Differential and Integral Equations

Peter J. Collins
Senior Research Fellow, St Edmund Hall, Oxford

Contents
Preface v
How to use this book xi
Prerequisites xiii
0 Some Preliminaries 1
1 Integral Equations and Picard’s Method 5
1.1 Integral equations and their relationship to diferential equations . . . . 5
1.2 Picard’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Existence and Uniqueness 19
2.1 First-order differential equations in a single independent variable . . . . 20
2.2 Two simultaneous equations in a single variable . . . . . . . . . . . . . . 26
2.3 A second-order equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 The Homogeneous Linear Equation and Wronskians 33
3.1 Some linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Wronskians and the linear independence of solutions of the second-order
homogeneous linear equation . . . . . . . . . . . . . . . . . . . . . . . . 36
4 The Non-Homogeneous Linear Equation 41
4.1 The method of variation of parameters . . . . . . . . . . . . . . . . . . . 43
4.2 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 First-Order Partial Differential Equations 59
5.1 Characteristics and some geometrical considerations . . . . . . . . . . . 60
5.2 Solving characteristic equations . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Fitting boundary conditions to general solutions . . . . . . . . . . . . . 70
5.5 Parametric solutions and domains of definition . . . . . . . . . . . . . . 75
5.6 A geometric interpretation of an analytic condition . . . . . . . . . . . . 83
6 Second-Order Partial Differential Equations 85
6.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Reduction to canonical form . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Problems involving boundary conditions . . . . . . . . . . . . . . . . . . 103
6.5 Appendix: technique in the use of the chain rule . . . . . . . . . . . . . 113
7 The Diffusion and Wave Equations and the Equation of Laplace 115
7.1 The equations to be considered . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 One-dimensional heat conduction . . . . . . . . . . . . . . . . . . . . . . 119
7.3 Transverse waves in a finite string . . . . . . . . . . . . . . . . . . . . . . 123
7.4 Separated solutions of Laplace’s equation in polar co-ordinates and
Legendre’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5 The Dirichlet problem and its solution for the disc . . . . . . . . . . . . 133
7.6 Radially symmetric solutions of the two-dimensional wave equation and
Bessel’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.7 Existence and uniqueness of solutions, well-posed problems . . . . . . . 138
7.8 Appendix: proof of the Mean Value Theorem for harmonic functions . . 144
8 The Fredholm Alternative 149
8.1 A simple case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2 Some algebraic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.3 The Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . 155
8.4 A worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9 Hilbert–Schmidt Theory 165
9.1 Eigenvalues are real and eigenfunctions corresponding to distinct
eigenvalues are orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.2 Orthonormal families of functions and Bessel’s inequality . . . . . . . . . 168
9.3 Some results about eigenvalues deducible from Bessel’s inequality . . . . 169
9.4 Description of the sets of all eigenvalues and all eigenfunctions . . . . . . 173
9.5 The Expansion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10 Iterative Methods and Neumann Series 181
10.1 An example of Picard’s method . . . . . . . . . . . . . . . . . . . . . . . 181
10.2 Powers of an integral operator . . . . . . . . . . . . . . . . . . . . . . . . 183
10.3 Iterated kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.4 Neumann series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.5 A remark on the convergence of iterative methods . . . . . . . . . . . . . 188
11 The Calculus of Variations 189
11.1 The fundamental problem . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11.2 Some classical examples from mechanics and geometry . . . . . . . . . . 191
11.3 The derivation of Euler’s equation for the fundamental problem . . . . . 196
11.4 The special case F = F(y, y

) . . . . . . . . . . . . . . . . . . . . . . . . 198
11.5 When F contains higher derivatives of y . . . . . . . . . . . . . . . . . . 201
11.6 When F contains more dependent functions . . . . . . . . . . . . . . . . 203
11.7 When F contains more independent variables . . . . . . . . . . . . . . . 208
11.8 Integral constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.9 Non-integral constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.10 Varying boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 218
12 The Sturm–Liouville Equation 225
12.1 Some elementary results on eigenfunctions and eigenvalues . . . . . . . . 226
12.2 The Sturm–Liouville Theorem . . . . . . . . . . . . . . . . . . . . . . . . 229
12.3 Derivation from a variational principle . . . . . . . . . . . . . . . . . . . 233
12.4 Some singular equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
12.5 The Rayleigh–Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . . 239
13 Series Solutions 243
13.1 Power series and analytic functions . . . . . . . . . . . . . . . . . . . . . 245
13.2 Ordinary and regular singular points . . . . . . . . . . . . . . . . . . . . 248
13.3 Power series solutions near an ordinary point . . . . . . . . . . . . . . . 250
13.4 Extended power series solutions near a regular singular point: theory . . 258
13.5 Extended power series solutions near a regular singular point: practice . 262
13.6 The method of Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . 275
13.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
13.8 Appendix: the use of complex variables . . . . . . . . . . . . . . . . . . 282
14 Transform Methods 287
14.1 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
14.2 Applications of the Fourier transform . . . . . . . . . . . . . . . . . . . . 292
14.3 The Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
14.4 Applications of the Laplace transform . . . . . . . . . . . . . . . . . . . 302
14.5 Applications involving complex analysis . . . . . . . . . . . . . . . . . . 309
14.6 Appendix: similarity solutions . . . . . . . . . . . . . . . . . . . . . . . . 323
15 Phase-Plane Analysis 327
15.1 The phase-plane and stability . . . . . . . . . . . . . . . . . . . . . . . . 327
15.2 Linear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
15.3 Some non-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
15.4 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Appendix: the solution of some elementary ordinary differential
equations 353
Bibliography 363
Index 369


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