Solving ordinary differential equations: 1 Nonstiff problems
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin u.a.
Springer
1993
|
| Edition: | 2. rev. ed. |
| Series: | Springer series in computational mathematics
8 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 528 S. graph. Darst. |
| ISBN: | 3540566708 0387566708 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text |
E. HAIRER S. R NORSETT G. WANNER SOLVING ORDINARY DIFFERENTIAL EQUATIONS
NONSTIFF PROBLEMS SECOND REVISED EDITION WITH 135 FIGURES
SPRINGER-VERLAG BERLIN HEIDELBERG NEW YORK LONDON PARIS TOKYO HONG KONG
BARCELONA BUDAPEST CONTENTS CHAPTER I. CLASSICAL MATHEMATICAL THEORY 1.1
TERMINOLOGY 2 1.2 THE OLDEST DIFFERENTIAL EQUATIONS 4 NEWTON 4 LEIBNIZ
AND DIE BERNOULLI BROUEIERS 6 VARIATIONAL CALCULUS 7 CLAIRAUT 9 EXERCISES
10 1.3 ELEMENTARY INTEGRATION METHODS 12 FIRST ORDER EQUATIONS 12 SECOND
ORDER EQUATIONS 13 EXERCISES 14 1.4 LINEAR DIFFERENTIAL EQUATIONS 16
EQUATIONS WITH CONSTANT COEFFICIENTS 16 VARIATION OF CONSTANTS 18
EXERCISES 19 1.5 EQUATIONS WITH WEAK SINGULARITIES 20 LINEAR EQUATIONS
20 NONLINEAR EQUATIONS 23 EXERCISES 24 1.6 SYSTEMS OF EQUATIONS 26 THE
VIBRATING STRING AND PROPAGATION OF SOUND 26 FOURIER 29 LAGRANGIAN
MECHANICS 30 HAMILTONIAN MECHANICS 32 EXERCISES 34 1.7 A GENERAL
EXISTENCE THEOREM 35 CONVERGENCE OF EULER'S METHOD 35 EXISTENCE THEOREM
OF PEANO 41 EXERCISES 43 1.8 EXISTENCE THEORY USING ITERATION METHODS
AND TAYLOR SERIES 44 PICARD-LINDELOEF ITERATION 45 TAYLOR SERIES 46
RECURSIVE COMPUTATION OF TAYLOR COEFFICIENTS 47 1 IXERCISES 49 X
CONTENTS 1.9 EXISTENCE THEORY FOR SYSTEMS OF EQUATIONS 51 VECTOR
NOTATION 52 SUBORDINATE MATRIX NORMS 53 EXERCISES 55 1.10 DIFFERENTIAL
INEQUALITIES 56 INTRODUCTION 56 THE FUNDAMENTAL THEOREMS 57 ESTIMATES
USING ONE-SIDED LIPSCHITZ CONDITIONS 60 EXERCISES 62 1.11 SYSTEMS OF
LINEAR DIFFERENTIAL EQUATIONS 64 RESOLVENT AND WRONSKIAN 65
INHOMOGENEOUS LINEAR EQUATIONS 66 THE
ABEL-LIOUVILLE-JACOBI-OSTROGRADSKII IDENTITY 66 EXERCISES 67 1.12
SYSTEMS WITH CONSTANT COEFFICIENTS 69 LINEARIZATION 69 DIAGONALIZATION
69 THE SCHUR DECOMPOSITION 70 NUMERICAL COMPUTATIONS 72 THE JORDAN
CANONICAL FORM 73 GEOMETRIE REPRESENTATION 77 EXERCISES 78 1.13
STABILITY 80 INTRODUCTION 80 THE ROUTH-HURWITZ CRITERION 81
COMPUTATIONAL CONSIDERATIONS 85 LIAPUNOV FUNCTIONS 86 STABILITY OF
NONLINEAR SYSTEMS 87 STABILITY OF NON-AUTONOMOUS SYSTEMS 88 EXERCISES 89
1.14 DERIVATIVES WITH RESPECT TO PARAMETERS AND INITIAL VALUES . 92 THE
DERIVATIVE WITH RESPECT TO A PARAMETER 93 DERIVATIVES WITH RESPECT TO
INITIAL VALUES 95 THE NONLINEAR VARIATION-OF-CONSTANTS FORMULA 96 FLOWS
AND VOLUME-PRESERVING FLOWS 97 CANONICAL EQUATIONS AND SYMPLECTIC
MAPPINGS 100 EXERCISES 104 1.15 BOUNDARY VALUE AND EIGENVALUE PROBLEMS
105 BOUNDARY VALUE PROBLEMS 105 STURM-LIOUVILLE EIGENVALUE PROBLEMS 107
EXERCISES 110 1.16 PERIODIC SOLUTIONS, LIMIT CYCLES, STRANGE ATTRACTORS
111 VAN DER POL'S EQUATION 111 CHEMICAL REACTIONS 115 LIMIT CYCLES IN
HIGHER DIMENSIONS, HOPF BIFURCATION 117 STRANGE ATTRACTORS 120 THE UPS
AND DOWNS OF THE LORENZ MODEL 123 FEIGENBAUM CASCADES 124 EXERCISES 126
CONTENTS XI CHAPTER II. RUNGE-KUTTA AND EXTRAPOLATION METHODS 11.1 THE
FIRST RUNGE-KUTTA METHODS 132 GENERAL FORMULATION OF RUNGE-KUTTA METHODS
134 DISCUSSION OF METHODS OF ORDER 4 135 "OPTIMAL" FORMULAS 139
NUMERICAL EXAMPLE 140 EXERCISES 141 11.2 ORDER CONDITIONS FOR
RUNGE-KUTTA METHODS 14 3 THE DERIVATIVES OF THE TRUE SOLUTION 145
CONDITIONS FOR ORDER 3 145 TREES AND ELEMENTARY DIFFERENTIALS 145 THE
TAYLOR EXPANSION OF THE TRUE SOLUTION 148 FAAE DI BRUNO'S FORMULA 149 THE
DERIVATIVES OF THE NUMERICAL SOLUTION 151 THE ORDER CONDITIONS 153
EXERCISES 154 11.3 ERROR ESTIMATION AND CONVERGENCE FOR RK METHODS 156
RIGOROUS ERROR BOUNDS 156 THE PRINCIPAL ERROR TERM 158 ESTIMATION OF THE
GLOBAL ERROR 159 EXERCISES 163 11.4 PRACTICAL ERROR ESTIMATION AND STEP
SIZE SELECTION 164 RICHARDSON EXTRAPOLATION 164 EMBEDDED RUNGE-KUTTA
FORMULAS 165 AUTOMATIC STEP SIZE CONTROL 167 STARTING STEP SIZE 169
NUMERICAL EXPERIMENTS 170 EXERCISES 172 11.5 EXPLICIT RUNGE-KUTTA
METHODS OF HIGHER ORDER 173 THE BUTCHER BARRIERE 173 6-STAGE, 5TH ORDER
PROCESSES 175 EMBEDDED FORMULAS OF ORDER 5 176 HIGHER ORDER PROCESSES
179 EMBEDDED FORMULAS OF HIGH ORDER 180 AN 8TH ORDER EMBEDDED METHOD 181
EXERCISES 185 11.6 DENSE OUTPUT, DISCONTINUITIES, DERIVATIVES 188 DENSE
OUTPUT 188 CONTINUOUS DORMAND & PRINCE PAIRS 191 DENSE OUTPUT FOR DOP853
194 EVENT LOCATION 195 DISCONTINUOUS EQUATIONS 196 NUMERICAL COMPUTATION
OF DERIVATIVES WITH RESPECT TO INITIAL VALUES AND PARAMETERS 200
EXERCISES 202 11.7 IMPLICIT RUNGE-KUTTA METHODS 204 EXISTENCE OF A
NUMERICAL SOLUTION 206 THE METHODS OF KUNTZMANN AND BUTCHER OF ORDER 2S
208 IRK METHODS BASED ON LOBATTO QUADRATURE 210 XII CONTENTS COLLOCATION
METHODS 211 EXERCISES 214 11.8 ASYMPTOTIC EXPANSION OF THE GLOBAL ERROR
216 THE GLOBAL ERROR 216 VARIABLE H 218 NEGATIVE H 219 PROPERTIES OF THE
ADJOINT METHOD 220 SYMMETRIE METHODS 221 EXERCISES 223 11.9
EXTRAPOLATION METHODS 224 DEFINITION OF THE METHOD 224 THE AITKEN -
NEVILLE ALGORITHM 226 THE GRAGG OR GBS METHOD 228 ASYMPTOTIC EXPANSION
FOR ODD INDICES 231 EXISTENCE OF EXPLICIT RK METHODS OF ARBITRARY ORDER
232 ORDER AND STEP SIZE CONTROL 233 DENSE OUTPUT FOR THE GBS METHOD 237
CONTROL OF THE INTERPOLATION ERROR 240 EXERCISES 241 11.10 NUMERICAL
COMPARISONS 244 PROBLEMS 244 PERFORMANCE OF THE CODES 249 A "STRETCHED"
ERROR ESTIMATOR FOR DOP853 254 EFFECT OF STEP-NUMBER SEQUENCE IN ODEX
256 11.11 PARALLEL METHODS 257 PARALLEL RUNGE-KUTTA METHODS 258 PARALLEL
ITERATED RUNGE-KUTTA METHODS 259 EXTRAPOLATION METHODS 261 INCREASING
RELIABILITY 261 EXERCISES 263 11.12 COMPOSITION OF B-SERIES 264
COMPOSITION OF RUNGE-KUTTA METHODS 264 B-SERIES 266 ORDER CONDITIONS FOR
RUNGE-KUTTA METHODS 269 BUTCHER'S "EFFECTIVE ORDER" 270 EXERCISES 272
11.13 HIGHER DERIVATIVE METHODS 274 COLLOCATION METHODS 275
HERMITE-OBRESCHKOFF METHODS 277 FEHLBERG METHODS 278 GENERAL THEORY OF
ORDER CONDITIONS 280 EXERCISES 281 11.14 NUMERICAL METHODS FOR SECOND
ORDER DIFFERENTIAL EQUATIONS 283 NYSTROEM METHODS 284 THE DERIVATIVES OF
THE EXACT SOLUTION 286 THE DERIVATIVES OF THE NUMERICAL SOLUTION 288 THE
ORDER CONDITIONS 290 ON THE CONSTRUCTION OF NYSTROEM METHODS 291 AN
EXTRAPOLATION METHOD FOR Y" = F(X,Y) 294 PROBLEMS FOR NUMERICAL
COMPARISONS 296 CONTENTS XIII PERFORMANCE OF THE CODES 298 EXERCISES 300
11.15 P-SERIES FOR PARTITIONED DIFFERENTIAL EQUATIONS 302 DERIVATIVES OF
THE EXACT SOLUTION, P-TREES 303 P-SERIES .' 306 ORDER CONDITIONS FOR
PARTITIONED RUNGE-KUTTA METHODS 307 FURTHER APPLICATIONS OF P-SERIES 308
EXERCISES 311 11.16 SYMPLECTIC INTEGRATION METHODS 312 SYMPLECTIC
RUNGE-KUTTA METHODS 315 AN EXAMPLE FROM GALACTIC DYNAMICS 319
PARTITIONED RUNGE-KUTTA METHODS 326 SYMPLECTIC NYSTROEM METHODS 330
CONSERVATION OF THE HAMILTONIAN; BACKWARD ANALYSIS 333 EXERCISES 337
11.17 DELAY DIFFERENTIAL EQUATIONS 339 EXISTENCE 339 CONSTANT STEP SIZE
METHODS FOR CONSTANT DELAY 341 VARIABLE STEP SIZE METHODS 342 STABILITY
343 AN EXAMPLE FROM POPULATION DYNAMICS 345 INFECTIOUS DISEASE MODELLING
347 AN EXAMPLE FROM ENZYME KINETICS 248 A MATHEMATICAL MODEL IN
IMMUNOLOGY 349 INTEGRO-DIFFERENTIAL EQUATIONS 351 EXERCISES 352 CHAPTER
III. MULTISTEP METHODS AND GENERAL LINEAR METHODS 111.1 CLASSICAL LINEAR
MULTISTEP FORMULAS 356 EXPLICIT ADAMS METHODS 357 IMPLICIT ADAMS METHODS
359 NUMERICAL EXPERIMENT 361 EXPLICIT NYSTROEM METHODS 362 MILNE-SIMPSON
METHODS 363 METHODS BASED ON DIFFERENTIATION (BDF) 364 EXERCISES 366
111.2 LOCAL ERROR AND ORDER CONDITIONS 368 LOCAL ERROR OF A MULTISTEP
METHOD 368 ORDER OF A MULTISTEP METHOD 370 ERROR CONSTANT 372
IRREDUCIBLE METHODS 374 THE PEANO KERNEL OF A MULTISTEP METHOD 375
EXERCISES 377 111.3 STABILITY AND THE FIRST DAHLQUIST BARRIER 378
STABILITY OF THE BDF-FORMULAS 380 HIGHEST ATTAINABLE ORDER OF STABLE
MULTISTEP METHODS 383 EXERCISES 387 XIV CONTENTS 111.4 CONVERGENCE OF
MULTISTEP METHODS 391 FORMULATION AS ONE-STEP METHOD 393 PROOF OF
CONVERGENCE 395 EXERCISES * 396 111.5 VARIABLE STEP SIZE MULTISTEP
METHODS 397 VARIABLE STEP SIZE ADAMS METHODS 397 RECURRENCE RELATIONS
FOR GJ(N), $J(N) AND $J(N) 399 VARIABLE STEP SIZE BDF 400 GENERAL
VARIABLE STEP SIZE METHODS AND THEIR ORDERS 401 STABILITY 402
CONVERGENCE 407 EXERCISES 409 111.6 NORDSIECK METHODS 410 EQUIVALENCE
WITH MULTISTEP METHODS 412 IMPLICIT ADAMS METHODS 417 BDF-METHODS 419
EXERCISES 420 111.7 IMPLEMENTATION AND NUMERICAL COMPARISONS 421 STEP
SIZE AND ORDER SELECTION 421 SOME AVAILABLE CODES 423 NUMERICAL
COMPARISONS 427 111.8 GENERAL LINEAR METHODS 430 A GENERAL INTEGRATION
PROCEDURE 431 STABILITY AND ORDER 436 CONVERGENCE 438 ORDER CONDITIONS
FOR GENERAL LINEAR METHODS 441 CONSTRUCTION OF GENERAL LINEAR METHODS
443 EXERCISES 445 111.9 ASYMPTOTIC EXPANSION OF THE GLOBAL ERROR 448 AN
INSTRUCTIVE EXAMPLE 448 ASYMPTOTIC EXPANSION FOR STRICTIY STAHLE METHODS
(8.4) 450 WEAKLY STAHLE METHODS 454 THE ADJOINT METHOD 457 SYMMETRIE
METHODS 459 EXERCISES 460 111.10 MULTISTEP METHODS FOR SECOND ORDER
DIFFERENTIAL EQUATIONS 461 EXPLICIT STOERMER METHODS 462 IMPLICIT STOERMER
METHODS 464 NUMERICAL EXAMPLE 465 GENERAL FORMULATION 467 CONVERGENCE
468 ASYMPTOTIC FORMULA FOR THE GLOBAL ERROR 471 ROUNDING ERRORS 472
EXERCISES 473 APPENDIX. FORTRAN CODES 475 DRIVER FOR THE CODE DOPRI5 475
SUBROUTINE DOPRI5 477 SUBROUTINE DOP853 481 SUBROUTINE ODEX 482 CONTENTS
XV SUBROUTINE ODEX2 484 DRIVER FOR THE CODE RETARD 486 SUBROUTINE RETARD
488 BIBLIOGRAPHY 491 SYMBOL INDEX 521 SUBJECT INDEX 523 |
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| discipline | Mathematik |
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| indexdate | 2024-09-06T00:29:07Z |
| institution | BVB |
| isbn | 3540566708 0387566708 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005408199 |
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| physical | XV, 528 S. graph. Darst. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
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| publisher | Springer |
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| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spelling | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner 2. rev. ed. Berlin u.a. Springer 1993 XV, 528 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 8 Springer series in computational mathematics ... Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Hairer, Ernst 1949- Sonstige (DE-588)139445188 oth Nørsett, Syvert P. 1944-2024 Sonstige (DE-588)13758718X oth Wanner, Gerhard 1942- Sonstige (DE-588)13944534X oth (DE-604)BV000612774 1 Springer series in computational mathematics 8 (DE-604)BV000012004 8 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005408199&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Solving ordinary differential equations Springer series in computational mathematics Numerisches Verfahren (DE-588)4128130-5 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4020929-5 |
| title | Solving ordinary differential equations |
| title_auth | Solving ordinary differential equations |
| title_exact_search | Solving ordinary differential equations |
| title_full | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner |
| title_fullStr | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner |
| title_full_unstemmed | Solving ordinary differential equations 1 Nonstiff problems E. Hairer ; S. P. Nørsett ; G. Wanner |
| title_short | Solving ordinary differential equations |
| title_sort | solving ordinary differential equations nonstiff problems |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Numerisches Verfahren Gewöhnliche Differentialgleichung |
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