Lectures on Differential and Integral Equations2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Chapter 1．THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS1

1．Successive Approximations1

1．Existence and uniqueness of the solution of the ordinary differential equation of the first order1

2．Remark on approximate solutions6

3．Integration constants8

4．Solution by power series expansion10

5．Differential equations containing parameters．Perturbation theory14

6．Existence and uniqueness of the solution of a system of differential equations17

2．Linear Differential Equations of the nth Order21

7．Singular points for linear differential equations21

8．Fundamental system of solutions23

9．Wronskian．Liouville＇s formula27

10．Lagrange＇s method of variation of constants and D＇Alembert＇s method of reduction of order29

11．Linear differential equations with constant coefficients31

3．Second Order Differential Equations of the Fuchs Type37

12．Regular singular points．Fuchs＇theorem37

13．Gauss differential equations45

14．Legendre differential equations48

15．Bessel differential equations51

Chapter 2．THE BOUNDARY VALUE PROBLEM FOR LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER61

1．Boundary Value Problem61

16．Boundary value problem of the Sturm-Liouville type61

17．Green＇s function．Reduction to integral equations64

18．Periodic solutions．Generalized Green＇s function68

2．Hilbert-Schmidt Theory of Integral Equations with Symmetric Kernels76

19．The Ascoli-Arzelà theorem76

20．Existence proof for the eigenvalues80

21．The Bessel inequality．The Hilbert-Schmidt expansion theorem83

22．Approximations of eigenvalues．Rayleigh＇s principle and the Kryloff-Weinstein theorem91

23．Inhomogeneous integral equations96

24．Hermite，Laguerre and Legendre polynomials100

3．Asymptotic Expression of Eigenvalues and Eigenfunctions，Liouville＇s Method110

25．The Liouville transformation110

26．Asymptotic expressions of eigenvalues and eigenfunctions112

Chapter 3．FREDHOLM INTEGRAL EQUATIONS115

1．Fredholm Alternative Theorem115

27．The case when ∫ba∫ba|K(s，t)|2 ds dt<1115

28．The general case118

29．Fredholm＇s alternative theorem125

2．The Schmidt Expansion Theorem and the Mercer Expansion Theorem127

30．Operator-theoretical notations127

31．The Schmidt expansion theorem128

32．Application to Fredholm integral equation of the first kind131

33．Positive definite kernels．Mercer＇s expansion theorem132

3．Singular Integral Equations139

34．Discontinuous kernels140

35．Examples．Band spectrum141

Chapter 4．VOLTERRA INTEGRAL EQUATIONS145

1．Volterra Integral Equations of the Second Kind145

36．Existence and uniqueness of solutions145

37．Resolvent kernels147

38．Application to linear differential equations149

39．The singular kernel P(s，t)/(s，t)a151

2．Volterra Integral Equations of the First Kind153

40．Reduction to integral equations of the second kind153

41．Abel integral equations154

Chapter 5．THE GENERAL EXPANSION THEOREM(WEYL-STONE-TITCHMARSH-KODAIRA＇S THEOREM)159

1．Classification of Singular Boundary Points160

42．Green＇s formula160

43．Limit point case and limit circle case162

44．Definition of m1(λ) and m2(λ)170

2．The General Expansion Theorem173

45．Application of the Hilbert-Schmidt expansion theorem173

46．Helly＇s theorem and Poisson＇s integral formula177

47．The Weyl-Stone-Titchmarsh-Kodaira theorem183

48．Density matrix190

3．Examples192

49．The Fourier series expansion192

50．The Fourier integral theorem194

51．The Hermite function expansion196

52．The Hankel integral theorem199

53．The Fourier-Bessel series expansion202

54．The Laguerre function expansion205

Chapter 6．NON-LINEAR INTEGRAL EQUATIONS209

55．Non-linear Volterra integral equations209

56．Non-linear Fredholm integral equations210

Appendix．FROM THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE213

A theorem on normal family of regular functions(Part 44)213

Hurwitz＇s theorem(Part 47)213

The Poisson integral formula(Part 46)214

BIBLIOGRAPHY217

INDEX219