《PARTIAL DIFFERENTIAL EQUATIONS》

CHAPTER Ⅰ．DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS1

1．1．Some definitions and examples1

1．2．The classification of equations and their solutions6

1．3．Power series solutions and existence theorems12

1．4．Transformations of variables；tensors20

CHAPTER Ⅱ．LINEAR EQUATIONS OF THE FIRST ORDER24

2．1．Homogeneous linear equations24

2．2．The quasi-linear equation of the first order29

2．3．Systems of linear homogeneous equations36

2．4．Adjoint systems40

CHAPTER Ⅲ．NON-LINEAR EQUATIONS OF THE FIRST ORDER47

3．1．Geometric theory of the characteristics47

3．2．Complete integrals55

3．3．The Hamilton-Jacobi theorem58

3．4．Involutory systems62

3．5．Jacobi＇s integration method66

CHAPTER Ⅳ．LINEAR EQUATIONS OF THE SECOND ORDER70

4．1．Classification；the fundamental tensor71

4．2．Riemannian geometry74

4．3．Green＇s formula80

4．4．Flat space．Equations with constant coefficients84

4．5．Geodesics and geodesic distance88

CHAPTER Ⅴ．SELF-ADJOINT ELLIPTIC EQUATIONS98

5．1．The Dirichlet integral99

5．2．A maximum principle102

5．3．The local fundamental solution104

5．4．Volume and surface potentials110

5．5．Closed Riemannian spaces116

5．6．The formulation of boundary value problems120

CHAPTER Ⅵ．LINEAR INTEGRAL-EQUATIONS125

6．1．Fredholm＇s first theorem125

6．2．Fredholm＇s second theorem130

6．3．Fredholm＇s third theorem133

6．4．Iterated kernels135

6．5．Symmetric kernels140

6．6．Eigenfunction expansions144

CHAPTER Ⅶ．BOUNDARY VALUE PROBLEMS147

7．1．Poisson＇a equation and the fundamental solution in the large147

7．2．Solution of the boundary value problems151

7．3．Representation formulae156

7．4．The kernel function162

CHAPTER Ⅷ．EIGENFUNCTIONS169

8．1．Harmonic functions169

8．2．Harmonic domain functionals173

8．3．The Poisson equation in a closed space177

8．4．Dirichlet＇s problem for the Poisson equation182

8．5．Eigenfunction expansions185

8．6．Initial value problems190

CHAPTER Ⅸ．NORMAL HYPERBOLIC EQUATIONS195

9．1．Characteristic surfaces195

9．2．Bicharacteristies200

9．3．Discontinuities and singularities205

9．4．The propagation of waves208

9．5．The initial value problem212

CHAPTER Ⅹ．INTEGRATION OF THE WAVE EQUATION217

10．1．The Riemann-Liouville integral217

10．2．The fractional hyperbolic potential220

10．3．The Cauchy problem224

10．4．Verification of the solution226

10．5．Lorentz spaces of even dimension233

10．6．Lorentz spaces of odd dimension236

10．7．The equation in a Riemann space238

BIBLIOGRAPHY244

INDEX246