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线性与非线性积分方程:方法及应用

样章
作者：
Abdul-Majid Wazwaz
定价：
128.00 元
版面字数：
790.000千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2011-01-20
ISBN：
978-7-04-031694-0
物料号：
31694-00
出版时间：
2011-06-20
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
积分方程

本书是一本同时介绍线性和非线性积分方程的教材，分成两部分，各部分自成体系。第一部分主要对第一类、第二类线性积分方程进行了系统、深入的分析并提 供各种解法；第二部分主要讲述非线性积分方程求解及其应用，针对不适定fredholm问题、分歧点和奇异点等问题进行了系统的分析，并提供易于理解的处 理方法。

本书通过大量的例子讲述线性与非线性积分方程最新发展起来的高效解法，无须要求读者对抽象理论本身有很深的理解，同时也讨论了某些经 典方法一些有价值的改进。书中对这些方法都给出了很好的解释，并通过对这些方法进行对比，使得读者能够快速地掌握并选择可行且高效的方法。本书提供了大量 的习题，并在书后附有答案。

本书可作为应用数学、工程学及其相关专业的高年级本科生和研究生教材，也可供相关领域的工程师参考。

  • front matter
  • Part I hspace betweenumberspace Linear Integral Equations
  • 1 Preliminaries
    • 1.1 Taylor Series
    • 1.2 Ordinary Differential Equations
      • 1.2.1 First Order Linear Differential Equations
      • 1.2.2 Second Order Linear Differential Equations
      • 1.2.3 The Series Solution Method
    • 1.3 Leibnitz Rule for Differentiation of Integrals
    • 1.4 Reducing Multiple Integrals to Single Integrals
    • 1.5 Laplace Transform
      • 1.5.1 Properties of Laplace Transforms
    • 1.6 Infinite Geometric Series
    • References
  • 2 Introductory Concepts of Integral Equations
    • 2.1 Classification of Integral Equations
      • 2.1.1 Fredholm Integral Equations
      • 2.1.2 Volterra Integral Equations
      • 2.1.3 Volterra-Fredholm Integral Equations
      • 2.1.4 Singular Integral Equations
    • 2.2 Classification of Integro-Differential Equations
      • 2.2.1 Fredholm Integro-Differential Equations
      • 2.2.2 Volterra Integro-Differential Equations
      • 2.2.3 Volterra-Fredholm Integro-Differential Equations
    • 2.3 Linearity and Homogeneity
      • 2.3.1 Linearity Concept
      • 2.3.2 Homogeneity Concept
    • 2.4 Origins of Integral Equations
    • 2.5 Converting IVP to Volterra Integral Equation
      • 2.5.1 Converting Volterra Integral Equation to IVP
    • 2.6 Converting BVP to Fredholm Integral Equation
      • 2.6.1 Converting Fredholm Integral Equation to BVP
    • 2.7 Solution of an Integral Equation
    • References
  • 3 Volterra Integral Equations
    • 3.1 Introduction
    • 3.2 Volterra Integral Equations of the Second Kind
      • 3.2.1 The Adomian Decomposition Method
      • 3.2.2 The Modified Decomposition Method
      • 3.2.3 The Noise Terms Phenomenon
      • 3.2.4 The Variational Iteration Method
      • 3.2.5 The Successive Approximations Method
      • 3.2.6 The Laplace Transform Method
      • 3.2.7 The Series Solution Method
    • 3.3 Volterra Integral Equations of the First Kind
      • 3.3.1 The Series Solution Method
      • 3.3.2 The Laplace Transform Method
      • 3.3.3 Conversion to a Volterra Equation of the Second Kind
    • References
  • 4 Fredholm Integral Equations
    • 4.1 Introduction
    • 4.2 Fredholm Integral Equations of the Second Kind
      • 4.2.1 The Adomian Decomposition Method
      • 4.2.2 The Modified Decomposition Method
      • 4.2.3 The Noise Terms Phenomenon
      • 4.2.4 The Variational Iteration Method
      • 4.2.5 The Direct Computation Method
      • 4.2.6 The Successive Approximations Method
      • 4.2.7 The Series Solution Method
    • 4.3 Homogeneous Fredholm Integral Equation
      • 4.3.1 The Direct Computation Method
    • 4.4 Fredholm Integral Equations of the First Kind
      • 4.4.1 The Method of Regularization
      • 4.4.2 The Homotopy Perturbation Method
    • References
  • 5 Volterra Integro-Differential Equations
    • 5.1 Introduction
    • 5.2 Volterra Integro-Differential Equations of the Second Kind
      • 5.2.1 The Adomian Decomposition Method
      • 5.2.2 The Variational Iteration Method
      • 5.2.3 The Laplace Transform Method
      • 5.2.4 The Series Solution Method
      • 5.2.5 Converting Volterra Integro-Differential Equations to Initial Value Problems
      • 5.2.6 Convertingtmspace +thinmuskip .1667em Volterratmspace +thickmuskip .2777em Integro-Differentialtmspace +thickmuskip .2777em Equation to Volterra Integral Equation
    • 5.3 Volterra Integro-Differential Equations of the First Kind
      • 5.3.1 Laplace Transform Method
      • 5.3.2 The Variational Iteration Method
    • References
  • 6 Fredholm Integro-Differential Equations
    • 6.1 Introduction
    • 6.2 Fredholm Integro-Differential Equations of the Second Kind
      • 6.2.1 The Direct Computation Method
      • 6.2.2 The Variational Iteration Method
      • 6.2.3 The Adomian Decomposition Method
      • 6.2.4 The Series Solution Method
    • References
  • 7 Abel's Integral Equation and Singular Integral Equations
    • 7.1 Introduction
    • 7.2 Abel's Integral Equation
      • 7.2.1 The Laplace Transform Method
    • 7.3 The Generalized Abel's Integral Equation
      • 7.3.1 The Laplace Transform Method
      • 7.3.2 The Main Generalized Abel Equation
    • 7.4 The Weakly Singular Volterra Equations
      • 7.4.1 The Adomian Decomposition Method
      • 7.4.2 The Successive Approximations Method
      • 7.4.3 The Laplace Transform Method
    • References
  • 8 Volterra-Fredholm Integral Equations
    • 8.1 Introduction
    • 8.2 The Volterra-Fredholm Integral Equations
      • 8.2.1 The Series Solution Method
      • 8.2.2 The Adomian Decomposition Method
    • 8.3 The Mixed Volterra-Fredholm Integral Equations
      • 8.3.1 The Series Solution Method
      • 8.3.2 The Adomian Decomposition Method
    • 8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables
      • 8.4.1 The Modified Decomposition Method
    • References
  • 9 Volterra-Fredholm Integro-Differential Equations
    • 9.1 Introduction
    • 9.2 The Volterra-Fredholm Integro-Differential Equation
      • 9.2.1 The Series Solution Method
      • 9.2.2 The Variational Iteration Method
    • 9.3 The Mixed Volterra-Fredholm Integro-Differential Equations
      • 9.3.1 The Direct Computation Method
      • 9.3.2 The Series Solution Method
    • 9.4 The Mixed Volterra-Fredholm Integro-Differential Equations in Two Variables
      • 9.4.1 The Modified Decomposition Method
    • References
  • 10 Systems of Volterra Integral Equations
    • 10.1 Introduction
    • 10.2 Systems of Volterra Integral Equations of the hspace*1.7mm Second Kind
      • 10.2.1 The Adomian Decomposition Method
      • 10.2.2 The Laplace Transform Method
    • 10.3 Systems of Volterra Integral Equations of the First Kind
      • 10.3.1 The Laplace Transform Method
      • 10.3.2 Conversion to a Volterra System of the hspace*2.5mm Second Kind
    • 10.4 Systems of Volterra Integro-Differential Equations
      • 10.4.1 The Variational Iteration Method
      • 10.4.2 The Laplace Transform Method
    • References
  • 11 Systems of Fredholm Integral Equations
    • 11.1 Introduction
    • 11.2 Systems of Fredholm Integral Equations
      • 11.2.1 The Adomian Decomposition Method
      • 11.2.2 The Direct Computation Method
    • 11.3 Systems of Fredholm Integro-Differential Equations
      • 11.3.1 The Direct Computation Method
      • 11.3.2 The Variational Iteration Method
    • References
  • 12 Systems of Singular Integral Equations
    • 12.1 Introduction
    • 12.2 Systems of Generalized Abel Integral Equations
      • 12.2.1 Systems of Generalized Abel Integral Equations in hspace*2.5mm Two Unknowns
      • 12.2.2 Systems of Generalized Abel Integral Equations in hspace*2.5mm Three Unknowns
    • 12.3 Systems of the Weakly Singular Volterra Integral hspace*1.7mm Equations
      • 12.3.1 The Laplace Transform Method
      • 12.3.2 The Adomian Decomposition Method
    • References
  • Part IIhspace betweenumberspace Nonlinear Integral Equations
  • 13 Nonlinear Volterra Integral Equations
    • 13.1 Introduction
    • 13.2 Existence of the Solution for Nonlinear Volterra Integral hspace*1.7mm Equations
    • 13.3 Nonlinear Volterra Integral Equations of the Second Kind
      • 13.3.1 The Successive Approximations Method
      • 13.3.2 The Series Solution Method
      • 13.3.3 The Adomian Decomposition Method
    • 13.4 Nonlinear Volterra Integral Equations of the First Kind
      • 13.4.1 The Laplace Transform Method
      • 13.4.2 Conversion to a Volterra Equation of the hspace*2.5mm Second Kind
    • 13.5 Systems of Nonlinear Volterra Integral Equations
      • 13.5.1 Systems of Nonlinear Volterra Integral Equations of hspace*2.5mm the Second Kind
      • 13.5.2 Systems of Nonlinear Volterra Integral Equations of hspace*2.5mm the First Kind
    • References
  • 14 Nonlinear Volterra Integro-Differential Equations
    • 14.1 Introduction
    • 14.2 Nonlinear Volterra Integro-Differential Equations of the hspace*1.7mm Second Kind
      • 14.2.1 The Combined Laplace Transform-Adomian hspace*2.5mm Decomposition Method
      • 14.2.2 The Variational Iteration Method
      • 14.2.3 The Series Solution Method
    • 14.3 Nonlinear Volterra Integro-Differential Equations of the hspace*1.7mm First Kind
      • 14.3.1 The Combined Laplace Transform-Adomian hspace*2.5mm Decomposition Method
      • 14.3.2 Conversion to Nonlinear Volterra Equation of the hspace*2.5mm Second Kind
    • 14.4 Systems of Nonlinear Volterra Integro-Differential hspace*1.7mm Equations
      • 14.4.1 The Variational Iteration Method
      • 14.4.2 The Combined Laplace Transform-Adomian hspace*2.5mm Decomposition Method
    • References
  • 15 Nonlinear Fredholm Integral Equations
    • 15.1 Introduction
    • 15.2 Existence of the Solution for Nonlinear Fredholm Integral hspace*1.7mm Equations
      • 15.2.1 Bifurcation Points and Singular Points
    • 15.3 Nonlinear Fredholm Integral Equations of the hspace*1.7mm Second Kind
      • 15.3.1 The Direct Computation Method
      • 15.3.2 The Series Solution Method
      • 15.3.3 The Adomian Decomposition Method
      • 15.3.4 The Successive Approximations Method
    • 15.4 Homogeneous Nonlinear Fredholm Integral Equations
      • 15.4.1 The Direct Computation Method
    • 15.5 Nonlinear Fredholm Integral Equations of the First Kind
      • 15.5.1 The Method of Regularization
      • 15.5.2 The Homotopy Perturbation Method
    • 15.6 Systems of Nonlinear Fredholm Integral Equations
      • 15.6.1 The Direct Computation Method
      • 15.6.2 The Modified Adomian Decomposition Method
    • References
  • 16 Nonlinear Fredholm Integro-Differential Equations
    • 16.1 Introduction
    • 16.2 Nonlinear Fredholm Integro-Differential Equations
      • 16.2.1 The Direct Computation Method
      • 16.2.2 The Variational Iteration Method
      • 16.2.3 The Series Solution Method
    • 16.3 Homogeneous Nonlinear Fredholm Integro-Differential hspace*1.7mm Equations
      • 16.3.1 The Direct Computation Method
    • 16.4 Systems of Nonlinear Fredholm Integro-Differential hspace*1.7mm Equations
      • 16.4.1 The Direct Computation Method
      • 16.4.2 The Variational Iteration Method
    • References
  • 17 Nonlinear Singular Integral Equations
    • 17.1 Introduction
    • 17.2 Nonlinear Abel's Integral Equation
      • 17.2.1 The Laplace Transform Method
    • 17.3 The Generalized Nonlinear Abel Equation
      • 17.3.1 The Laplace Transform Method
      • 17.3.2 The Main Generalized Nonlinear Abel Equation
    • 17.4 The Nonlinear Weakly-Singular Volterra Equations
      • 17.4.1 The Adomian Decomposition Method
    • 17.5 Systems of Nonlinear Weakly-Singular Volterra Integral hspace*1.7mm Equations
      • 17.5.1 The Modified Adomian Decomposition Method
    • References
  • 18 Applications of Integral Equations
    • 18.1 Introduction
    • 18.2 Volterra's Population Model
      • 18.2.1 The Variational Iteration Method
      • 18.2.2 The Series Solution Method
      • 18.2.3 The Pad'e Approximants
    • 18.3 Integral Equations with Logarithmic Kernels
      • 18.3.1 Second Kind Fredholm Integral Equation with a hspace*2.5mm Logarithmic Kernel
      • 18.3.2 First Kind Fredholm Integral Equation with a hspace*2.5mm Logarithmic Kernel
      • 18.3.3 Another First Kind Fredholm Integral Equation hspace*2.5mm with a Logarithmic Kernel
    • 18.4 The Fresnel Integrals
    • 18.5 The Thomas-Fermi Equation
    • 18.6 Heat Transfer and Heat Radiation
      • 18.6.1 Heat Transfer: Lighthill Singular Integral Equation
      • 18.6.2 Heat Radiation in a Semi-Infinite Solid
    • References
  • Appendix A hspace*18mm Table of Indefinite Integrals
    • A.1 Basic Forms
    • A.2 Trigonometric Forms
    • A.3 Inverse Trigonometric Forms
    • A.4 Exponential and Logarithmic Forms
    • A.5 Hyperbolic Forms
    • A.6 Other Forms
  • Appendix B hspace*18mm Integrals Involving Irrational Algebraic hspace*18mm Functions
    • B.1 Integrals Involving $frac t^n sqrt x-t $, $n$ is an integer, $n geqslant 0$
    • B.2 Integrals Involving $frac t^frac n 2 sqrt x-t $, $n$ is an odd integer, $n geqslant 1$
  • Appendix C hspace*18mm Series Representations
    • C.1 Exponential Functions Series
    • C.2 Trigonometric Functions
    • C.3 Inverse Trigonometric Functions
    • C.4 Hyperbolic Functions
    • C.5 Inverse Hyperbolic Functions
    • C.6 Logarithmic Functions
  • Appendix D hspace*18mm The Error and the Complementary Error Functions
    • D.1 The Error Function
    • D.2 The Complementary Error Function
  • Appendix E hspace*18mm Gamma Function
  • Appendix F hspace*18mm Infinite Series
    • F.1 Numerical Series
    • F.2 Trigonometric Series
  • Appendix G hspace*18mm The Fresnel Integrals
    • G.1 The Fresnel Cosine Integral
    • G.2 The Fresnel Sine Integral
  • Answers
  • Index
  • 版权

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线性与非线性积分方程-方法及应用（海外版）

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