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CHAPTER Ⅰ．DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS 1

1．1．Some definitions and examples 1

1．2．The classification of equations and their solutions 6

1．3．Power series solutions and existence theorems 12

1．4．Transformations of variables；tensors 20

CHAPTER Ⅱ．LINEAR EQUATIONS OF THE FIRST ORDER 24

2．1．Homogeneous linear equations 24

2．2．The quasi-linear equation of the first order 29

2．3．Systems of linear homogeneous equations 36

2．4．Adjoint systems 40

CHAPTER Ⅲ．NON-LINEAR EQUATIONS OF THE FIRST ORDER 47

3．1．Geometric theory of the characteristics 47

3．2．Complete integrals 55

3．3．The Hamilton-Jacobi theorem 58

3．4．Involutory systems 62

3．5．Jacobi＇s integration method 66

CHAPTER Ⅳ．LINEAR EQUATIONS OF THE SECOND ORDER 70

4．1．Classification；the fundamental tensor 71

4．2．Riemannian geometry 74

4．3．Green＇s formula 80

4．4．Flat space．Equations with constant coefficients 84

4．5．Geodesics and geodesic distance 88

CHAPTER Ⅴ．SELF-ADJOINT ELLIPTIC EQUATIONS 98

5．1．The Dirichlet integral 99

5．2．A maximum principle 102

5．3．The local fundamental solution 104

5．4．Volume and surface potentials 110

5．5．Closed Riemannian spaces 116

5．6．The formulation of boundary value problems 120

CHAPTER Ⅵ．LINEAR INTEGRAL-EQUATIONS 125

6．1．Fredholm＇s first theorem 125

6．2．Fredholm＇s second theorem 130

6．3．Fredholm＇s third theorem 133

6．4．Iterated kernels 135

6．5．Symmetric kernels 140

6．6．Eigenfunction expansions 144

CHAPTER Ⅶ．BOUNDARY VALUE PROBLEMS 147

7．1．Poisson＇a equation and the fundamental solution in the large 147

7．2．Solution of the boundary value problems 151

7．3．Representation formulae 156

7．4．The kernel function 162

CHAPTER Ⅷ．EIGENFUNCTIONS 169

8．1．Harmonic functions 169

8．2．Harmonic domain functionals 173

8．3．The Poisson equation in a closed space 177

8．4．Dirichlet＇s problem for the Poisson equation 182

8．5．Eigenfunction expansions 185

8．6．Initial value problems 190

CHAPTER Ⅸ．NORMAL HYPERBOLIC EQUATIONS 195

9．1．Characteristic surfaces 195

9．2．Bicharacteristies 200

9．3．Discontinuities and singularities 205

9．4．The propagation of waves 208

9．5．The initial value problem 212

CHAPTER Ⅹ．INTEGRATION OF THE WAVE EQUATION 217

10．1．The Riemann-Liouville integral 217

10．2．The fractional hyperbolic potential 220

10．3．The Cauchy problem 224

10．4．Verification of the solution 226

10．5．Lorentz spaces of even dimension 233

10．6．Lorentz spaces of odd dimension 236

10．7．The equation in a Riemann space 238

BIBLIOGRAPHY 244

INDEX 246