Computational methods in ordinary differential equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London [u.a.]
Wiley
1974
|
| Edition: | Repr. |
| Series: | Introductory mathematics for scientists and engineers
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 278 S. |
| ISBN: | 0471511943 |
Staff View
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Record in the Search Index
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| adam_text | Contents
1 Preliminaries
1.1 Notation 1
1.2 Prerequisites ........ 1 •
1.3 Initial value problems for first order ordinary differential
equations ......... 2
1.4 Initial value problems for systems of first order ordinary
differential equations 3
1.5 Reduction of higher order differential equations to first
order systems ........ 4
1.6 First order linear systems with constant coefficients . . 5
1.7 Linear difference equations with constant coefficients . 7
1.8 Iterative methods for non linear equations 10
2 Linear multistep methods I: basic theory
2.1 The general linear multistep method . . . .11
2.2 Derivation through Taylor expansions .... 13
2.3 Derivation through numerical integration ... 15
2.4 Derivation through interpolation 17
2.5 Convergence 21
2.6 Order and error constant 23
2.7 Local and global truncation error. .... 27
2.8 Consistency and zero stability 30
2.9 Attainable order of zero stable methods ... 37
2.10 Specification of linear multistep methods 40
3 Linear multistep methods II: application
3.1 Problems in applying linear multistep methods . . 44
3.2 Starting values 45
3.3 A bound for the local truncation error .... 48
3.4 A bound for the global truncation error ... 55
3.5 Discussion on error bounds 59
3.6 Weak stability theory 64
xiii
xiv Contents
3.7 General methods for finding intervals of absolute and
relative stability........ 11
3.8 Comparison of explicit and implicit linear multistep
methods ......... 84
3.9 Predictor corrector methods ..... 85
3.10 The local truncation error of predictor corrector methods:
Milne s device ........ 88
3.11 Weak stability of predictor corrector methods . . 96
3.12 Step control policy for predictor corrector methods . 101
3.13 Choice of predictor corrector methods . . . .102
3.14 Implementation of predictor corrector methods: Gear s
method 108
4 Runge Kutta methods
4.1 Introduction . . . . . . . .114
4.2 Order and convergence of the general explicit one step
method . . . . . . . . .115
4.3 Derivation of classical Runge Kutta methods . .116
4.4 Runge Kutta methods of order greater than four . . 121
4.5 Error bounds for Runge Kutta methods . . .123
4.6 Error estimates for Runge Kutta methods . . . 130
4.7 Weak stability theory for Runge Kutta methods . . 135
4.8 Runge Kutta methods with special properties . . 140
4.9 Comparison with predictor corrector methods . . 144
4.10 Implicit Runge Kutta methods ..... 149
4.11 Block methods 156
5 Hybrid methods
5.1 Hybrid formulae 162
5.2 Hybrid predictor corrector methods . . . .167
5.3 The local truncation error of hybrid methods. . . 170
5.4 Weak stability of hybrid methods 181
5.5 Generalizations. . . . . . . .184
5.6 Comparison with linear multistep and Runge Kutta
methods 185
6 Extrapolation methods
6.1 Polynomial extrapolation ...... 186
6.2 Application to initial value problems in ordinary differen¬
tial equations . . . . . . . .189
Contents xv
6.3 Existence of asymptotic expansions :Gragg s method . 190
6.4 Weak stability 194
6.5 Rational extrapolation: the GBS method . . .195
7 Methods for special problems
7.1 Introduction 198
7.2 Obrechkoff methods 199
7.3 Problems with oscillatory solutions .... 205
7.4 Problems whose solutions possess singularities . . 209
8 First order systems and the problem of stiffness
8.1 Applicability to systems . . . . . .217
8.2 Applicability of linear multistep methods . . .219
8.3 Applicability of Runge Kutta methods. . . . 225
8.4 Applicability of the methods of chapters 5, 6, and 7 . 228
8.5 Stiff systems 228
8.6 The problem of stability for stiff systems . . . 233
8.7 Some properties of rational approximations to the expo¬
nential 236
8.8 The problem of implicitness for stiff systems . . . 238
8.9 Linear multistep methods for stiff systems . . . 240
8.10 Runge Kutta methods for stiff systems. . . . 243
8.11 Other methods for stiff systems ..... 245
8.12 Application to partial differential equations . . . 249
9 Linear multistep methods for a special class of second order
differential equations
9.1 Introduction 252
9.2 Order, consistency, zero stability, and convergence . 253
9.3 Specification of methods ...... 254
9.4 Application to initial value problems . . . .256
9.5 Application to boundary value problems . . . 259
Appendix: The shooting method for two point boundary value
problems 262
References 265
Index .... ... 275
|
| adam_txt |
Contents
1 Preliminaries
1.1 Notation 1
1.2 Prerequisites . 1 •
1.3 Initial value problems for first order ordinary differential
equations . 2
1.4 Initial value problems for systems of first order ordinary
differential equations 3
1.5 Reduction of higher order differential equations to first
order systems . 4
1.6 First order linear systems with constant coefficients . . 5
1.7 Linear difference equations with constant coefficients . 7
1.8 Iterative methods for non linear equations 10
2 Linear multistep methods I: basic theory
2.1 The general linear multistep method . . . .11
2.2 Derivation through Taylor expansions . 13
2.3 Derivation through numerical integration . 15
2.4 Derivation through interpolation 17
2.5 Convergence 21
2.6 Order and error constant 23
2.7 Local and global truncation error. . 27
2.8 Consistency and zero stability 30
2.9 Attainable order of zero stable methods . 37
2.10 Specification of linear multistep methods 40
3 Linear multistep methods II: application
3.1 Problems in applying linear multistep methods . . 44
3.2 Starting values 45
3.3 A bound for the local truncation error . 48
3.4 A bound for the global truncation error . 55
3.5 Discussion on error bounds 59
3.6 Weak stability theory 64
xiii
xiv Contents
3.7 General methods for finding intervals of absolute and
relative stability. 11
3.8 Comparison of explicit and implicit linear multistep
methods . 84
3.9 Predictor corrector methods . 85
3.10 The local truncation error of predictor corrector methods:
Milne's device . 88
3.11 Weak stability of predictor corrector methods . . 96
3.12 Step control policy for predictor corrector methods . 101
3.13 Choice of predictor corrector methods . . . .102
3.14 Implementation of predictor corrector methods: Gear's
method 108
4 Runge Kutta methods
4.1 Introduction . . . . . . . .114
4.2 Order and convergence of the general explicit one step
method . . . . . . . . .115
4.3 Derivation of classical Runge Kutta methods . .116
4.4 Runge Kutta methods of order greater than four . . 121
4.5 Error bounds for Runge Kutta methods . . .123
4.6 Error estimates for Runge Kutta methods . . . 130
4.7 Weak stability theory for Runge Kutta methods . . 135
4.8 Runge Kutta methods with special properties . . 140
4.9 Comparison with predictor corrector methods . . 144
4.10 Implicit Runge Kutta methods . 149
4.11 Block methods 156
5 Hybrid methods
5.1 Hybrid formulae 162
5.2 Hybrid predictor corrector methods . . . .167
5.3 The local truncation error of hybrid methods. . . 170
5.4 Weak stability of hybrid methods 181
5.5 Generalizations. . . . . . . .184
5.6 Comparison with linear multistep and Runge Kutta
methods 185
6 Extrapolation methods
6.1 Polynomial extrapolation . 186
6.2 Application to initial value problems in ordinary differen¬
tial equations . . . . . . . .189
Contents xv
6.3 Existence of asymptotic expansions :Gragg's method . 190
6.4 Weak stability 194
6.5 Rational extrapolation: the GBS method . . .195
7 Methods for special problems
7.1 Introduction 198
7.2 Obrechkoff methods 199
7.3 Problems with oscillatory solutions . 205
7.4 Problems whose solutions possess singularities . . 209
8 First order systems and the problem of stiffness
8.1 Applicability to systems . . . . . .217
8.2 Applicability of linear multistep methods . . .219
8.3 Applicability of Runge Kutta methods. . . . 225
8.4 Applicability of the methods of chapters 5, 6, and 7 . 228
8.5 Stiff systems 228
8.6 The problem of stability for stiff systems . . . 233
8.7 Some properties of rational approximations to the expo¬
nential 236
8.8 The problem of implicitness for stiff systems . . . 238
8.9 Linear multistep methods for stiff systems . . . 240
8.10 Runge Kutta methods for stiff systems. . . . 243
8.11 Other methods for stiff systems . 245
8.12 Application to partial differential equations . . . 249
9 Linear multistep methods for a special class of second order
differential equations
9.1 Introduction 252
9.2 Order, consistency, zero stability, and convergence . 253
9.3 Specification of methods . 254
9.4 Application to initial value problems . . . .256
9.5 Application to boundary value problems . . . 259
Appendix: The shooting method for two point boundary value
problems 262
References 265
Index . . 275 |
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| author_facet | Lambert, John D. 1932- |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Repr. |
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| id | DE-604.BV021876701 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:03:36Z |
| indexdate | 2024-07-09T20:46:31Z |
| institution | BVB |
| isbn | 0471511943 |
| language | English |
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| physical | XV, 278 S. |
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| series2 | Introductory mathematics for scientists and engineers |
| spelling | Lambert, John D. 1932- Verfasser (DE-588)141742909 aut Computational methods in ordinary differential equations J. D. Lambert Repr. London [u.a.] Wiley 1974 XV, 278 S. txt rdacontent n rdamedia nc rdacarrier Introductory mathematics for scientists and engineers Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 Differentialgleichung (DE-588)4012249-9 s Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Datenverarbeitung (DE-588)4011152-0 s 2\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015092301&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lambert, John D. 1932- Computational methods in ordinary differential equations Numerisches Verfahren (DE-588)4128130-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd Datenverarbeitung (DE-588)4011152-0 gnd Differentialgleichung (DE-588)4012249-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4042805-9 (DE-588)4011152-0 (DE-588)4012249-9 (DE-588)4020929-5 |
| title | Computational methods in ordinary differential equations |
| title_auth | Computational methods in ordinary differential equations |
| title_exact_search | Computational methods in ordinary differential equations |
| title_exact_search_txtP | Computational methods in ordinary differential equations |
| title_full | Computational methods in ordinary differential equations J. D. Lambert |
| title_fullStr | Computational methods in ordinary differential equations J. D. Lambert |
| title_full_unstemmed | Computational methods in ordinary differential equations J. D. Lambert |
| title_short | Computational methods in ordinary differential equations |
| title_sort | computational methods in ordinary differential equations |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd Datenverarbeitung (DE-588)4011152-0 gnd Differentialgleichung (DE-588)4012249-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Numerisches Verfahren Numerische Mathematik Datenverarbeitung Differentialgleichung Gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015092301&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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