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OverviewFull Product DetailsAuthor: Victor Henner (Perm State University, Russia, and University of Louisville) , Tatyana Belozerova (Perm State University, Russia) , Alexander NepomnyashchyPublisher: Taylor & Francis Ltd Imprint: Chapman & Hall/CRC Weight: 0.740kg ISBN: 9781032475080ISBN 10: 1032475080 Pages: 396 Publication Date: 29 March 2023 Audience: General/trade , General Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction Basic definitions Examples First-order equations Linear first-order equations General solution Initial condition Quasilinear first-order equations Characteristic curves Examples Second-order equations Classification of second-order equations Canonical forms Hyperbolic equations Elliptic equations Parabolic equations The Sturm-Liouville Problem General consideration Examples of Sturm-Liouville Problems One-Dimensional Hyperbolic Equations Wave Equation Boundary and Initial Conditions Longitudinal Vibrations of a Rod and Electrical Oscillations Rod oscillations: Equations and boundary conditions Electrical Oscillations in a Circuit Traveling Waves: D'Alembert Method Cauchy problem for nonhomogeneous wave equation D'Alembert's formula The Green's function Well-posedness of the Cauchy problem Finite intervals: The Fourier Method for Homogeneous Equations The Fourier Method for Nonhomogeneous Equations The Laplace Transform Method: simple cases Equations with Nonhomogeneous Boundary Conditions The Consistency Conditions and Generalized Solutions Energy in the Harmonics Dispersion of waves Cauchy problem in an infinite region Propagation of a wave train One-Dimensional Parabolic Equations Heat Conduction and Diffusion: Boundary Value Problems Heat conduction Diffusion equation One-dimensional parabolic equations and initial and boundary conditions The Fourier Method for Homogeneous Equations Nonhomogeneous Equations The Green's function and Duhamel's principle The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions Large time behavior of solutions Maximum principle The heat equation in an infinite region Elliptic equations Elliptic differential equations and related physical problems Harmonic functions Boundary conditions Example of an ill-posed problem Well-posed boundary value problems Maximum principle and its consequences Laplace equation in polar coordinates Laplace equation and interior BVP for circular domain Laplace equation and exterior BVP for circular domain Poisson equation: general notes and a simple case Poisson Integral Application of Bessel functions for the solution of Poisson equations in a circle Three-dimensional Laplace equation for a cylinder Three-dimensional Laplace equation for a ball Axisymmetric case Non-axisymmetric case BVP for Laplace Equation in a Rectangular Domain The Poisson Equation with Homogeneous Boundary Conditions Green's function for Poisson equations Homogeneous boundary conditions Nonhomogeneous boundary conditions Some other important equations Helmholtz equation Schrӧdinger equation Two Dimensional Hyperbolic Equations Derivation of the Equations of Motion Boundary and Initial Conditions Oscillations of a Rectangular Membrane The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions Small Transverse Oscillations of a Circular Membrane The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions Axisymmetric Oscillations of a Membrane The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions Forced Axisymmetric Oscillations The Fourier Method for Equations with Nonhomogeneous Boundary Conditions Two-Dimensional Parabolic Equations Heat Conduction within a Finite Rectangular Domain The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange) The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary conditions Heat Conduction within a Circular Domain The Fourier Method for the Homogeneous Heat Equation The Fourier Method for the Nonhomogeneous Heat Equation Heat conduction in an Infinite Medium Heat Conduction in a Semi-Infinite Medium Nonlinear equations Burgers equation Kink solution Symmetries of the Burgers equation General solution of the Cauchy problem. Interaction of kinks Korteweg-de Vries equation Symmetry properties of the KdV equation Cnoidal waves Solitons Bilinear formulation of the KdV equation Hirota's method Multisoliton solutions Nonlinear Schrӧdinger equation Symmetry properties of NSE Solitary waves Appendix A. Fourier Series, Fourier and Laplace Transforms Appendix B. Bessel and Legendre Functions Appendix C. Sturm-Liouville problem and auxiliary functions for one and two dimensions Appendix D. D1. The Sturm-Liouville problem for a circle D2. The Sturm-Liouville problem for the rectangle Appendix E. E1. The Laplace and Poisson equations for a rectangular domain with nonhomogeneous boundary conditions. E2. The heat conduction equations with nonhomogeneous boundary conditions.ReviewsAuthor InformationVictor 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. Tab Content 6Author Website:Countries AvailableAll regions |
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