Differential and Integral Equations

Contents......Page 8
Preface......Page 6
How to use this book......Page 12
Prerequisites......Page 14
0 Some Preliminaries......Page 16
1.1 Integral equations and their relationship to differential equations......Page 20
1.2 Picard’s method......Page 26
2 Existence and Uniqueness......Page 34
2.1 First-order differential equations in a single independent variable......Page 35
2.2 Two simultaneous equations in a single variable......Page 41
2.3 A second-order equation......Page 44
3 The Homogeneous Linear Equation and Wronskians......Page 48
3.1 Some linear algebra......Page 49
3.2 Wronskians and the linear independence of solutions of the second-order homogeneous linear equation......Page 51
4 The Non-Homogeneous Linear Equation......Page 56
4.1 The method of variation of parameters......Page 58
4.2 Green’s functions......Page 63
5 First-Order Partial Differential Equations......Page 74
5.1 Characteristics and some geometrical considerations......Page 75
5.2 Solving characteristic equations......Page 77
5.3 General solutions......Page 82
5.4 Fitting boundary conditions to general solutions......Page 85
5.5 Parametric solutions and domains of definition......Page 90
5.6 A geometric interpretation of an analytic condition......Page 98
6 Second-Order Partial Differential Equations......Page 100
6.1 Characteristics......Page 102
6.2 Reduction to canonical form......Page 108
6.3 General solutions......Page 115
6.4 Problems involving boundary conditions......Page 118
6.5 Appendix: technique in the use of the chain rule......Page 128
7 The Diffusion and Wave Equations and the Equation of Laplace......Page 130
7.1 The equations to be considered......Page 131
7.2 One-dimensional heat conduction......Page 134
7.3 Transverse waves in a finite string......Page 138
7.4 Separated solutions of Laplace’s equation in polar co-ordinates and Legendre’s equation......Page 141
7.5 The Dirichlet problem and its solution for the disc......Page 148
7.6 Radially symmetric solutions of the two-dimensional wave equation and Bessel’s equation......Page 151
7.7 Existence and uniqueness of solutions, well-posed problems......Page 153
7.8 Appendix: proof of the Mean Value Theorem for harmonic functions......Page 159
8.1 A simple case......Page 164
8.2 Some algebraic preliminaries......Page 169
8.3 The Fredholm Alternative Theorem......Page 170
8.4 A worked example......Page 175
9 Hilbert–Schmidt Theory......Page 180
9.1 Eigenvalues are real and eigenfunctions corresponding to distinct eigenvalues are orthogonal......Page 181
9.2 Orthonormal families of functions and Bessel’s inequality......Page 183
9.3 Some results about eigenvalues deducible from Bessel’s inequality......Page 184
9.4 Description of the sets of all eigenvalues and all eigenfunctions......Page 188
9.5 The Expansion Theorem......Page 191
10.1 An example of Picard’s method......Page 196
10.2 Powers of an integral operator......Page 198
10.3 Iterated kernels......Page 199
10.4 Neumann series......Page 201
10.5 A remark on the convergence of iterative methods......Page 203
11.1 The fundamental problem......Page 204
11.2 Some classical examples from mechanics and geometry......Page 206
11.3 The derivation of Euler’s equation for the fundamental problem......Page 211
11.4 The special case F = F(y, y')......Page 213
11.5 When F contains higher derivatives of y......Page 216
11.6 When F contains more dependent functions......Page 218
11.7 When F contains more independent variables......Page 223
11.8 Integral constraints......Page 226
11.9 Non-integral constraints......Page 230
11.10 Varying boundary conditions......Page 233
12 The Sturm–Liouville Equation......Page 240
12.1 Some elementary results on eigenfunctions and eigenvalues......Page 241
12.2 The Sturm–Liouville Theorem......Page 244
12.3 Derivation from a variational principle......Page 248
12.4 Some singular equations......Page 250
12.5 The Rayleigh–Ritz method......Page 254
13 Series Solutions......Page 258
13.1 Power series and analytic functions......Page 260
13.2 Ordinary and regular singular points......Page 263
13.3 Power series solutions near an ordinary point......Page 265
13.4 Extended power series solutions near a regular singular point: theory......Page 273
13.5 Extended power series solutions near a regular singular point: practice......Page 277
13.6 The method of Frobenius......Page 290
13.7 Summary......Page 295
13.8 Appendix: the use of complex variables......Page 297
14 Transform Methods......Page 302
14.1 The Fourier transform......Page 303
14.2 Applications of the Fourier transform......Page 307
14.3 The Laplace transform......Page 312
14.4 Applications of the Laplace transform......Page 317
14.5 Applications involving complex analysis......Page 324
14.6 Appendix: similarity solutions......Page 338
15.1 The phase-plane and stability......Page 342
15.2 Linear theory......Page 349
15.3 Some non-linear systems......Page 358
15.4 Linearisation......Page 365
Appendix: the solution of some elementary ordinary differential equations......Page 368
Bibliography......Page 378
E......Page 384
L......Page 385
S......Page 386
W......Page 387