Introduction to partial differential equations: a computational approach
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2005]
|
| Edition: | Corrected second printing |
| Series: | Texts in applied mathematics
29 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 385 - 387 |
| Physical Description: | xv, 392 Seiten Diagramme 24 cm |
| ISBN: | 354022551X 0387983279 |
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MARC
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| 100 | 1 | |a Tveito, Aslak |d 1961- |0 (DE-588)1012216276 |4 aut | |
| 245 | 1 | 0 | |a Introduction to partial differential equations |b a computational approach |c Aslak Tveito ; Ragnar Winther |
| 250 | |a Corrected second printing | ||
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [2005] | |
| 264 | 4 | |c © 2005 | |
| 300 | |a xv, 392 Seiten |b Diagramme |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Texts in applied mathematics |v 29 | |
| 500 | |a Literaturverz. S. 385 - 387 | ||
| 650 | 4 | |a Équations aux dérivées partielles | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Texts in applied mathematics |v 29 |w (DE-604)BV002476038 |9 29 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-013108242 | ||
Record in the Search Index
| _version_ | 1804133266811256832 |
|---|---|
| adam_text | 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 Diiference 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
....................... 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
Generai
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 Poincaré
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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| discipline | Mathematik |
| edition | Corrected second printing |
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| id | DE-604.BV019782329 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T20:05:59Z |
| institution | BVB |
| isbn | 354022551X 0387983279 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013108242 |
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| owner_facet | DE-355 DE-BY-UBR DE-11 DE-83 |
| physical | xv, 392 Seiten Diagramme 24 cm |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Tveito, Aslak 1961- (DE-588)1012216276 aut Introduction to partial differential equations a computational approach Aslak Tveito ; Ragnar Winther Corrected second printing Berlin ; Heidelberg Springer [2005] © 2005 xv, 392 Seiten Diagramme 24 cm txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 29 Literaturverz. S. 385 - 387 Équations aux dérivées partielles Differential equations, Partial Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Winther, Ragnar (DE-588)1278891749 aut Erscheint auch als Online-Ausgabe 0-387-98327-9 Texts in applied mathematics 29 (DE-604)BV002476038 29 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013108242&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tveito, Aslak 1961- Winther, Ragnar Introduction to partial differential equations a computational approach Texts in applied mathematics Équations aux dérivées partielles Differential equations, Partial Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4044779-0 |
| 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 | Équations aux dérivées partielles Differential equations, Partial Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Équations aux dérivées partielles Differential equations, Partial Numerisches Verfahren Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013108242&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
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