Solving differential equations in R:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Heidelberg [und 3 weitere]
Springer
2012
|
| Schriftenreihe: | Use R!
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XVI, 248 Seiten Illustrationen, Diagramme |
| ISBN: | 9783642280696 9783642280702 |
Internformat
MARC
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| 100 | 1 | |a Soetaert, Karline |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Solving differential equations in R |c Karline Soetaert ; Jeff Cash ; Francesca Mazzia |
| 264 | 1 | |a Heidelberg [und 3 weitere] |b Springer |c 2012 | |
| 300 | |a XVI, 248 Seiten |b Illustrationen, Diagramme | ||
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
1 DIFFERENTIAL EQUATIONS 1
1.1 BASIC THEORY O F ORDINARY DIFFERENTIAL EQUATIONS 1
1.1.1 FIRST ORDER DIFFERENTIAL EQUATIONS 1
1.1.2 ANALYTIC AND NUMERICAL SOLUTIONS 2
1.1.3 HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS 3
1.1.4 INITIAL AND BOUNDARY VALUES 4
1.1.5 EXISTENCE AND UNIQUENESS OF ANALYTIC SOLUTIONS 5
1.2 NUMERICAL METHODS 6
1.2.1 THE EULER METHOD 6
1.2.2 IMPLICIT METHODS 7
1.2.3 ACCURACY AND CONVERGENCE O F NUMERICAL METHODS 8
1.2.4 STABILITY AND CONDITIONING 9
1.3 OTHER TYPES O F DIFFERENTIAL EQUATIONS 11
1.3.1 PARTIAL DIFFERENTIAL EQUATIONS 11
1.3.2 DIFFERENTIAL ALGEBRAIC EQUATIONS 12
1.3.3 DELAY DIFFERENTIAL EQUATIONS 13
REFERENCES 13
2 INITIAL VALUE PROBLEMS 15
2.1 RUNGE-KUTTA METHODS 15
2.1.1 EXPLICIT RUNGE-KUTTA FORMULAE 15
2.1.2 DERIVING A RUNGE-KUTTA FORMULA 17
2.1.3 IMPLICIT RUNGE-KUTTA FORMULAE 22
2.2 LINEAR MULTISTEP METHODS 22
2.2.1 CONVERGENCE, STABILITY AND CONSISTENCY 23
2.2.2 ADAMS METHODS 25
2.2.3 BACKWARD DIFFERENTIATION FORMULAE 27
2.2.4 VARIABLE ORDER - VARIABLE COEFFICIENT FORMULAE FOR LINEAR
MULTISTEP METHODS 29
2.3 BOUNDARY VALUE METHODS 30
2.4 MODIFIED EXTENDED BACKWARD DIFFERENTIATION FORMULAE 31
XI
HTTP://D-NB.INFO/1018408258
IMAGE 2
X I I C O N T E N T S
2.5 STIFF PROBLEMS 32
2.5.1 STIFFNESS DETECTION 33
2.5.2 NON-STIFFNESS TEST 34
2.6 IMPLEMENTING IMPLICIT METHODS 34
2.6.1 FIXED-POINT ITERATION TO CONVERGENCE 34
2.6.2 CHORD ITERATION 35
2.6.3 PREDICTOR-CORRECTOR METHODS 36
2.6.4 NEWTON ITERATION FOR IMPLICIT RUNGE-KUTTA METHODS 36
2.7 CODES TO SOLVE INITIAL VALUE PROBLEMS 37
2.7.1 CODES TO SOLVE NON-STIFF PROBLEMS 38
2.7.2 CODES TO SOLVE STIFF PROBLEMS 38
2.7.3 CODES THAT SWITCH BETWEEN STIFF AND NON-STIFF SOLVERS 38
REFERENCES 39
3 SOLVING ORDINARY DIFFERENTIAL EQUATIONS IN R 41
3.1 IMPLEMENTING INITIAL VALUE PROBLEMS IN R 41
3.1.1 A DIFFERENTIAL EQUATION COMPRISING ONE VARIABLE 42
3.1.2 MULTIPLE VARIABLES: THE LORENZ MODEL 44
3.2 RUNGE-KUTTA METHODS 45
3.2.1 RIGID BODY EQUATIONS 47
3.2.2 ARENSTORF ORBITS 49
3.3 LINEAR MULTISTEP METHODS 51
3.3.1 SEVEN MOVING STARS 52
3.3.2 A STIFF CHEMICAL EXAMPLE 5 6
3.4 DISCONTINUOUS EQUATIONS, EVENTS 59
3.4.1 PHARMACOKINETIC MODELS 60
3.4.2 A BOUNCING BALL 64
3.4.3 TEMPERATURE IN A CLIMATE-CONTROLLED ROOM 66
3.5 METHOD SELECTION 68
3.5.1 THE VAN DER POL EQUATION 7 0
3.6 EXERCISES 75
3.6.1 GETTING STARTED WITH IVP 75
3.6.2 THE ROBERTSON PROBLEM 76
3.6.3 DISPLAYING RESULTS IN A PHASE-PLANE GRAPH 76
3.6.4 EVENTS AND ROOTS 78
3.6.5 STIFF PROBLEMS 79
REFERENCES 79
4 DIFFERENTIAL ALGEBRAIC EQUATIONS 81
4.1 INTRODUCTION 81
4.1.1 THE INDEX O F A DAE 82
4.1.2 A SIMPLE EXAMPLE 83
4.1.3 DAES IN HESSENBERG FORM 84
4.1.4 HIDDEN CONSTRAINTS AND THE INITIAL CONDITIONS 85
4.1.5 THE PENDULUM PROBLEM 86
IMAGE 3
C O N T E N T S X I I I
4.2 SOLVING DAES 87
4.2.1 SEMI-IMPLICIT DAES OF INDEX 1 87
4.2.2 GENERAL IMPLICIT DAES O F INDEX 1 88
4.2.3 DISCRETIZATION ALGORITHMS 89
4.2.4 DAES O F HIGHER INDEX 90
4.2.5 INDEX O F A DAE VARIABLE 93
REFERENCES 94
5 SOLVING DIFFERENTIAL ALGEBRAIC EQUATIONS IN R 95
5.1 DIFFERENTIAL ALGEBRAIC EQUATION SOLVERS IN R 95
5.2 A SIMPLE DAE O F INDEX 2 9 6
5.2.1 SOLVING THE DAES IN GENERAL IMPLICIT FORM 97
5.2.2 SOLVING THE DAES IN LINEARLY IMPLICIT FORM 98
5.3 A NONLINEAR IMPLICIT ODE 98
5.4 A DAE O F INDEX 3: THE PENDULUM PROBLEM 100
5.5 MULTIBODY SYSTEMS 101
5.5.1 THE CAR AXIS PROBLEM 102
5.6 ELECTRICAL CIRCUIT MODELS 106
5.6.1 THE TRANSISTOR AMPLIFIER 107
5.7 EXERCISES I L L
5.7.1 A SIMPLE DAE I L L
5.7.2 THE ROBERTSON PROBLEM I L L
5.7.3 THE PENDULUM PROBLEM REVISITED I L L
5.7.4 THE AKZO NOBEL PROBLEM 112
REFERENCES 115
6 DELAY DIFFERENTIAL EQUATIONS 117
6.1 DELAY DIFFERENTIAL EQUATIONS 117
6.1.1 DDES WITH DELAYS O F THE DEPENDENT VARIABLES 118
6.1.2 DDES WITH DELAYS OF THE DERIVATIVES 118
6.2 DIFFICULTIES WHEN SOLVING DDES 119
6.2.1 DISCONTINUITIES IN DDES 119
6.2.2 SMALL AND VANISHING DELAYS 120
6.3 NUMERICAL METHODS FOR SOLVING DDES 121
REFERENCES 121
7 SOLVING DELAY DIFFERENTIAL EQUATIONS IN R 1 23
,7.1 DELAY DIFFERENTIAL EQUATION SOLVERS IN R 123
7.2 TWO SIMPLE EXAMPLES 124
7.2.1 DDES INVOLVING SOLUTION DELAY TERMS 124
7.2.2 DDES INVOLVING DERIVATIVE DELAY TERMS 124
7.3 CHAOTIC PRODUCTION O F WHITE BLOOD CELLS 125
7.4 A DDE INVOLVING A ROOT FUNCTION 127
7.5 VANISHING TIME DELAYS 128
7.6 PREDATOR-PREY DYNAMICS WITH HARVESTING 130
IMAGE 4
X I V C O N T E N T S
7.7 EXERCISES 132
7.7.1 THE LEMMING MODEL 132
7.7.2 OBERLE AND PESCH 132
7.7.3 AN EPIDEMIOLOGICAL MODEL 133
7.7.4 A NEUTRAL DDE 134
7.7.5 DELAYED CELLULAR NEURAL NETWORKS WITH IMPULSES 134 REFERENCES 135
8 PARTIAL DIFFERENTIAL EQUATIONS 137
8.1 PARTIAL DIFFERENTIAL EQUATIONS 137
8.1.1 ALTERNATIVE FORMULATIONS 138
8.1.2 POLAR, CYLINDRICAL AND SPHERICAL COORDINATES 140
8.1.3 BOUNDARY CONDITIONS 141
8.2 SOLVING PDES 142
8.3 DISCRETISING DERIVATIVES 143
8.3.1 BASIC DIFFUSION SCHEMES 144
8.3.2 BASIC ADVECTION SCHEMES 145
8.3.3 FLUX-CONSERVATIVE DISCRETISATIONS 147
8.3.4 MORE COMPLEX ADVECTION SCHEMES 148
8.4 THE METHOD O F LINES 152
8.5 THE FINITE DIFFERENCE METHOD 153
REFERENCES 153
9 SOLVING PARTIAL DIFFERENTIAL EQUATIONS IN R 157
9.1 METHODS FOR SOLVING PDES IN R 157
9.1.1 NUMERICAL APPROXIMATIONS 157
9.1.2 SOLUTION METHODS 159
9.2 SOLVING PARABOLIC, ELLIPTIC AND HYPERBOLIC PDES IN R 160
9.2.1 THE HEAT EQUATION 160
9.2.2 THE WAVE EQUATION 163
9.2.3 POISSON AND LAPLACE'S EQUATION 166
9.2.4 THE ADVECTION EQUATION 168
9.3 MORE COMPLEX EXAMPLES 170
9.3.1 THE BRUSSELATOR IN ONE DIMENSION 170
9.3.2 THE BRUSSELATOR IN TWO DIMENSIONS 173
9.3.3 LAPLACE EQUATION IN POLAR COORDINATES 174
9.3.4 THE TIME-DEPENDENT 2-D SINE-GORDON EQUATION 176 9.3.5 THE
NONLINEAR SCHRODINGER EQUATION 179
9.4 EXERCISES 181
9.4.1 THE GRAY-SCOTT EQUATION 181
9.4.2 A MACROSCOPIC MODEL OF TRAFFIC 182
9.4.3 A VIBRATING STRING 183
9.4.4 A PEBBLE IN A BUCKET O F WATER 184
9.4.5 COMBUSTION IN 2-D 184
REFERENCES 185
IMAGE 5
CONTENTS X V
10 BOUNDARY VALUE PROBLEMS 187
10.1 TWO-POINT BOUNDARY VALUE PROBLEMS 187
10.2 CHARACTERISTICS O F BOUNDARY VALUE PROBLEMS 188
10.2.1 UNIQUENESS O F SOLUTIONS 188
10.2.2 ISOLATION O F SOLUTIONS 189
10.2.3 STIFFNESS O F BOUNDARY VALUE PROBLEMS AND DICHOTOMY 189
10.2.4 CONDITIONING O F BOUNDARY VALUE PROBLEMS 190
10.2.5 SINGULAR PROBLEMS 191
10.3 BOUNDARY CONDITIONS 192
10.3.1 SEPARATED BOUNDARY CONDITIONS 192
10.3.2 DEFINING GOOD BOUNDARY CONDITIONS 193
10.3.3 PROBLEMS DEFINED ON AN INFINITE INTERVAL 193
10.4 METHODS OF SOLUTION 194
10.5 SHOOTING METHODS FOR TWO-POINT B VPS 194
10.5.1 THE LINEAR CASE 194
10.5.2 THE NONLINEAR CASE 195
10.5.3 MULTIPLE SHOOTING 196
10.6 FINITE DIFFERENCE METHODS 197
10.6.1 A LOW ORDER METHOD FOR SECOND ORDER EQUATIONS . . . . 197 10.6.2
OTHER LOW ORDER METHODS 198
10.6.3 HIGHER ORDER METHODS BASED ON COLLOCATION RUNGE-KUTTA SCHEMES 199
10.6.4 HIGHER ORDER METHODS BASED ON MONO IMPLICIT RUNGE-KUTTA FORMULAE
200
10.6.5 HIGHER ORDER METHODS BASED ON LINEAR MULTISTEP FORMULAE 201
10.6.6 DEFERRED CORRECTION 202
10.7 CODES FOR THE NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS 203
REFERENCES 203
11 SOLVING BOUNDARY VALUE PROBLEMS IN R 207
11.1 BOUNDARY VALUE PROBLEM SOLVERS IN R 207
11.2 A SIMPLE BVP EXAMPLE 208
11.2.1 IMPLEMENTING THE BVP IN FIRST ORDER FORM 208
11.2.2 IMPLEMENTING THE BVP IN SECOND ORDER FORM 209
11.3 A MORE COMPLEX BVP EXAMPLE 210
11.4 MORE COMPLEX INITIAL OR END CONDITIONS 214
11.5 SOLVING A BOUNDARY VALUE PROBLEM USING CONTINUATION 216
11.5.1 MANUAL CONTINUATION 216
11.5.2 AUTOMATIC CONTINUATION 219
11.6 B VPS WITH UNKNOWN CONSTANTS 220
11.6.1 THE ELASTICA PROBLEM 221
11.6.2 NON-SEPARATED BOUNDARY CONDITIONS 222
11.6.3 AN UNKNOWN INTEGRATION INTERVAL 225
IMAGE 6
XVI CONTENTS
11.7 INTEGRAL CONSTRAINTS 228
11.8 STURM-LIOUVILLE PROBLEMS 229
11.9 A REACTION TRANSPORT PROBLEM 230
11.10 EXERCISES 233
11.10.1 A STIFF BOUNDARY VALUE PROBLEM 233
11.10.2 THE MATHIEU EQUATION 234
11.10.3 ANOTHER SWIRLING FLOW PROBLEM 234
11.10.4 ANOTHER REACTION TRANSPORT PROBLEM 236
REFERENCES 237
A APPENDIX 239
A. 1 BUTCHER TABLEAUX FOR SOME RUNGE-KUTTA METHODS : 239
A.2 COEFFICIENTS FOR SOME LINEAR MULTISTEP FORMULAE 239
A.3 IMPLEMENTED INTEGRATION METHODS FOR SOLVING INITIAL VALUE PROBLEMS
IN R 241
A.4 OTHER INTEGRATION METHODS IN R 242
REFERENCES 242
INDEX 245 |
| any_adam_object | 1 |
| author | Soetaert, Karline Cash, Jeff R. Mazzia, Francesca 1967- |
| author_GND | (DE-588)173793231 |
| author_facet | Soetaert, Karline Cash, Jeff R. Mazzia, Francesca 1967- |
| author_role | aut aut aut |
| author_sort | Soetaert, Karline |
| author_variant | k s ks j r c jr jrc f m fm |
| building | Verbundindex |
| bvnumber | BV040345849 |
| classification_rvk | MR 2100 SK 500 SK 920 ST 250 |
| ctrlnum | (OCoLC)802287223 (DE-599)DNB1018408258 |
| dewey-full | 518.63 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.63 |
| dewey-search | 518.63 |
| dewey-sort | 3518.63 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Soziologie Mathematik |
| format | Book |
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| id | DE-604.BV040345849 |
| illustrated | Illustrated |
| indexdate | 2024-08-21T00:03:38Z |
| institution | BVB |
| isbn | 9783642280696 9783642280702 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025200005 |
| oclc_num | 802287223 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-11 DE-824 DE-83 DE-521 DE-B768 DE-20 DE-523 DE-634 DE-188 |
| owner_facet | DE-19 DE-BY-UBM DE-11 DE-824 DE-83 DE-521 DE-B768 DE-20 DE-523 DE-634 DE-188 |
| physical | XVI, 248 Seiten Illustrationen, Diagramme |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series2 | Use R! |
| spelling | Soetaert, Karline Verfasser aut Solving differential equations in R Karline Soetaert ; Jeff Cash ; Francesca Mazzia Heidelberg [und 3 weitere] Springer 2012 XVI, 248 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Use R! Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf R Programm (DE-588)4705956-4 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Programmiersprache (DE-588)4047409-4 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Partielle Differentialgleichung (DE-588)4044779-0 s Datenverarbeitung (DE-588)4011152-0 s Programmiersprache (DE-588)4047409-4 s DE-604 Differentialgleichung (DE-588)4012249-9 s R Programm (DE-588)4705956-4 s Cash, Jeff R. Verfasser aut Mazzia, Francesca 1967- Verfasser (DE-588)173793231 aut Erscheint auch als Online-Ausgabe 10.1007/978-3-642-28070-2 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3949847&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025200005&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Soetaert, Karline Cash, Jeff R. Mazzia, Francesca 1967- Solving differential equations in R Differentialgleichung (DE-588)4012249-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd R Programm (DE-588)4705956-4 gnd Datenverarbeitung (DE-588)4011152-0 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Programmiersprache (DE-588)4047409-4 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4044779-0 (DE-588)4705956-4 (DE-588)4011152-0 (DE-588)4020929-5 (DE-588)4047409-4 |
| title | Solving differential equations in R |
| title_auth | Solving differential equations in R |
| title_exact_search | Solving differential equations in R |
| title_full | Solving differential equations in R Karline Soetaert ; Jeff Cash ; Francesca Mazzia |
| title_fullStr | Solving differential equations in R Karline Soetaert ; Jeff Cash ; Francesca Mazzia |
| title_full_unstemmed | Solving differential equations in R Karline Soetaert ; Jeff Cash ; Francesca Mazzia |
| title_short | Solving differential equations in R |
| title_sort | solving differential equations in r |
| topic | Differentialgleichung (DE-588)4012249-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd R Programm (DE-588)4705956-4 gnd Datenverarbeitung (DE-588)4011152-0 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Programmiersprache (DE-588)4047409-4 gnd |
| topic_facet | Differentialgleichung Partielle Differentialgleichung R Programm Datenverarbeitung Gewöhnliche Differentialgleichung Programmiersprache |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3949847&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025200005&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT soetaertkarline solvingdifferentialequationsinr AT cashjeffr solvingdifferentialequationsinr AT mazziafrancesca solvingdifferentialequationsinr |