　　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.

内容简介

　　《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。

内页插图

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

精彩书摘

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.
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