Numerical Solution of Sturm-Liouville Problems

Pryce John D.Numerical Solution of Sturm-Liouville Problems (Numerical Mathematics and Scientific Computation)(OUP,1993)(ISBN 0198534159)(600dpi)(337p) ......Page 4
Copyright ......Page 5
Contents ix ......Page 10
Preface  v  ......Page 6
1 Introductory background 1 ......Page 15
1.1.1 Basic properties of functions of a real variable 2 ......Page 16
1.1.3 Basic assumptions about Sturm-Liouville problems 4 ......Page 18
1.2.1 A simple physical example 6 ......Page 20
1.2.2 Further examples of Sturm-Liouville problems 12 ......Page 26
1.3 Summary 18 ......Page 32
2.1 Existence, uniqueness, linearity 19 ......Page 33
2.2 Self-adjointness 21 ......Page 35
2.3 Comparison, perturbation and conditioning theorems 26 ......Page 40
2.4.1 Parameters in the coefficient functions 28 ......Page 42
2.4.2 Dependence on the BCs and endpoints 30 ......Page 44
2.5.1 Transformation to SLDE with same eigenvalues 33 ......Page 47
2.5.2 Transformation to SLDE with related eigenvalues 34 ......Page 48
2.5.3 Reduction of SLDE to nonlinear first-order equation 35 ......Page 49
2.6.1 The JWKB approximation 40 ......Page 54
2.6.3 Estimating eigenvalues I 42 ......Page 56
2.6.4 Estimating eigenvalues II 43 ......Page 57
2.6.5 Estimating eigenvalues III 45 ......Page 59
2.7 Summary 51 ......Page 65
3 Simple matrix methods 52 ......Page 66
3.1.1 Method A. Simple centred differences 54 ......Page 68
3.1.2 Method B. Numerov’s method 56 ......Page 70
3.1.3 Extrapolation 57 ......Page 71
3.1.4 Correction methods 60 ......Page 74
3.2.1 Simple differences and Numerov 62 ......Page 76
3.2.3 Correction methods, general BCs 64 ......Page 78
3.3 Sturm-Liouville problems not in Liouville normal form 68 ......Page 82
3.4 Algorithms for tridiagonal matrix eigenvalues 69 ......Page 83
3.5 Convergence theory 72 ......Page 86
3.6 Summary 74 ......Page 88
4 Variational methods 75 ......Page 89
4.1.1 The variational principle 76 ......Page 90
4.1.2 Using a finite element basis 79 ......Page 93
4.2 Convergence theory and convergence improvement 81 ......Page 95
4.2.1 Extrapolation and correction 82 ......Page 96
4.2.2 Numerical example 83 ......Page 97
4.3 Perturbation of a problem with known eigenfunctions 84 ......Page 98
4.4 Summary 87 ......Page 101
5.1 Basic idea 88 ......Page 102
5.1.1 Improvements 90 ......Page 104
5.2.1 The scaled Priifer equations 92 ......Page 106
5.2.2 Scaled Priifer miss-distance and shooting method 97 ......Page 111
5.2.3 The normalized eigenfunction 99 ......Page 113
5.2.4 Rescaling at jumps in S 100 ......Page 114
5.3 Implementation of scaled Priifer algorithms 102 ......Page 116
5.3.1 Choice of scaling function 104 ......Page 118
5.3.2 Exponential growth 107 ......Page 121
5.3.3 Choice of matching point 108 ......Page 122
5.3.4 Eigenvalue error estimation and control 112 ......Page 126
5.4 Summary 115 ......Page 129
6 Pruess methods 117 ......Page 131
6.1.1 Basic convergence results 118 ......Page 132
6.1.2 Using extrapolation 120 ......Page 134
6.2 Advantages of Pruess methods 121 ......Page 135
6.3.1 Advancing a step 122 ......Page 136
6.3.3 Error estimation and mesh selection 129 ......Page 143
6.4 Summary 134 ......Page 148
7 Singular SLPs: theory 136 ......Page 150
7.1 Singular problems 137 ......Page 151
7.1.1 Admissible functions 138 ......Page 152
7.1.2 The spectrum 140 ......Page 154
7.1.3 Classifying the spectrum 143 ......Page 157
7.2 Classification of singular endpoints 146 ......Page 160
7.2.1 Limit-point and limit-circle endpoints 147 ......Page 161
7.2.2 Form of boundary conditions for limit-circle problems 151 ......Page 165
7.2.3 Oscillatory and nonoscillatory behaviour 153 ......Page 167
7.3.1 Large and small solutions at an endpoint 158 ......Page 172
7.3.3 LC case: the Friedrichs extension 159 ......Page 173
7.4.1 Theorems about convergence 162 ......Page 176
7.4.2 Estimating the rate of convergence 167 ......Page 181
7.5 Summary 172 ......Page 186
8 Singular SLPs: numerical treatment 173 ......Page 187
8.1 Methods implemented in current software 174 ......Page 188
8.2 The Fulton-Pruess endpoint classification algorithm 178 ......Page 192
8.3 About the numerical experiments 182 ......Page 196
8.4.1 Test: convergence using truncated BC u — 0 184 ......Page 198
8.4.2 Test: non-Friedrichs boundary conditions 186 ......Page 200
8.5.1 Choice of truncated interval 190 ......Page 204
8.5.2 Infinite wells 193 ......Page 207
8.5.3 Performance comparisons 194 ......Page 208
8.6 Summary 197 ......Page 211
9 Computing and manipulating eigenfunctions 198 ......Page 212
9.1 Perturbation theory 199 ......Page 213
9.2 Convergence theory 203 ......Page 217
9.3.1 Exponential growth 208 ......Page 222
9.3.2 Computing small tails 211 ......Page 225
9.3.3 Eigenfunction error estimation 212 ......Page 
9.4 Presenting eigenfunctions to the user 216 ......Page 230
9.5 Summary 218 ......Page 232
10 The computation of resonances 219 ......Page 233
10.1 Notation and informal description 220 ......Page 234
10.2 Theory of the time delay 224 ......Page 238
10.3.1 Bracketing the resonance 226 ......Page 240
10.3.2 Iteration to find bracketing interval 227 ......Page 241
10.3.3 Asymptotic correction of r(A) 230 ......Page 244
10.3.4 Computation of r00 232 ......Page 246
10.3.5 Interpolation between A_ and A+ 233 ......Page 247
10.4 An implementation 234 ......Page 248
10.5 Numerical results and comparisons 235 ......Page 249
10.6 Summary 241 ......Page 255
11 Further topics 242 ......Page 256
11.1 ‘Other’ methods for the SLP 243 ......Page 257
11.2 Exploiting parallelism 247 ......Page 261
11.3.1 The bounds game 249 ......Page 263
11.3.2 Two-sided bounds by Rayleigh-Ritz 251 ......Page 265
11.4.1 Simon’s theory 253 ......Page 267
11.4.2 Practical estimates 258 ......Page 272
11.5 Non-separated boundary conditions 260 ......Page 274
11.6 Problems with more general A-dependence 262 ......Page 276
11.7 Multiparameter eigenproblems 265 ......Page 279
11.8 Vector Sturm-Liouville problems 267 ......Page 281
11.9 Computation of the Weyl m(A) function 270 ......Page 284
11.10 Computation of the spectral density function 271 ......Page 285
11.11 Summary 274 ......Page 288
12 Conclusion 275......Page 289
A Eigenvalues of Paine problems 278 ......Page 292
B List of test problems 279 ......Page 293
B.l Notes to the problems 280 ......Page 294
B.2 The problem set 281 ......Page 295
B.3 Specification of Benchmark 296 ......Page 310
C Available Sturm-Liouville software 297 ......Page 311
Index 315 329......Page 329
cover 1......Page 1
back cover 323......Page 337