Introduction to computation and modeling for differential equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, N.J.
Wiley
2008
|
| Schlagworte: | |
| Online-Zugang: | Table of contents only Publisher description Contributor biographical information Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XVI, 235 S. |
| ISBN: | 9780470270851 |
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| 245 | 1 | 0 | |a Introduction to computation and modeling for differential equations |c Lennart Edsberg |
| 264 | 1 | |a Hoboken, N.J. |b Wiley |c 2008 | |
| 300 | |a XVI, 235 S. | ||
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| 650 | 4 | |a Équations différentielles - Modèles mathématiques | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Mathematisches Modell | |
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Datensatz im Suchindex
| _version_ | 1805082899367591936 |
|---|---|
| adam_text |
Titel: Introduction to computation and modeling for differential equations
Autor: Edsberg, Lennart
Jahr: 2008
Contents
List of Figures
Preface
1. Introduction
xv
xi
1
1.1 What Is a Differential Equation? / 1
1.2 Examples of an Ordinary and a Partial Differential Equation / 2
1.3 Numerical Analysis, a Necessity for Scientific Computing / 5
1.4 Outline of the Contents of This Book / 8
Bibliography / 10
2. Ordinary Differential Equations 11
2.1 Problem Classification / 11
2.2 Linear Systems of ODEs with Constant Coefficients / 16
2.3 Some Stability Concepts for ODEs / 20
2.3.1 Stability for a Solution Trajectory of an ODE System / 20
2.3.2 Stability for Critical Points of ODE Systems / 23
2.4 Some ODE Models in Science and Engineering / 26
2.4.1 Newton's Second Law / 27
2.4.2 Hamilton's Equations / 27
2.4.3 Electrical Networks / 28
2.4.4 Chemical Kinetics / 28
vii
W/7 CONTENTS
2.4.5 Control Theory / 29
2.4.6 Compartment Models / 29
2.5 Some Examples from Applications / 30
Bibliography / 36
3. Numerical Methods for Initial Value Problems 37
3.1 Graphical Representation of Solutions / 38
3.2 Basic Principles of Numerical Approximation of ODEs / 40
3.3 Numerical Solution of IVPs with Euler's Method / 41
3.3.1 Euler's Explicit Method: Accuracy / 43
3.3.2 Euler's Explicit Method: Improving the Accuracy / 46
3.3.3 Euler's Explicit Method: Stability / 47
3.3.4 Euler's Implicit Method / 53
3.3.5 The Trapezoidal Method / 54
3.4 Higher-Order Methods for the IVP / 55
3.4.1 Runge-Kutta Methods / 55
3.4.2 Linear Multistep Methods / 59
3.5 The Variational Equation and Parameter Fitting in IVPs / 61
Bibliography / 63
4. Numerical Methods for Boundary Value Problems 65
4.1 Applications / 67
4.2 Difference Methods for BVPs / 72
4.2.1 A Model Problem for BVPs / 72
4.2.2 Accuracy / 78
4.2.3 Spurious Oscillations / 79
4.2.4 Linear Two-Point BVPs / 82
4.2.5 Nonlinear Two-Point BVPs / 82
4.2.6 The Shooting Method / 84
4.3 Ansatz Methods for BVPs / 86
Bibliography / 90
5. Partial Differential Equations 91
5.1 Classical PDE Problems / 92
5.2 Differential Operators Used for PDEs / 96
5.3 Some PDEs in Science and Engineering / 99
5.3.1 Navier-Stokes Equations in Fluid Dynamics / 100
CONTENTS IX
5.3.2 The Convection-Diffusion-Reaction Equations / 101
5.3.3 The Heat Equation / 101
5.3.4 The Diffusion Equation / 102
5.3.5 Maxwell's Equations for the Electromagnetic Field / 102
5.3.6 Acoustic Waves / 103
5.3.7 Schrödinger's Equation in Quantum Mechanics / 103
5.3.8 Navier's Equations in Structural Mechanics / 104
5.3.9 Black-Scholes Equation in Financial Mathematics / 104
5.4 Initial and Boundary Conditions for PDEs / 105
5.5 Numerical Solution of PDEs, Some General Comments / 106
Bibliography / 106
6. Numerical Methods for Parabolic Partial Differential Equations 107
6.1 Applications / 108
6.2 An Introductory Example of Discretization /III
6.3 The Method of Lines for Parabolic PDEs / 114
6.3.1 Solving the Model Problem with MoL / 114
6.3.2 Various Types of Boundary Conditions / 119
6.3.3 An Example of the Use of MoL for a Mixed Boundary
Condition / 120
6.4 Generalizations of the Heat Equation / 121
6.4.1 The Heat Equation with Variable Conductivity / 121
6.4.2 The Convection-Diffusion-Reaction PDE / 122
6.4.3 The General Nonlinear Parabolic PDE / 122
6.5 Ansatz Methods for the Model Equation / 123
Bibliography / 125
7. Numerical Methods for Elliptic Partial Differential Equations 127
7.1 Applications / 129
7.2 The Finite Difference Method / 135
7.3 Discretization of a Problem with Different BCs / 138
7.4 The Finite Element Method / 140
Bibliography / 146
8. Numerical Methods for Hyperbolic Partial Differential Equations 147
8.1 Applications / 153
8.2 Numerical Solution of Hyperbolic PDEs / 155
X CONTENTS
8.3 Introduction to Numerical Stability for Hyperbolic PDEs / 161
Bibliography / 162
9. Mathematical Modeling with Differential Equations 163
9.1 Laws of Nature / 164
9.2 Constitutive Equations / 166
9.2.1 Equations in Heat Transfer Problems / 167
9.2.2 Equations in Mass Diffusion Problems / 167
9.2.3 Equations in Mechanical Moment Diffusion Problems / 168
9.2.4 Equations in Elastic Solid Mechanics Problems / 168
9.2.5 Equations in Chemical Reaction Engineering Problems / 168
9.2.6 Equations in Electrical Engineering Problems / 169
9.3 Conservative Laws / 169
9.3.1 Some Examples of Lumped Models / 170
9.3.2 Some Examples of Distributed Models / 172
9.4 Scaling of Differential Equations to Dimensionless Form / 175
Bibliography / 179
A. Appendix: Some Numerical and Mathematical Tools / 181
A.l Newton's Method for Systems of Nonlinear Algebraic Equations / 181
A.l.l Square Systems / 181
A.1.2 Overdetermined Systems / 184
A.2 Some Facts about Linear Difference Equations / 185
A.3 Derivation of Difference Approximations / 189
A.4 The Interpretations of Div and Curl / 192
A.5 Numerical Solution of Algebraic Systems of Equations / 194
A.5.1 Direct Methods / 194
A.5.2 Iterative Methods for Linear Systems of Equations / 197
A.6 Some Results for Fourier Transforms / 202
B. Appendix: Software for Scientific Computing / 205
B.l Matlab® / 206
B.2 Comsol Multiphysics® / 212
Bibliography / 214
C. Appendix: Computer Exercises to Support the Chapters / 215
Index
229 |
| adam_txt |
Titel: Introduction to computation and modeling for differential equations
Autor: Edsberg, Lennart
Jahr: 2008
Contents
List of Figures
Preface
1. Introduction
xv
xi
1
1.1 What Is a Differential Equation? / 1
1.2 Examples of an Ordinary and a Partial Differential Equation / 2
1.3 Numerical Analysis, a Necessity for Scientific Computing / 5
1.4 Outline of the Contents of This Book / 8
Bibliography / 10
2. Ordinary Differential Equations 11
2.1 Problem Classification / 11
2.2 Linear Systems of ODEs with Constant Coefficients / 16
2.3 Some Stability Concepts for ODEs / 20
2.3.1 Stability for a Solution Trajectory of an ODE System / 20
2.3.2 Stability for Critical Points of ODE Systems / 23
2.4 Some ODE Models in Science and Engineering / 26
2.4.1 Newton's Second Law / 27
2.4.2 Hamilton's Equations / 27
2.4.3 Electrical Networks / 28
2.4.4 Chemical Kinetics / 28
vii
W/7 CONTENTS
2.4.5 Control Theory / 29
2.4.6 Compartment Models / 29
2.5 Some Examples from Applications / 30
Bibliography / 36
3. Numerical Methods for Initial Value Problems 37
3.1 Graphical Representation of Solutions / 38
3.2 Basic Principles of Numerical Approximation of ODEs / 40
3.3 Numerical Solution of IVPs with Euler's Method / 41
3.3.1 Euler's Explicit Method: Accuracy / 43
3.3.2 Euler's Explicit Method: Improving the Accuracy / 46
3.3.3 Euler's Explicit Method: Stability / 47
3.3.4 Euler's Implicit Method / 53
3.3.5 The Trapezoidal Method / 54
3.4 Higher-Order Methods for the IVP / 55
3.4.1 Runge-Kutta Methods / 55
3.4.2 Linear Multistep Methods / 59
3.5 The Variational Equation and Parameter Fitting in IVPs / 61
Bibliography / 63
4. Numerical Methods for Boundary Value Problems 65
4.1 Applications / 67
4.2 Difference Methods for BVPs / 72
4.2.1 A Model Problem for BVPs / 72
4.2.2 Accuracy / 78
4.2.3 Spurious Oscillations / 79
4.2.4 Linear Two-Point BVPs / 82
4.2.5 Nonlinear Two-Point BVPs / 82
4.2.6 The Shooting Method / 84
4.3 Ansatz Methods for BVPs / 86
Bibliography / 90
5. Partial Differential Equations 91
5.1 Classical PDE Problems / 92
5.2 Differential Operators Used for PDEs / 96
5.3 Some PDEs in Science and Engineering / 99
5.3.1 Navier-Stokes Equations in Fluid Dynamics / 100
CONTENTS IX
5.3.2 The Convection-Diffusion-Reaction Equations / 101
5.3.3 The Heat Equation / 101
5.3.4 The Diffusion Equation / 102
5.3.5 Maxwell's Equations for the Electromagnetic Field / 102
5.3.6 Acoustic Waves / 103
5.3.7 Schrödinger's Equation in Quantum Mechanics / 103
5.3.8 Navier's Equations in Structural Mechanics / 104
5.3.9 Black-Scholes Equation in Financial Mathematics / 104
5.4 Initial and Boundary Conditions for PDEs / 105
5.5 Numerical Solution of PDEs, Some General Comments / 106
Bibliography / 106
6. Numerical Methods for Parabolic Partial Differential Equations 107
6.1 Applications / 108
6.2 An Introductory Example of Discretization /III
6.3 The Method of Lines for Parabolic PDEs / 114
6.3.1 Solving the Model Problem with MoL / 114
6.3.2 Various Types of Boundary Conditions / 119
6.3.3 An Example of the Use of MoL for a Mixed Boundary
Condition / 120
6.4 Generalizations of the Heat Equation / 121
6.4.1 The Heat Equation with Variable Conductivity / 121
6.4.2 The Convection-Diffusion-Reaction PDE / 122
6.4.3 The General Nonlinear Parabolic PDE / 122
6.5 Ansatz Methods for the Model Equation / 123
Bibliography / 125
7. Numerical Methods for Elliptic Partial Differential Equations 127
7.1 Applications / 129
7.2 The Finite Difference Method / 135
7.3 Discretization of a Problem with Different BCs / 138
7.4 The Finite Element Method / 140
Bibliography / 146
8. Numerical Methods for Hyperbolic Partial Differential Equations 147
8.1 Applications / 153
8.2 Numerical Solution of Hyperbolic PDEs / 155
X CONTENTS
8.3 Introduction to Numerical Stability for Hyperbolic PDEs / 161
Bibliography / 162
9. Mathematical Modeling with Differential Equations 163
9.1 Laws of Nature / 164
9.2 Constitutive Equations / 166
9.2.1 Equations in Heat Transfer Problems / 167
9.2.2 Equations in Mass Diffusion Problems / 167
9.2.3 Equations in Mechanical Moment Diffusion Problems / 168
9.2.4 Equations in Elastic Solid Mechanics Problems / 168
9.2.5 Equations in Chemical Reaction Engineering Problems / 168
9.2.6 Equations in Electrical Engineering Problems / 169
9.3 Conservative Laws / 169
9.3.1 Some Examples of Lumped Models / 170
9.3.2 Some Examples of Distributed Models / 172
9.4 Scaling of Differential Equations to Dimensionless Form / 175
Bibliography / 179
A. Appendix: Some Numerical and Mathematical Tools / 181
A.l Newton's Method for Systems of Nonlinear Algebraic Equations / 181
A.l.l Square Systems / 181
A.1.2 Overdetermined Systems / 184
A.2 Some Facts about Linear Difference Equations / 185
A.3 Derivation of Difference Approximations / 189
A.4 The Interpretations of Div and Curl / 192
A.5 Numerical Solution of Algebraic Systems of Equations / 194
A.5.1 Direct Methods / 194
A.5.2 Iterative Methods for Linear Systems of Equations / 197
A.6 Some Results for Fourier Transforms / 202
B. Appendix: Software for Scientific Computing / 205
B.l Matlab® / 206
B.2 Comsol Multiphysics® / 212
Bibliography / 214
C. Appendix: Computer Exercises to Support the Chapters / 215
Index
229 |
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| spelling | Edsberg, Lennart Verfasser aut Introduction to computation and modeling for differential equations Lennart Edsberg Hoboken, N.J. Wiley 2008 XVI, 235 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Équations différentielles - Informatique Équations différentielles - Modèles mathématiques Datenverarbeitung Mathematisches Modell Differential equations Data processing Differential equations Mathematical models Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Mathematisches Modell (DE-588)4114528-8 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 http://www.loc.gov/catdir/enhancements/fy0806/2007046848-t.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0806/2007046848-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0829/2007046848-b.html Contributor biographical information HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016580881&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Edsberg, Lennart Introduction to computation and modeling for differential equations Équations différentielles - Informatique Équations différentielles - Modèles mathématiques Datenverarbeitung Mathematisches Modell Differential equations Data processing Differential equations Mathematical models Differentialgleichung (DE-588)4012249-9 gnd Mathematisches Modell (DE-588)4114528-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4114528-8 (DE-588)4128130-5 |
| title | Introduction to computation and modeling for differential equations |
| title_auth | Introduction to computation and modeling for differential equations |
| title_exact_search | Introduction to computation and modeling for differential equations |
| title_exact_search_txtP | Introduction to computation and modeling for differential equations |
| title_full | Introduction to computation and modeling for differential equations Lennart Edsberg |
| title_fullStr | Introduction to computation and modeling for differential equations Lennart Edsberg |
| title_full_unstemmed | Introduction to computation and modeling for differential equations Lennart Edsberg |
| title_short | Introduction to computation and modeling for differential equations |
| title_sort | introduction to computation and modeling for differential equations |
| topic | Équations différentielles - Informatique Équations différentielles - Modèles mathématiques Datenverarbeitung Mathematisches Modell Differential equations Data processing Differential equations Mathematical models Differentialgleichung (DE-588)4012249-9 gnd Mathematisches Modell (DE-588)4114528-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Équations différentielles - Informatique Équations différentielles - Modèles mathématiques Datenverarbeitung Mathematisches Modell Differential equations Data processing Differential equations Mathematical models Differentialgleichung Numerisches Verfahren |
| url | http://www.loc.gov/catdir/enhancements/fy0806/2007046848-t.html http://www.loc.gov/catdir/enhancements/fy0806/2007046848-d.html http://www.loc.gov/catdir/enhancements/fy0829/2007046848-b.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016580881&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT edsberglennart introductiontocomputationandmodelingfordifferentialequations |