Partial Differential Equations

Analytical Methods and Applications
 
 
Routledge Cavendish (Verlag)
  • erschienen am 20. November 2019
  • |
  • 392 Seiten
 
E-Book | ePUB ohne DRM | Systemvoraussetzungen
978-0-429-80441-0 (ISBN)
 

Partial Differential Equations: Analytical Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners' course for graduate students. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor.

This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The "teaching-by-examples" approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The book's level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses.

Highlights:

    • Offers a complete first course on PDEs

    • The text's flexible structure promotes varied syllabi for courses

    • Written with a teach-by-example approach which offers numerous examples and applications

    • Includes additional topics such as the Sturm-Liouville problem, Fourier and Laplace transforms, and special functions

    • The text's graphical material makes excellent use of modern software packages

    • Features numerous examples and applications which are suitable for readers studying the subject remotely or independently

    • Englisch
    • London
    • |
    • Großbritannien
    Taylor & Francis Ltd
    • Für höhere Schule und Studium
    92 schwarz-weiße Abbildungen, 4 schwarz-weiße Tabellen
    • 7,03 MB
    978-0-429-80441-0 (9780429804410)
    weitere Ausgaben werden ermittelt

    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.

    Introduction

    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

    Schr¿dinger 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 Schr¿dinger 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.

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