall Transverse Oscillations of a Circular MembraneThe Fourier Method for Homogeneous Equations with Homogeneous Boundary ConditionsAxisymmetric Oscillations of a MembraneThe Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary ConditionsForced Axisymmetric OscillationsThe Fourier Method for Equations with Nonhomogeneous Boundary ConditionsTwo-Dimensional Parabolic EquationsHeat Conduction within a Finite Rectangular DomainThe Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange)The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary conditionsHeat Conduction within a Circular DomainThe Fourier Method for the Homogeneous Heat Equation The Fourier Method for the Nonhomogeneous Heat EquationHeat conduction in an Infinite MediumHeat Conduction in a Semi-Infinite MediumNonlinear equationsBurgers equationKink solutionSymmetries of the Burgers equationGeneral solution of the Cauchy problem.Interaction of kinksKorteweg-de Vries equationSymmetry properties of the KdV equationCnoidal wavesSolitonsBilinear formulation of the KdV equationHirota's methodMultisoliton solutionsNonlinear Schrӧdinger equationSymmetry properties of NSESolitary wavesAppendix A. Fourier Series, Fourier and Laplace TransformsAppendix B. Bessel and Legendre FunctionsAppendix C. Sturm-Liouville problem and auxiliary functions for one and two dimensionsAppendix D.D1. The Sturm-Liouville problem for a circleD2. The Sturm-Liouville problem for the rectangleAppendix E.E1. The Laplace and Poisson equations for a rectangular domain with nonhomogeneous boundary conditions. E2. The heat conduction equations with nonhomogeneous boundary conditions.