Preface ...................................................... xiii
Chapter 1 Introduction
References ...................................................... 5
Chapter 2 Shooting Methods
2.1 Introduction ............................................... 6
2.2 Two-Point Boundary Value Problem Statement ................. 6
2.3 Reduction of nth-Order Equation ............................ 8
2.4 Analytical Solution ........................................ 9
2.5 Numerical Solution ........................................ 10
2.6 Stability ................................................. 11
References ..................................................... 15
Chapter 3 Method of Adjoints
3.1 Introduction .............................................. 17
3.2 Linear Two-Point Boundary Value Problem ................... 17
3.3 Discussion ................................................ 22
3.4 Example: Linear System .................................... 23
3.5 Example: Problem of Section 2.4 ........................... 25
3.6 Nonlinear Two-Point Boundary Value Problems ............... 27
3.7 Discussion ................................................ 31
3.8 Implicit Boundary Conditions .............................. 32
3.9 Example: Implicit Boundary Conditions .................... 41
3.10 Numerical Examples: Method of Adjoints .................... 43
References ..................................................... 49
Chapter 4 Method of Complementary Functions
4.1 Introduction .............................................. 50
4.2 Linear Combination of Solutions ........................... 51
4.3 Method of Complementary Functions ......................... 56
4.4 Example ................................................... 60
4.5 Discussion ................................................ 61
4.6 Alternative Derivations, Methods of Adjoints and
Complementary Functions ................................... 63
4.7 Comparison of the Two Methods ............................. 67
4.8 Practical Numerical Experience ............................ 69
4.9 Numerical Linear Independence ............................. 71
4.10 Gram-Schmidt Process ...................................... 73
4.11 Orthonormalization and Differential Equations ............. 78
4.12 Conte's Method ............................................ 83
References ..................................................... 86
Chapter 5 Quasilinearization
5.1 Introduction .............................................. 87
5.2 Quasilinearization ........................................ 87
5.3 Discussion ................................................ 90
5.4 Example ................................................... 93
5.5 Convergence of the Quasilinearization Method .............. 97
References .................................................... 108
Chapter 6 Newton-Raphson Method and Two-Point Boundary
Value Problems
6.1 Introduction ............................................. 110
6.2 Newton-Raphson Method, Finite Set of Nonlinear
Equations ................................................ 1ll
6.3 Kantorovich Theorem ...................................... 114
6.4 Kantorovich Theorem for a Finite Set of Nonlinear
Equations ................................................ 121
6.5 Newton-Raphson Method as Contraction Mapping ............. 123
6.6 Comparison of Kantorovich's Theorem and Contraction
Mapping .................................................. 125
6.7 Method of Adjoints and Newton-Raphson Method ............. 128
6.8 Numerical Results ........................................ 131
6.9 Modified Method of Adjoints .............................. 138
6.10 Friedrichs' Solution of P(x) = 0 ......................... 140
6.11 Quasilinearization and Newton-Raphson Method ............. 143
References .................................................... 151
Chapter 7 Continuation
7.1 Introduction ............................................. 153
7.2 Continuation Method in Two-Point Boundary Value
Problems ................................................. 154
7.3 Discussion of the Continuation Method .................... 155
7.4 Numerical Results ........................................ 158
7.5 Justification for the Continuation Method in Two-Point
Boundary Value Problems .................................. 164
7.6 Continuation Method of Ficken ............................ 169
7.7 Continuation in Quasilinearization ....................... 172
7.8 Discussion of the Procedure .............................. 174
7.9 Numerical Examples ....................................... 176
7.10 A Perturbation Technique ................................. 185
7.11 Justification of the Perturbation Method ................. 186
7.12 Numerical Experience ..................................... 191
7.13 Discussion ............................................... 198
References .................................................... 199
Chapter 8 Finite Difference Methods and Related Topics
8.1 Introduction ............................................. 201
8.2 Finite Difference Method ................................. 202
8.3 Discussion of Finite Difference Method ................... 204
8.4 Problems in Setting Up the Finite Difference
Equations ................................................ 205
8.5 Some Matrix Theory ....................................... 212
8.6 Numerical Examples ....................................... 214
8.7 Parallel or Multiple Shooting ............................ 217
8.8 Quasilinearization and Finite Difference Equations ....... 221
8.9 Boundary Value Version of Initial Value Problem .......... 226
References .................................................... 230
Chapter 9 Second-Order Newton-Raphson Methods
9.1 Introduction ............................................. 232
9.2 Second-Order Newton-Raphson Methods ...................... 233
9.3 Sets of Equations ........................................ 235
9.4 Second-Order Methods for P(x) = 0 ........................ 242
9.5 Calculation of the Partial Derivatives ................... 245
9.6 Numerical Experience ..................................... 256
References .................................................... 264
Appendix ...................................................... 265
Author Index .................................................. 271
Subject Index ................................................. 273
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