偏微分方程与孤波理论

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前言

Partial Differential Equations and Solitary Waves Theory is designed to serve as atext and a reference. The book is designed to be accessible to advanced undergrad-uate and beginning graduate students as well as research monograph to researchersin applied mathematics, science and engineering. This text is different from othertexts in that it explains classical methods in a non abstract ,way and it introducesand explains how the newly developed methods provide more concise methods toprovide efficient results.Partial Differential Equations and Solitary Waves Theory is designed to focusreaders' attentions on these recently developed valuable techniques that have proventheir effectiveness and reliability over existing classical methods. Moreover, this textalso explains the necessary classical methods because the aim is that new methodswould complement the traditional methods in order to improve the understanding ofthe material.The book avoids approaching the subject through the compact and classicalmethods that make the material impossible to be grasped, especially by studentswho do not have the background in these abstract concepts. Compact theorems andabstract handling of the material are not presented in this text.The book was developed as a result of many years of experience in teachingpartial differential equations and conducting research work in this field. The authorhas taken account on his teaching experience, research work as well as valuablesuggestions received from students and scholars from a wide variety of audience.Numerous examples and exercises, ranging in level from easy to difficult, but con-sistent with the material, are given in each section to give the reader the knowledge,practice and skill in partial differential equations and solitary waves theory. There isplenty of material in this text to be covered in two semesters for senior undergradu-ates and beginning graduates of Mathematics, Science, and Engineering.

内容概要

作者：(美国) 佤斯瓦茨 (Wazwaz.A.M.)

书籍目录

Part I Partial Differential Equations1 Basic Concepts1.1 Introduction1.2 Definitions1.2.1 Definition of a PDE1.2.2 Order of a PDE1.2.3 Linear and Nonlinear PDEs1.2.4 Some Linear Partial Differential Equations1.2.5 Some Nonlinear Partial Differential Equations..1.2.6 Homogeneous and Inhomogeneot, s PDEs1.2.7 Solution of a PDE1.2.8 Boundary Conditions1.2.9 Initial Conditions1.2.10 Well-posed PDEs1.3 Classifications of a Second-order PDEReferences2 First-order Partial Differential Equations2.1 Introduction2.2 Adomian Decomposition Method2.3 The Noise Terms Phenomenon2.4 The Modified Decomposition Method2.5 The Variational Iteration Method2.6 Method of Characteristics2.7 Systems of Linear PDEs by Adomian Method2.8 Systems of Linear PDEs by Variational Iteration MethodReferences3 One Dimensional Heat Flow3.1 Introduction3.2 The Adomian Decomposition Method3.2.1 Homogeneous Heat Equations3.2.2 Inhomogeneous Heat Equations3.3 The Variational Iteration Method3.3.1 Homogeneous Heat Equations3.3.2 Inhomogeneous Heat Equations3.4 Method of Separation of Variables3.4.1 Analysis of the Method3.4.2 Inhomogeneous Boundary Conditions3.4.3 Equations with Lateral Heat LossReferences4 Higher Dimensional Heat Flow4.1 Introduction4.2 Adomian Decomposition Method4.2.1 Two Dimensional Heat Flow4.2.2 Three Dimensional Heat Flow4.3 Method of Separation of Variables4.3.1 Two Dimensional Heat Flow4.3.2 Three Dimensional Heat FlowReferences5 One Dimensional Wave Equation5.1 Introduction5.2 Adomian Decomposition Method5.2.1 Homogeneous Wave Equations5.2.2 Inhomogeneous Wave Equations5.2.3 Wave Equation in an Infinite Domain5.3 The Variational Iteration Method5.3.1 Homogeneous Wave Equations5.3.2 Inhomogeneous Wave Equations5.3.3 Wave Equation in an Infinite Domain5.4 Method of Separation of Variables5.4.1 Analysis of the Method5.4.2 Inhomogeneous Boundary Conditions5.5 Wave Equation in an Infinite Domain: D'Alembert SolutionReferences6 Higher Dimensional Wave Equation6.1 Introduction6.2 Adomian Decomposition Method6.2.1 Two Dimensional Wave Equation6.2.2 Three Dimensional Wave Equation6.3 Method of Separation of Variables6.3.1 Two Dimensional Wave Equation6.3.2 Three Dimensional Wave EquationReferences7 Laplace's Equation7.1 Introduction7.2 Adomian Decomposition Method7.2.1 Two Dimensional Laplace's Equation ...7.3 The Variational Iteration Method7.4 Method of Separation of Variables7.4.1 Laplace's Equation in Two Dimensions..7.4.2 Laplace's Equation in Three Dimensions7.5 Laplace's Equation in Polar Coordinates7.5.1 Laplace's Equation for a Disc7.5.2 Laplace's Equation for an AnnulusReferences8 Nonlinear Partial Differential Equations8.1 Introduction8.2 Adomian Decomposition Method8.2.1 Calculation of Adomian Polynomials ...8.2.2 Alternative Algorithm for Calculating Adomian Polynomials8.3 Nonlinear ODEs by Adomian Method8.4 Nonlinear ODEs by VIM8.5 Nonlinear PDEs by Adomian Method8.6 Nonlinear PDEs by VIM8.7 Nonlinear PDEs Systems by Adomian Method..8.8 Systems of Nonlinear PDEs by VIMReferences9 Linear and Nonlinear Physical Models9.1 Introduction9.2 The Nonlinear Advection Problem9.3 The Goursat Problem9.4 The Klein-Gordon Equation9.4.1 Linear Klein-Gordon Equation9.4.2 Nonlinear Klein-Gordon Equation9.4.3 The Sine-Gordon Equation9.5 The Burgers Equation9.6 The Telegraph Equation9.7 Schrodinger Equation9.7.1 The Linear Schrodinger Equation9.7.2 The Nonlinear Schrodinger Equation9.8 Korteweg-deVries Equation9.9 Fourth-order Parabolic Equation9.9.1 Equations with Constant Coefficients9.9.2 Equations with Variable CoefficientsReferences10 Numerical Applications and Pade Approximants10.1 Introduction10.2 Ordinary Differential Equations10.2.1 Perturbation Problems10.2.2 Nonperturbed Problems10.3 Partial Differential Equations10.4 The Pade Approximants10.5 Pad6 Approximants and Boundary Value ProblemsReferences11 Solitons and Compaetons11.1 Introduction11.2 Solitons11.2.1 The KdV Equation11.2.2 The Modified KdV Equation11.2.3 The Generalized KdV Equation11.2.4 The Sine-Gordon Equation11.2.5 The Boussinesq Equation11.2.6 The Kadomtsev-Petviashvili Equation11.3 Compactons11.4 The Defocusing Branch of K(n,n)ReferencesPart HSolitray Waves Theory12 Solitary Waves Theory12.1 Introduction12.2 Definitions12.2.1 Dispersion and Dissipation12.2.2 Types of Travelling Wave Solutions12.2.3 Nonanalytic Solitary Wave Solutions12.3 Analysis of the Methods12.3.1 The Tanh-coth Method12.3.2 The Sine-cosine Method12.3.3 Hirota's Bilinear Method12.4 Conservation LawsReferences

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《偏微分方程与孤波理论》由高等教育出版社出版发行。

作者简介

《偏微分方程与孤波理论》讲述了：Partial Differential Equations and Solitary Waves Theory is a self-containedbook divided into two parts: Part I is a coherent survey bringing together newlydeveloped methods for solving PDEs. While some traditional techniques are pre-sented, this part does not require thorough understanding of abstract theories orcompact concepts. Well-selected worked examples and exercises shall guide thereader through the text. Part II provides an extensive exposition of the solitarywaves theory. This part handles nonlinear evolution equations by methods suchas Hirota's bilinear method or the tanh-coth method. A self-contained treatmentis presented to discuss complete integrability of a wide class of nonlinear equa-tions. This part presents in an accessible manner a systematic presentation ofsolitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons.

While the whole book can be used as a text for advanced undergraduate andgraduate students in applied mathematics, physics and engineering, Part II willbe most useful for graduate students and researchers in mathematics, engineer-ing, and other related fields.

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