Solving differential problems by multistep initial and boundary value methods:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam
Gordon and Breach
1998
|
| Series: | Stability and control
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 418 S. graph. Darst. |
| ISBN: | 9056991078 |
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MARC
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| 264 | 1 | |a Amsterdam |b Gordon and Breach |c 1998 | |
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| 650 | 7 | |a BOUNDARY VALUE PROBLEMS |2 nasat | |
| 650 | 7 | |a DIFFERENTIAL EQUATIONS |2 nasat | |
| 650 | 4 | |a Boundary value problems | |
| 650 | 4 | |a Differential equations |x Numerical solutions. | |
| 650 | 4 | |a Initial value problems | |
| 650 | 0 | 7 | |a Anfangswertproblem |0 (DE-588)4001991-3 |2 gnd |9 rswk-swf |
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Record in the Search Index
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| adam_text | IMAGE 1
SOLVING DIFFERENTIAL PROBLEMS BY MULTISTEP
INITIAL AND BOUNDARY VALUE METHODS
L. BRUGNANO AND D. TRIGIANTE UNIVERSITAE DI FIRENZE, ITALY
GORDON AND BREACH SCIENCE PUBLISHERS AUSTRALIA * CANADA * CHINA * FRANCE
* GERMANY * INDIA * JAPAN LUXEMBOURG * MALAYSIA * THE NETHERLANDS *
RUSSIA * SINGAPORE SWITZERLAND * THAILAND * UNITED KINGDOM
IMAGE 2
CONTENTS
INTRODUCTION TO THE SERIES XI
PREFACE XIII
1 DIFFERENTIAL EQUATIONS 1
1.1 FROM CONTINUOUS TO DISCRETE 1
1.2 STABILITY CONCEPTS 3
1.3 LINEARIZATION 4
1.4 TOTAL STABILITY 8
1.5 HOPF BIFURCATION 9
1.6 SUMMARY AND PARADIGMS 12
NOTES 14
2 LINEAR DIFFERENCE EQUATIONS WITH CONSTANT COEFNCIENTS 15
2.1 PRELIMINARIES AND NOTATIONS 15
2.2 THE CASE OF SIMPLE ROOTS 16
2.3 THE CASE OF MULTIPLE ROOTS *. . . 23
2.4 THE NONHOMOGENEOUS CASE 27
2.4.1 DIFFERENCE EQUATIONS IN MATRIX FORM 30
2.5 STABILITY OF SOLUTIONS 31
2.6 FOLLOWING A PARTICULAR SOLUTION 36
2.6.1 PROOF OF THEOREM 2.6.1 39
2.7 SYSTEMS OF LINEAR DIFFERENCE EQUATIONS 42
2.7.1 LINEAR SYSTEMS WITH CONSTANT MATRIX 43
2.7.2 THE GENERAL LINEAR CASE 44
2.7.3 DIFFERENCE EQUATIONS WITH MATRIX COEFFICIENTS 45
NOTES . 49
3 POLYNOMIALS AND TOEPLITZ MATRICES 51
3.1 LOCATION OF ZEROS 51
3.1.1 CONDITIONS CHARACTERIZING THE TYPES OF POLYNOMIALS 53
3.2 TOEPLITZ BAND MATRICES (T-MATRICES) 56
3.3 INFINITE T-MATRICES 56
3.3.1 INVERSE OF INFINITE T-MATRICES 57
3.3.2 BOUNDARY LOCUS 61
3.4 FINITE T-MATRICES . . -. 1 64
V
IMAGE 3
VI CONTENTS
3.4.1 SPECTRUM OF A FAMILY OF FINITE T-MATRICES 64
3.4.2 COMPONENTWISE BOUNDS FOR THE INVERSES OF FINITE T-MATRICES . 72
3.5 SUMMARY 76
NOTES . 77
4 NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS 79
4.1 PRELIMINARIES 79
4.2 LINEAR MULTISTEP FORMULAE (LMF) 82
4.3 LMF IN MATRIX FORM 85
4.4 CONVERGENCE 87
4.4.1 CONVERGENCE OF INITIAL VALUE METHODS 88
4.4.2 CONVERGENCE OF BOUNDARY VALUE METHODS 93
4.5 OFC^-STABILITY 97
4.6 FIXED-H STABILITY FOR INITIAL VALUE METHODS 97
4.7 FIXED-H STABILITY FOR BOUNDARY VALUE METHODS 100
4.7.1 BOUNDARY LOCUS AND RELATED QUESTIONS 102
4.8 AFCJFCJ-STABILITY VERSUS 0FC X FC 2 -STABILITY 105
4.9 CORRECT USE OF A METHOD 107
4.9.1 CONDITIONING OF T-MATRICES AND BVMS 111
4.10 STIFF PROBLEMS 113
4.11 RELATIVE STABILITY AND UNSTABLE PROBLEMS 114
4.11.1 EXISTENCE OF SOLUTIONS . 120
NOTES 120
5 GENERALIZED BACKWARD DIFFERENTIATION FORMULAE 121
5.1 BDF AND GENERALIZED BDF 121
5.2 DERIVATION OF GBDF 125
5.2.1 THE CASE OF A NONUNIFORM MESH 127
5.2.2 SOLVING VANDERMONDE SYSTEMS 128
5.3 THE ADDITIONAL CONDITIONS 128
5.3.1 STABILITY OF THE DISCRETE PROBLEM 135
5.4 THE INTEGRATION OF SYSTEMS OF EQUATIONS 135
5.4.1 STABILITY ANALYSIS FOR SYSTEMS OF EQUATIONS 137
5.4.2 THE BEHAVIOR ON THE IMAGINARY AXIS 139
NOTES 140
6 GENERALIZED ADAMS METHODS 143
6.1 ADAMS-MOULTON METHODS 143
6.1.1 DERIVATION OF THE ADAMS-MOULTON FORMULAE 144
6.2 REVERSE ADAMS METHODS 146
6.3 GENERALIZED ADAMS METHODS (GAMS) 148
6.3.1 THE CASE OF A NONUNIFORM MESH 150
6.4 THE ADDITIONAL CONDITIONS 152
6.4.1 THE BEHAVIOR ON THE IMAGINARY AXIS 154
NOTES 157
IMAGE 4
CONTENTS VII
7 SYMMETRIE SCHEINES 159
7.1 GENERAL PROPERTIES OF SYMMETRIE SCHEINES . . . . 159
7.2 EXTENDED TRAPEZOIDAL RULES (ETRS) 162
7.3 EXTENDED TRAPEZOIDAL RULES OF SECOND KIND (ETR 2 S) 164
7.3.1 THE CASE OF A NONUNIFORM MESH 168
7.3.2 THE ADDITIONAL CONDITIONS 168
7.3.3 UNSYMMETRIC ETR 2 S 170
7.4 TOP ORDER METHODS (TOMS) 171
7.4.1 THE ADDITIONAL CONDITIONS 174
7.4.2 VARIABLE STEPSIZE 175
V 7.4.3 SOLVING CONFLUENT VANDERMONDE SYSTEMS 176
7.5 NUMERICAL EXAMPLES 177
7.5.1 RELATIVE STABILITY REGIONS OF SYMMETRIE SCHEMES 178
NOTES 183
8 HAMILTONIAN PROBLEMS 185
8.1 INTRODUCTION 185
8.2 SYMPLECTIC METHODS 188
8.3 DISCRETE PROBLEMS 194
8.4 DISCRETE VARIATIONAL PRINCIPLE 198
8.5 TIME REVERSAL SYMMETRY AND ADDITIONAL METHODS 201
8.5.1 PROOFOF LEMMA 8.5.1 206
8.6 DISCRETE MAPS 208
8.7 NUMERICAL METHODS 210
NOTES 212
9 BOUNDARY VALUE PROBLEMS 213
9.1 INTRODUCTION . ( 213
9.2 SENSITIVITY ANALYSIS AND CLASSIFICATION OF PROBLEMS 216
9.3 TIME REVERSAL SYMMETRY 217
9.4 CONDITIONING OF LINEAR PROBLEMS 222
9.4.1 DISCRETE BVPS 226
9.5 NUMERICAL METHODS 226
9.5.1 THE CONTRIBUTION OF SPURIOUS ROOTS 229
9.6 APPROXIMATING CONTINUOUS BVPS BY MEANS OF BVMS 232
9.6.1 NUMERICAL EXAMPLES 233
NOTES 236
10 MESH SELECTION STRATEGIES 237
10.1 CLASSIFICATION OF CONTINUOUS PROBLEMS AND STIFFNESS 237
10.1.1 THE SCALAR CASE 237
10.1.2 SYSTEMS OF EQUATIONS 239
10.1.3 111 CONDITIONED PROBLEMS 242
10.1.4 NONHOMOGENEOUS PROBLEMS 245
10.2 DISCRETE PROBLEMS 247
10.3 MESH SELECTION 248
IMAGE 5
VIII CONTENTS
10.3.1 CONTROL OF THE PARAMETERS/TD AND 7^ 253
10.3.2 ESTIMATE OF THE PRECISION SET 254
10.4 MINIMIZATION OF THE GLOBAL ERROR 256
10.4.1 MONITORING THE TRUNCATION ERRORS 260
10.5 STABILITY AND EQUIDISTRIBUTION 261
10.6 THE NONHOMOGENEOUS CASE 262
10.7 THE IVP CASE 264
10.8 NUMERICAL EXAMPLES 271
NOTES 277
11 BLOCK B V MS 279
11.1 INTRODUCTION 279
11.2 MATRIX FORM 280
11.3 BLOCK VERSION OF BVMS 282
11.4 CHOOSING THE ADDITIONAL METHODS 283
11.5 B 2 VMS AND RUNGE-KUTTA SCHEMES 286
11.5.1 B 2 VMS VERSUS RK SCHEMES 288
11.5.2 CHOOSING THE BLOCKSIZE OF A B 2 VM 289
11.5.3 STABILITY PROPERTIES OF B 2 VMS 292
11.6 BLOCK BVMS AND GENERAL LINEAR METHODS 295
11.6.1 STABILITY PROPERTIES OF B 2 VM 2 S - 297
NOTES 299
12 PARALLEL IMPLEMENTATION OF B 2 V MS 301
12.1 INTRODUCTION 301
12.2 THE PARALLEL ALGORITHM 302
12.2.1 SUPPLEMENTARY CONSIDERATIONS 305
12.3 PARALLEL SOLUTION OF TWO-POINT BVPS 306
12.4 EXPECTED SPEED-UP 311
12.4.1 THE IVP CASE 311
12.4.2 THE BVP CASE 312
12.5 PARALLEL SOLUTION OF THE REDUCED SYSTEM 313
12.5.1 THE IVP CASE 314
12.5.2 THE BVP CASE 317
12.5.3 NUMERICAL EXAMPLES 320
NOTES 322
13 EXTENSIONS AND APPLICATIONS TO SPECIAL PROBLEMS 325
13.1 THE METHOD OF LINES 325
13.1.1 SOME EXAMPLES 326
13.2 DIFFERENTIAL ALGEBRAIC EQUATIONS 332
13.2.1 NUMERICAL EXAMPLES 336
13.3 DELAY DIFFERENTIAL EQUATIONS 337
13.3.1 NUMERICAL EXAMPLES 340
13.4 MULTIDERIVATIVE BVMS 343
13.5 NONLINEAR PROBLEMS 344
IMAGE 6
CONTENTS IX
NOTES 348
A MATRICES 349
A.L FUNCTIONS OF MATRICES 349
A.2 M-MATRICES 353
A.3 THE KRONECKER PRODUCT 354
A.3.1 USE OF KRONECKER PRODUCT FOR SOLVING MATRIX EQUATIONS . . .. 357
A.4 HAMILTONIAN MATRICES 358
A.5 SYMPLECTIC MATRICES 360
B ANSWERS TO THE EXERCISES 363
B.L CHAPTER 1 363
B.2 CHAPTER 2 . . . 364
B.3 CHAPTER 3 370
B.4 CHAPTER 4 373
B.5 CHAPTER 5 380
B.6 CHAPTER 6 382
B.7 CHAPTER 7 384
B.8 CHAPTER 8 387
B.9 CHAPTER 9 390
B.10 CHAPTER 10 390
B.LL CHAPTER 11 391
B.12 CHAPTER 12 392
B.13 APPENDIX A 394
BIBLIOGRAPHY 399
INDEX 413
|
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| physical | XV, 418 S. graph. Darst. |
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| spelling | Brugnano, Luigi Verfasser aut Solving differential problems by multistep initial and boundary value methods L. Brugnano and D. Trigiante Amsterdam Gordon and Breach 1998 XV, 418 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Stability and control 6 ANALYSIS (MATHEMATICS) nasat BOUNDARY VALUE PROBLEMS nasat DIFFERENTIAL EQUATIONS nasat Boundary value problems Differential equations Numerical solutions. Initial value problems Anfangswertproblem (DE-588)4001991-3 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Mehrschrittverfahren (DE-588)4198527-8 gnd rswk-swf Anfangswertproblem (DE-588)4001991-3 s Mehrschrittverfahren (DE-588)4198527-8 s DE-604 Randwertproblem (DE-588)4048395-2 s Trigiante, Donato Verfasser aut Stability and control 6 (DE-604)BV010641392 6 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008232473&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brugnano, Luigi Trigiante, Donato Solving differential problems by multistep initial and boundary value methods Stability and control ANALYSIS (MATHEMATICS) nasat BOUNDARY VALUE PROBLEMS nasat DIFFERENTIAL EQUATIONS nasat Boundary value problems Differential equations Numerical solutions. Initial value problems Anfangswertproblem (DE-588)4001991-3 gnd Randwertproblem (DE-588)4048395-2 gnd Mehrschrittverfahren (DE-588)4198527-8 gnd |
| subject_GND | (DE-588)4001991-3 (DE-588)4048395-2 (DE-588)4198527-8 |
| title | Solving differential problems by multistep initial and boundary value methods |
| title_auth | Solving differential problems by multistep initial and boundary value methods |
| title_exact_search | Solving differential problems by multistep initial and boundary value methods |
| title_full | Solving differential problems by multistep initial and boundary value methods L. Brugnano and D. Trigiante |
| title_fullStr | Solving differential problems by multistep initial and boundary value methods L. Brugnano and D. Trigiante |
| title_full_unstemmed | Solving differential problems by multistep initial and boundary value methods L. Brugnano and D. Trigiante |
| title_short | Solving differential problems by multistep initial and boundary value methods |
| title_sort | solving differential problems by multistep initial and boundary value methods |
| topic | ANALYSIS (MATHEMATICS) nasat BOUNDARY VALUE PROBLEMS nasat DIFFERENTIAL EQUATIONS nasat Boundary value problems Differential equations Numerical solutions. Initial value problems Anfangswertproblem (DE-588)4001991-3 gnd Randwertproblem (DE-588)4048395-2 gnd Mehrschrittverfahren (DE-588)4198527-8 gnd |
| topic_facet | ANALYSIS (MATHEMATICS) BOUNDARY VALUE PROBLEMS DIFFERENTIAL EQUATIONS Boundary value problems Differential equations Numerical solutions. Initial value problems Anfangswertproblem Randwertproblem Mehrschrittverfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008232473&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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