线性与非线性积分方程

章节摘录

版权页：插图：Integral equations and in tegro-differential equations will be classified in to distinct types according to the limits of integration and the kernel K（x, t）.Alltypes of integral equations and in tegro differential equations will be classifiedand investigated in the forthcoming chapters.   In this chapter, we will review the most important concepts needed to study integral equations. The traditional methods, such as Taylor seriesmethod and the Laplace transform method, will be used in this text. More-over, the recently developed methods, that will be used thoroughly in this text, will determine the solution in a power series that will converge to an exact solution if such a solution exists. However, if exact solution does not exist, we use as many terms of the obtained series for numerical purposes to approximate the solution.

内容概要

作者：（美国）佤斯瓦茨（Abdul-Majid Wazwaz）

书籍目录

part i linear integral equations
1 preliminaries
　1.1 taylor series
　1.2 ordinary differential equations
　1.3 leibnitz rule for differentiation of integrals
　1.4 reducing multiple integrals to single integrals
　1.5 laplace transform
　1.6 infinite geometric series
　references
2 introductory concepts of integral equations
　2.1 classification of integral equations
　2.2 classification of integro-differential equations
　2.3 linearity and homogeneity
　2.4 origins of integral equations
　2.5 converting ivp to volterra integral equation
　2.6 converting bvp to fredholm integral equation
　2.7 solution of an integral equation
　references
3　volterra integral equations
　3.1 introduction
　3.2 volterra integral equations of the second kind
　3.3 volterra integral equations of the first kind references
4 fredholm integral equations
　4.1 introduction
　4.2 fredholm integral equations of the second kind
　4.3 homogeneous fredholm integral equation
　4.4 fredholm integral equations of the first kind
　references
5　volterra integro-differential equations
　5.1 introduction
　5.2 volterra integro-differential equations of the second
kind
　5.3 volterra integro-differential equations of the first
kind
　references
6 fredholm integro-differential equations
　6.1 introduction
　6.2 fredholm integro-differential equations of
　the second kind
　references
7　abel's integral equation and singular integral equations
　7.1 introduction
　7.2 abel's integral equation
　7.3 the generalized abel's integral equation
　7.4 the weakly singular volterra equations
　references
8　volterra-fredholm integral equations
　8.1 introduction
　8.2 the volterra-fredholm integral equations
　8.3 the mixed volterra-fredholm integral equations
　8.4 the mixed volterra-fredholm integral equations in two
variables
　references
9　volterra-fredholm integro-differential equations
　9.1 introduction
　9.2 the volterra-fredholm integro-differential equation
　9.3 the mixed volterra-fredholm integro-differential
equations
　9.4 the mixed volterra-fredholm integro-differential equations in
two variables
　references
10 systems of volterra integral equations
　10.1 introduction
　10.2 systems of volterra integral equations of the second
kind
　10.3 systems of volterra integral equations of the first
kind
　10.4 systems of volterra integro-differential equations
　references
11 systems of fredholm integral equations
　11.1 introduction
　11.2 systems of fredholm integral equations
　11.3 systems of fredholm integro-differential equations
　references
12 systems of singular integral equations
　12.1 introduction
　12.2 systems of generalized abel integral equations
　12.3 systems of the weakly singular volterra integral
equations
　references
　part ii nonlinear integral equations
13 nonlinear volterra integral equations
　13.1 introduction
　13.2 existence of the solution for nonlinear volterra integral
equations
　13.3 nonlinear volterra integral equations of the second
kind
　13.4 nonlinear volterra integral equations of the first kind
　13.5 systems of nonlinear volterra integral equations
　references
14 nonlinear volterra integro-differential equations
　14.1 introduction
　14.2 nonlinear volterra integro-differential equations of the
second kind
　14.3 nonlinear volterra integro-differential equations of the
first kind
　14.4 systems of nonlinear volterra integro-differential
equations
　references
15 nonlinear fredholm integral equations
　15.1 introduction
　15.2 existence of the solution for nonlinear fredholm integral
equations
　15.3 nonlinear fredholm integral equations of the second
kind
　15.4 homogeneous nonlinear fredholm integral equations
　15.5 nonlinear fredholm integral equations of the first kind
　15.6 systems of nonlinear fredholm integral equations
　references
16 nonlinear fredholm integro-differential equations
　16.1 introduction
　16.2 nonlinear fredholm integro-differential equations.
　16.3 homogeneous nonlinear fredholm integro-differential
equations
　16.4 systems of nonlinear fredholm integro-differential
equations
　references
17 nonlinear singular integral equations
　17.1 introduction
　17.2 nonlinear abel's integral equation
　17.3 the generalized nonlinear abel equation
　17.4 the nonlinear weakly-singular volterra equations
　17.5 systems of nonlinear weakly-singular volterra integral
equations
　references
18 applications of integral equations
　18.1 introduction
　18.2 volterra's population model
　18.3 integral equations with logarithmic kernels
　18.4 the fresnel integrals
　18.5 the thomas-fermi equation
　18.6 heat transfer and heat radiation
　references
appendix a 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 integrals involving irrational algebraic functions
　b.1 integrals involving n is an integer, n ≥ 0
　b.2 integrals involving n is an odd integer, n ≥ i
appendix c 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 the error and the complementary error
　functions
　d.1 the error function
　d.2 the complementary error function
appendix e gamma function
appendix f infinite series
　f.1 numerical series
　f.2 trigonometric series
appendix g the fresnel integrals
　g.1 the fresnel cosine integral
　g.2 the fresnel sine integral
answers
index

编辑推荐

《线性与非线性积分方程:方法及应用》：关键词：线性与非线性Volterra方程，线性与非线性Fredholm方程，线性与非线性奇异方程，积分方程组。Nonlinear Physical Science focuses on the recent a dvances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physical science and nonlinear mathematics with engineering applications.

作者简介

《线性与非线性积分方程:方法及应用》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。

《线性与非线性积分方程:方法及应用》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。《线性与非线性积分方程:方法及应用》提供了大量的习题，并在书后附有答案。

《线性与非线性积分方程:方法及应用》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。

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