Partial Differential Equations

Cover......Page 1
S Title......Page 2
Title......Page 3
ISBN 0 521098939......Page 4
CONTENTS......Page 5
PREFACE......Page 6
1.1 Lagrange's equation......Page 7
1.2 Two examples......Page 10
1.3 The general first order equation......Page 11
1.4 An example of the Lagrange-Charpit method......Page 14
1.5 An initial value problem......Page 15
1.6 Systems of semi-linear equations of the first order......Page 20
1.7 An application of the method of characteristics......Page 26
Exercises......Page 28
2.1 The general equation of the second order......Page 30
2.2 The Cauchy-Kowalewsky theorem......Page 32
2.3 The linear equation......Page 34
2.4 The quasi-linear equation......Page 36
2.5 The normal form of a half-linear equati......Page 39
2.6 The half-linear equation with three independent variables......Page 41
2.7 The half-linear equation in general......Page 44
Exercises......Page 48
3.1 Laplace's equation......Page 50
3.2 The equation of wave motions......Page 52
3.3 Characteristics as wave fronts......Page 53
3.5 The equation of heat......Page 56
3.6 Well-posed problems......Page 57
4.2 The equation U XY = 0......Page 60
4.3 The uniqueness theorem for u xy = 0......Page 63
4.4 The Cauchy problem for the half-linear equation of hyperbolic type......Page 64
4.5 Two other applications of Picard's method......Page 68
4.6 Duly inclined initial lines......Page 69
4.7 The equation of wave motions......Page 70
4.8 The uniqueness theorem......Page 76
4.9 The use of Fourier series......Page 77
4.10 The equation of telegraphy......Page 78
Exercises......Page 80
5.1 Adjoint linear operators......Page 83
5.2 Riemann's method......Page 84
5.1 Another form of Riemann's method......Page 86
5.5 A series formula for the Riemann-Green function......Page 89
5.6 The equation of telegraphy......Page 91
5.7 More examples of the Riemann-Green function......Page 92
Exercises......Page 94
6.1 Spherical waves......Page 96
6.2 Cylindrical waves......Page 97
6.3 Poisson's mean value solution......Page 99
6.4 The method of descent......Page 101
6.5 The uniqueness theorem......Page 102
6.6 The Euler-Poisson-Darboux equation......Page 104
6.7 Poisson's solutions......Page 107
6.8 The formulae of Volterra and Hobson......Page 108
Exercises......Page 110
7.1 A comparison with potential theory......Page 113
7.2 The Riesz integral of functional order......Page 115
7.3 The analytical continuation of Riesz's integra......Page 119
7.4 Cauchy's problem for the non-homogeneous wave equationin two dimensions......Page 123
7.5 The equation of wave motions in three d......Page 126
7.6 Babha's equation......Page 130
7.7 A mixed boundary and initial value problem......Page 131
Exercises......Page 134
8.1 Gravitation......Page 137
8.2 Green's equivalent layer......Page 139
8.3 Properties of the logarithmic potentials......Page 141
8.4 Some other logarithmic potentials......Page 144
8.5 Harmonic functions......Page 145
8.6 Dirichlet's principle......Page 150
8.7 A problem in electrostatics......Page 151
8.8 Green's function and the problem of Dirichlet......Page 152
8.9 Properties of Green's function......Page 153
8.10 The case of polynomial data......Page 155
8.11 Some examples of Green's function......Page 156
8.12 Poisson's integral......Page 159
8.13 The problem of Neumann......Page 162
8.14 Harnack's first theorem on convergence......Page 164
8.15 Harnack's inequality......Page 165
8.17 Functions harmonic in an annulus......Page 167
8.18 Unbounded domains......Page 170
8.19 Connexion with complex variable theory......Page 171
8.20 Conformal mapping......Page 172
8.21 The problem of Neumann......Page 173
8.22 Green's function and conformal mapping......Page 175
Exercises......Page 177
9.2 Subharmonic functions......Page 181
9.4 Perron's function......Page 184
9.5 .Barriers......Page 186
9.6 Some examples of barriers......Page 188
9.7 Discontinuous boundary data......Page 190
10.1 The linear equatio......Page 192
10.2 The reduced wave equation......Page 193
10.3 The elementary solution......Page 195
10.4 Boundary value problems......Page 199
10.5 The linear equation with constant coefficients......Page 202
10.6 The use of the elementary solution......Page 204
10.7 Divergent waves......Page 206
10.8 The half-plane problem......Page 208
10.9 A boundary and initial value problem......Page 210
Exercises......Page 211
11.2 Polynomial solutions......Page 213
11.3 Spherical harmonics......Page 216
11.4 Green's theorem......Page 218
11.6 Green's equivalent layer......Page 220
11.7 Green's function for a sphere......Page 222
11.8 The analytic character of harmonic functions......Page 224
11.9 The linear equation of elliptic type......Page 226
11.10 The equation with constant coefficients.......Page 227
11.11 The mean value theorem......Page 228
11.12 The solution of V2u- k2u =0 in polar coordinates......Page 230
11.13 The solution of V2u+k2u =0 in polar coordinates......Page 231
11.14 Helmholtz's formula......Page 233
11.15 The exterior problem of Dirichlet......Page 235
Exercises......Page 240
12.2 A formal solution of the equation of heat......Page 244
12.3 Use of integral transforms......Page 246
12.4 Use of Cauchy-Kowalewsky theorem......Page 250
12.5 An example due to Tikhonov......Page 252
12.6 The case of continuous initial data......Page 253
12.7 The existence and uniqueness theorem......Page 255
12.8 The equation of heat in two and three dimensions......Page 261
12.9 Boundary conditions......Page 262
12.10 The finite rod......Page 263
12.11 The semi-infinite rod......Page 264
12.12 The finite rod again......Page 268
12.13 The use of Fourier series......Page 271
Exercises......Page 274
Note 2. Dominant functions......Page 277
Note 4. Regular closed curves......Page 278
Note 6. Surfaces......Page 279
Note 8. Summability......Page 280
Note 9. Fourier series......Page 281
BOOKS FOR FURTHER READING......Page 283
INDEX......Page 285
Back Cover......Page 287