Collocation Methods for Volterra Integral and Related Functional Differential Equations

✍ Scribed by Hermann Brunner

• Publisher: Cambridge University Press
• Year: 2004
• Tongue: English
• Leaves: 613
• Series: Cambridge Monographs on Applied and Computational Mathematics
• Category: Library

✦ Synopsis

This is the first comprehensive introduction to collocation methods for the numerical solution of initial-value problems for ordinary differential equations, Volterra integral and integro-differential equations, and various classes of more general functional equations. It guides the reader from the "basics" to the current state-of-the-art level of the field, describes important problems and directions for future research, and highlights methods. The analysis includes numerous exercises and applications to the modelling of physical and biological phenomena.

✦ Table of Contents

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Content......Page 7
Preface......Page 11
Acknowledgements......Page 15
1.1.1 Collocation-based implicit Runge–Kutta methods......Page 17
1.1.2 Convergence and global order on I......Page 25
1.1.3 Local superconvergence results on Ih......Page 33
1.1.4 Nonlinear initial-value problems......Page 35
1.1.5 Collocation for ‘integrated’ ODEs......Page 37
1.1.6 Padé approximations to exp(z)......Page 42
1.2 Perturbed collocation methods......Page 45
1.3.1 Divergence of collocation solutions......Page 47
1.4 Higher-order ODEs......Page 50
1.5 Multistep collocation......Page 54
1.6 The discontinuous Galerkin method for ODEs......Page 56
1.8 The Peano theorems for interpolation and quadrature......Page 59
1.9 Preview: Collocation for Volterra equations......Page 62
1.10 Exercises......Page 63
1.11 Notes......Page 65
2.1.1 Linear VIEs of the second kind......Page 69
2.1.2 Linear convolution equations......Page 77
2.1.3 Systems of linear VIEs......Page 78
2.1.4 Linear VIEs of the first kind......Page 80
2.1.5 Nonlinear VIEs......Page 84
2.1.6 Volterra–Fredholm integral equations......Page 92
2.1.7 Volterra integral equations in IR......Page 94
2.1.8 Comparison theorems......Page 95
2.1.9 Discrete Volterra equations and and discrete Gronwall inequalities......Page 97
2.2.1 Meshes and piecewise polynomial spaces......Page 98
2.2.2 Piecewise polynomial collocation methods in …......Page 101
2.2.3 The fully discretised collocation equation......Page 104
2.2.4 Global convergence results......Page 108
2.2.5 Local superconvergence results......Page 113
2.2.6 Optimal orders for the discretised collocation solutions......Page 116
2.2.7 Divergence of collocation solutions in smoother collocation spaces......Page 119
2.2.8 Hermite-type collocation methods......Page 123
2.2.9 Multidimensional VIEs of the second kind......Page 124
2.2.10 Comparison with collocation for Fredholm integral equations......Page 126
2.2.11 Collocation for Volterra–Fredholm integral equations......Page 128
2.3.1 Global error analysis......Page 130
2.3.2 Local superconvergence results for nonlinear V2s......Page 131
2.3.3 Hammerstein-type VIEs: implicitly linear collocation......Page 132
2.4.1 The exact collocation equations for …......Page 136
2.4.2 Global convergence in …......Page 139
2.4.3 Collocation and global convergence in …......Page 144
2.4.4 Is local superconvergence on Ih possible?......Page 147
2.4.5 Fully discretised collocation for first-kind VIEs......Page 150
2.4.6 Direct versus indirect collocation......Page 151
2.4.7 Adjoint first-kind Volterra integral equations......Page 152
2.4.8 Nonlinear first-kind VIEs......Page 153
2.4.9 Collocation in smoother piecewise polynomial spaces......Page 154
2.5 Exercises and research problems......Page 155
2.6 Notes......Page 159
3.1.1 Linear VIDEs......Page 167
3.1.2 Neutral and higher-order VIDEs......Page 171
3.1.3 Nonlinear and non-standard VIDEs......Page 174
3.2.1 The exact collocation equations......Page 176
3.2.2 The fully discretised collocation equations......Page 180
3.2.3 Global convergence results......Page 184
3.2.4 Local superconvergence results......Page 189
3.2.5 Neutral and higher-order VIDEs......Page 191
3.3.1 Local superconvergence results......Page 199
3.3.2 Kernels of ‘non-standard’ form…......Page 201
3.4 Partial VIDEs: time-stepping......Page 202
3.5 Exercises and research problems......Page 204
3.6 Notes......Page 208
4.1.1 Definitions and notation......Page 212
4.1.2 Second-kind Volterra integral equations with non-vanishing delays......Page 214
4.1.3 First-kind VIEs with non-vanishing delays......Page 220
4.1.4 VIDEs with non-vanishing delays......Page 224
4.1.5 Nonlinear delay problems......Page 228
4.1.6 Volterra functional equations of neutral type......Page 230
4.2.1 Constrained and theta-invariant meshes......Page 233
4.2.2 Collocation and continuous implicit Runge–Kutta methods......Page 235
4.3.1 The exact collocation equations......Page 237
4.3.2 Global convergence results......Page 242
4.3.3 Local superconvergence results......Page 245
4.3.4 Nonlinear delay VIEs......Page 247
4.4.1 The collocation space …......Page 250
4.4.2 Direct versus indirect collocation......Page 252
4.5.1 The exact collocation equations......Page 253
4.5.2 Global convergence results......Page 256
4.5.3 Local superconvergence results......Page 258
4.5.4 Neutral VFIDEs......Page 259
4.6.1 DDEs with state-dependent delays......Page 261
4.7 Exercises and research problems......Page 262
4.8 Notes......Page 265
5.1.1 Volterra’s 1897 paper and some early history......Page 269
5.1.2 Linear differential equations with proportional delays......Page 272
5.1.3 Linear Volterra integral equations with proportional delays......Page 274
5.1.4 Volterra integro-differential equations with proportional delays......Page 276
5.1.5 Embedding techniques......Page 278
5.2 Collocation for DDEs with proportional delays......Page 282
5.2.1 Collocation and continuous Runge–Kutta methods......Page 283
5.2.2 Global convergence results: uniform Ih......Page 292
5.2.3 Attainable order at t=t1=h......Page 297
5.3 Second-kind VIEs with proportional delays......Page 300
5.3.1 The collocation equations for uniform meshes......Page 301
5.3.2 Two prominent DVIEs with proportional delay......Page 305
5.3.3 Global convergence results: uniform Ih......Page 308
5.3.4 A more general VIE with proportional delay......Page 314
5.3.5 Attainable order at t = t1 = h......Page 315
5.3.6 Local superconvergence analysis on uniform meshes......Page 317
5.3.7 Local superconvergence on geometric meshes......Page 318
5.4.1 Collocation in…......Page 320
5.5.1 The collocation equations and their discretisations......Page 324
5.5.2 Convergence results on uniform meshes......Page 334
5.5.3 Collocation on quasi-geometric meshes......Page 337
5.5.4 Superconvergence results on quasi-geometric meshes......Page 342
5.5.5 More general vanishing delays......Page 347
5.6 Exercises and research problems......Page 349
5.7 Notes......Page 353
6.1.1 The Mittag-Leffler function......Page 356
6.1.2 Linear VIEs of the second kind......Page 358
6.1.3 Nonlinear VIEs of the second kind......Page 367
6.1.4 Linear VIEs of the first kind......Page 368
6.1.5 Nonlinear VIEs of the first kind......Page 372
6.1.6 Weakly singular Volterra equations with non-vanishing delays......Page 374
6.1.7 Comparison theorems and Gronwall-type inequalities......Page 375
6.2.1 The exact collocation equations......Page 377
6.2.2 The fully discretised collocation equations......Page 382
6.2.3 Approximation of functions in Hölder spaces and graded meshes......Page 386
6.2.4 The error in product quadrature on graded meshes......Page 390
6.2.5 Global convergence results......Page 392
6.2.6 Global and local superconvergence results......Page 401
6.2.7 Fully discretised collocation and product integration methods......Page 404
6.2.8 Comparison with weakly singular Fredholm integral equations......Page 406
6.2.9 Hammerstein-type VIEs: Implicitly linear collocation......Page 407
6.3 Collocation for weakly singular first-kind VIE......Page 411
6.3.1 Collocation in…......Page 412
6.3.2 Collocation in…......Page 415
6.3.3 Convergence analysis; conjectures......Page 417
6.3.4 Fully discretised collocation......Page 422
6.4.1 Weakly singular VIEs of the second kind......Page 425
6.5.1 Collocation for delay equations of the second kind......Page 426
6.5.2 Collocation for weakly singular delay VIEs of the first kind......Page 428
6.6 Exercises and research problems......Page 429
6.7 Notes......Page 434
7.1.1 Linear weakly singular VIDEs......Page 440
7.1.2 Nonlinear VIDEs with weakly singular kernels......Page 445
7.1.3 Neutral and higher-order VIDEs......Page 446
7.1.4 Weakly singular VIDEs with delay arguments......Page 449
7.1.5 A generalisation of Gronwall’s Lemma......Page 450
7.2.1 The exact collocation equations......Page 451
7.2.2 The fully discretised collocation equations......Page 456
7.2.3 Global convergence results......Page 459
7.3 Hammerstein-type VIDEs with weakly singular kernels......Page 465
7.4 Higher-order weakly singular VIDEs......Page 466
7.5 Non-polynomial spline collocation methods......Page 471
7.6.1 Weakly singular VIDEs with non-vanishing delays......Page 472
7.7 Exercises and research problems......Page 473
7.8 Notes......Page 476
8.1 Basic theory of DAEs and IAEs......Page 479
8.1.1 DAEs: a brief introduction......Page 480
8.1.2 IAEs with smooth kernels......Page 486
8.1.3 IDAEs with smooth kernels......Page 490
8.1.4 IAEs and IDAEs with weakly singular kernels......Page 493
8.2.1 The collocation equations for index-1 problems......Page 495
8.2.2 Collocation for semi-explicit index-2 DAEs......Page 497
8.2.3 Numerically properly formulated DAEs......Page 498
8.3.1 The collocation equations for Volterra IAEs......Page 500
8.3.2 Convergence results......Page 502
8.4.1 The collocation equations for Volterra IDAEs......Page 505
8.4.2 Convergence results......Page 507
8.5.1 Collocation for weakly singular IAEs......Page 509
8.5.2 Collocation for weakly singular IDAEs......Page 511
8.6 Exercises and research problems......Page 513
8.7 Notes......Page 515
9.1 Semigroups and abstract resolvent theory......Page 519
9.2 C-algebra techniques and invertibility of approximating operator sequences......Page 520
9.3 Abstract DAEs......Page 521
References......Page 522
Index......Page 604