Introduction to partial differential equations: a computational approach
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2009
|
| Ausgabe: | 1. softcover printing |
| Schriftenreihe: | Texts in applied mathematics
29 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [385] - 387 |
| Beschreibung: | XV, 392 S. graph. Darst. |
| ISBN: | 9783540887041 |
Internformat
MARC
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| 245 | 1 | 0 | |a Introduction to partial differential equations |b a computational approach |c Aslak Tveito ; Ragnar Winther |
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| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
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Datensatz im Suchindex
| _version_ | 1804140648629010432 |
|---|---|
| adam_text | ASLAK TVEITO * RAGNAR WINTER INTRODUCTION TO PARTIAL DIFFERENTIAL
EQUATIONS A COMPUTATIONAL APPROACH 4Y SPRINGER CONTENTS 1 SETTING THE
SCENE 1 1.1 WHAT IS A DIFFERENTIAL EQUATION? 1 1.1.1 CONCEPTS 2 1.2 THE
SOLUTION AND ITS PROPERTIES 4 1.2.1 AN ORDINARY DIFFERENTIAL EQUATION 4
1.3 A NUMERICAL METHOD 6 1.4 CAUCHY PROBLEMS 10 1.4.1 FIRST-ORDER
HOMOGENEOUS EQUATIONS 11 1.4.2 FIRST-ORDER NONHOMOGENEOUS EQUATIONS 14
1.4.3 THE WAVE EQUATION 15 1.4.4 THE HEAT EQUATION 18 1.5 EXERCISES 20
1.6 PROJECTS 28 2 TWO-POINT BOUNDARY VALUE PROBLEMS 39 2.1 POISSON S
EQUATION IN ONE DIMENSION 40 2.1.1 GREEN S FUNCTION 42 2.1.2 SMOOTHNESS
OF THE SOLUTION 43 2.1.3 A MAXIMUM PRINCIPLE 44 2.2 A FINITE DIFFERENCE
APPROXIMATION 45 2.2.1 TAYLOR SERIES : 46 2.2.2 A SYSTEM OF ALGEBRAIC
EQUATIONS 47 2.2.3 GAUSSIAN ELIMINATION FOR TRIDIAGONAL LINEAR SYSTEMS
50 2.2.4 DIAGONAL DOMINANT MATRICES 53 XII CONTENTS 2.2.5 POSITIVE
DEFINITE MATRICES 55 2.3 CONTINUOUS AND DISCRETE SOLUTIONS 57 2.3.1
DIFFERENCE AND DIFFERENTIAL EQUATIONS 57 2.3.2 SYMMETRY 58 2.3.3
UNIQUENESS 61 2.3.4 A MAXIMUM PRINCIPLE FOR THE DISCRETE PROBLEM ... 61
2.3.5 CONVERGENCE OF THE DISCRETE SOLUTIONS 63 2.4 EIGENVALUE PROBLEMS
65 2.4.1 THE CONTINUOUS EIGENVALUE PROBLEM 65 2.4.2 THE DISCRETE
EIGENVALUE PROBLEM 68 2.5 EXERCISES 72 2.6 PROJECTS 82 3 THE HEAT
EQUATION 87 3.1 A BRIEF OVERVIEW 88 3.2 SEPARATION OF VARIABLES 90 3.3
THE PRINCIPLE OF SUPERPOSITION 92 3.4 FOURIER COEFFICIENTS 95 3.5 OTHER
BOUNDARY CONDITIONS 97 3.6 THE NEUMANN PROBLEM 98 3.6.1 THE EIGENVALUE
PROBLEM 99 3.6.2 PARTICULAR SOLUTIONS 100 3.6.3 A FORMAL SOLUTION 101
3.7 ENERGY ARGUMENTS 102 3.8 DIFFERENTIATION OF INTEGRALS 106 3.9
EXERCISES 108 3.10 PROJECTS 113 4 FINITE DIFFERENCE SCHEMES FOR THE HEAT
EQUATION 117 4.1 AN EXPLICIT SCHEME 119 4.2 FOURIER ANALYSIS OF THE
NUMERICAL SOLUTION 122 4.2.1 PARTICULAR SOLUTIONS 123 4.2.2 COMPARISON
OF THE ANALYTICAL AND DISCRETE SOLUTION 127 4.2.3 STABILITY
CONSIDERATIONS 129 4.2.4 THE ACCURACY OF THE APPROXIMATION 130 4.2.5
SUMMARY OF THE COMPARISON 131 4.3 VON NEUMANN S STABILITY ANALYSIS 132
4.3.1 PARTICULAR SOLUTIONS: CONTINUOUS AND DISCRETE .... 133 4.3.2
EXAMPLES 134 4.3.3 A NONLINEAR PROBLEM 137 4.4 AN IMPLICIT SCHEME I T
140 4.4.1 STABILITY ANALYSIS 143 4.5 NUMERICAL STABILITY BY ENERGY
ARGUMENTS 145 4.6 EXERCISES 148 CONTENTS XIII THE WAVE EQUATION 159 5.1
SEPARATION OF VARIABLES 160 5.2 UNIQUENESS AND ENERGY ARGUMENTS 163 5.3
A FINITE DIFFERENCE APPROXIMATION 165 5.3.1 STABILITY ANALYSIS 168 5.4
EXERCISES 170 MAXIMUM PRINCIPLES 175 6.1 A TWO-POINT BOUNDARY VALUE
PROBLEM 175 6.2 THE LINEAR HEAT EQUATION 178 6.2.1 THE CONTINUOUS CASE
180 6.2.2 UNIQUENESS AND STABILITY 183 6.2.3 THE EXPLICIT FINITE
DIFFERENCE SCHEME 184 6.2.4 THE IMPLICIT FINITE DIFFERENCE SCHEME 186
6.3 THE NONLINEAR HEAT EQUATION 188 6.3.1 THE CONTINUOUS CASE 189 6.3.2
AN EXPLICIT FINITE DIFFERENCE SCHEME 190 6.4 HARMONIC FUNCTIONS 191
6.4.1 MAXIMUM PRINCIPLES FOR HARMONIC FUNCTIONS .... 193 6.5 DISCRETE
HARMONIC FUNCTIONS 195 6.6 EXERCISES 201 POISSON S EQUATION IN TWO SPACE
DIMENSIONS 209 7.1 RECTANGULAR DOMAINS 209 7.2 POLAR COORDINATES 212
7.2.1 THE DISC 213 7.2.2 A WEDGE 216 7.2.3 A CORNER SINGULARITY 217 7.3
APPLICATIONS OF THE DIVERGENCE THEOREM 218 7.4 THE MEAN VALUE PROPERTY
FOR HARMONIC FUNCTIONS 222 7.5 A FINITE DIFFERENCE APPROXIMATION 225
7.5.1 THE FIVE-POINT STENCIL 225 7.5.2 AN ERROR ESTIMATE 228 7.6
GAUSSIAN ELIMINATION FOR GENERAL SYSTEMS 230 7.6.1 UPPER TRIANGULAR
SYSTEMS 230 7.6.2 GENERAL SYSTEMS 231 7.6.3 BANDED SYSTEMS 234 7.6.4
POSITIVE DEFINITE SYSTEMS 236 7.7 EXERCISES 237 ORTHOGONALITY AND
GENERAL FOURIER SERIES 245 8.1 THE FULL FOURIER SERIES . . . , 246 8.1.1
EVEN AND ODD FUNCTIONS 249 8.1.2 DIFFERENTIATION OF FOURIER SERIES 252
8.1.3 THE COMPLEX FORM 255 XIV CONTENTS 8.1.4 CHANGING THE SCALE 256 8.2
BOUNDARY VALUE PROBLEMS AND ORTHOGONAL FUNCTIONS . . . . 257 8.2.1 OTHER
BOUNDARY CONDITIONS 257 8.2.2 STURM-LIOUVILLE PROBLEMS 261 8.3 THE MEAN
SQUARE DISTANCE 264 8.4 GENERAL FOURIER SERIES 267, 8.5 A POINCARE
INEQUALITY 273 8.6 EXERCISES 276 9 CONVERGENCE OF FOURIER SERIES 285 9.1
DIFFERENT NOTIONS OF CONVERGENCE 285 9.2 POINTWISE CONVERGENCE 290 9.3
UNIFORM CONVERGENCE 296 9.4 MEAN SQUARE CONVERGENCE 300 9.5 SMOOTHNESS
AND DECAY OF FOURIER COEFFICIENTS 302 9.6 EXERCISES 307 10 THE HEAT
EQUATION REVISITED 313 10.1 COMPATIBILITY CONDITIONS 314 10.2 FOURIER S
METHOD: A MATHEMATICAL JUSTIFICATION 319 10.2.1 THE SMOOTHING PROPERTY
319 10.2.2 THE DIFFERENTIAL EQUATION 321 10.2.3 THE INITIAL CONDITION
323 10.2.4 SMOOTH AND COMPATIBLE INITIAL FUNCTIONS 325 10.3 CONVERGENCE
OF FINITE DIFFERENCE SOLUTIONS 327 10.4 EXERCISES 331 11
REACTION-DIFFUSION EQUATIONS 337 11.1 THE LOGISTIC MODEL OF POPULATION
GROWTH 337 11.1.1 A NUMERICAL METHOD FOR THE LOGISTIC MODEL 339 11.2
FISHER S EQUATION 340 11.3 A FINITE DIFFERENCE SCHEME FOR FISHER S
EQUATION 342 11.4 AN INVARIANT REGION 343 11.5 THE ASYMPTOTIC SOLUTION
346 11.6 ENERGY ARGUMENTS 349 11.6.1 AN INVARIANT REGION 350 11.6.2
CONVERGENCE TOWARDS EQUILIBRIUM 351 11.6.3 DECAY OF DERIVATIVES 352 11.7
BLOWUP OF SOLUTIONS 354 11.8 EXERCISES 357 11.9 PROJECTS 360 12
APPLICATIONS OF THE FOURIER TRANSFORM 365 12.1 THE FOURIER TRANSFORM 366
12.2 PROPERTIES OF THE FOURIER TRANSFORM 368 CONTENTS XV 12.3 THE
INVERSION FORMULA 372 12.4 THE CONVOLUTION 375 12.5 PARTIAL DIFFERENTIAL
EQUATIONS 377 12.5.1 THE HEAT EQUATION 377 12.5.2 LAPLACE S EQUATION IN
A HALF-PLANE 380 12.6 EXERCISES 382 REFERENCES 385 INDEX 389
|
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| author | Tveito, Aslak Winther, Ragnar |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.353 |
| dewey-search | 515.353 |
| dewey-sort | 3515.353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. softcover printing |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T22:03:19Z |
| institution | BVB |
| isbn | 9783540887041 |
| language | English |
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| physical | XV, 392 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Tveito, Aslak Verfasser aut Introduction to partial differential equations a computational approach Aslak Tveito ; Ragnar Winther 1. softcover printing Berlin [u.a.] Springer 2009 XV, 392 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 29 Literaturverz. S. [385] - 387 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Winther, Ragnar Verfasser aut Texts in applied mathematics 29 (DE-604)BV002476038 29 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018597977&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tveito, Aslak Winther, Ragnar Introduction to partial differential equations a computational approach Texts in applied mathematics Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4128130-5 |
| title | Introduction to partial differential equations a computational approach |
| title_auth | Introduction to partial differential equations a computational approach |
| title_exact_search | Introduction to partial differential equations a computational approach |
| title_full | Introduction to partial differential equations a computational approach Aslak Tveito ; Ragnar Winther |
| title_fullStr | Introduction to partial differential equations a computational approach Aslak Tveito ; Ragnar Winther |
| title_full_unstemmed | Introduction to partial differential equations a computational approach Aslak Tveito ; Ragnar Winther |
| title_short | Introduction to partial differential equations |
| title_sort | introduction to partial differential equations a computational approach |
| title_sub | a computational approach |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Partielle Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018597977&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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