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Partial Differential Equations with Numerical Methods Book

Partial Differential Equations with Numerical Methods
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Partial Differential Equations with Numerical Methods, The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elli, Partial Differential Equations with Numerical Methods
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  • Partial Differential Equations with Numerical Methods
  • Written by author Stig Larsson
  • Published by Springer-Verlag New York, LLC, 12/28/2008
  • The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elli
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1 Introduction 1

1.1 Background 1

1.2 Notation and Mathematical Preliminaries 4

1.3 Physical Derivation of the Heat Equation 7

1.4 Problems 12

2 A Two-Point Boundary Value Problem 15

2.1 The Maximum Principle 15

2.2 Green's Function 18

2.3 Variational Formulation 20

2.4 Problems 23

3 Elliptic Equations 25

3.1 Preliminaries 25

3.2 A Maximum Principle 26

3.3 Dirichlet's Problem for a Disc. Poisson's Integral 28

3.4 Fundamental Solutions. Green's Function 30

3.5 Variational Formulation of the Dirichlet Problem 32

3.6 A Neumann Problem 35

3.7 Regularity 37

3.8 Problems 38

4 Finite Difference Methods for Elliptic Equations 43

4.1 A Two-Point Boundary Value Problem 43

4.2 Poisson's Equation 46

4.3 Problems 49

5 Finite Element Methods for Elliptic Equations 51

5.1 A Two-Point Boundary Value Problem 51

5.2 A Model Problem in the Plane 57

5.3 Some Facts from Approximation Theory 60

5.4 Error Estimates 63

5.5 An A Posteriori Error Estimate 66

5.6 Numerical Integration 67

5.7 A Mixed Finite Element Method 71

5.8 Problems 73

6 The Elliptic Eigenvalue Problem 77

6.1 Eigenfunction Expansions 77

6.2 Numerical Solution of the Eigenvalue Problem 88

6.3 Problems 93

7 Initial-Value Problems for ODEs 95

7.1 The Initial Value Problem for a Linear System 95

7.2 Numerical Solution of ODEs 101

7.3 Problems 106

8 Parabolic Equations 109

8.1 The Pure Initial Value Problem 109

8.2 Solution by Eigenfunction Expansion 114

8.3 Variational Formulation. Energy Estimates 120

8.4 A Maximum Principle 122

8.5 Problems 124

9 Finite Difference Methods for Parabolic Problems 129

9.1 The Pure Initial Value Problem 129

9.2 The MixedInitial-Boundary Value Problem 138

9.3 Problems 146

10 The Finite Element Method for a Parabolic Problem 149

10.1 The Semidiscrete Galerkin Finite Element Method 149

10.2 Some Completely Discrete Schemes 156

10.3 Problems 159

11 Hyperbolic Equations 163

11.1 Characteristic Directions and Surfaces 163

11.2 The Wave Equation 166

11.3 First Order Scalar Equations 169

11.4 Symmetric Hyperbolic Systems 173

11.5 Problems 181

12 Finite Difference Methods for Hyperbolic Equations 185

12.1 First Order Scalar Equations 185

12.2 Symmetric Hyperbolic Systems 192

12.3 The Wendroff Box Scheme 196

12.4 Problems 198

13 The Finite Element Method for Hyperbolic Equations 201

13.1 The Wave Equation 201

13.2 First Order Hyperbolic Equations 205

13.3 Problems 216

14 Some Other Classes of Numerical Methods 217

14.1 Collocation methods 217

14.2 Spectral Methods 218

14.3 Finite Volume Methods 219

14.4 Boundary Element Methods 221

14.5 Problems 223

A Some Tools from Mathematical Analysis 225

A.1 Abstract Linear Spaces 225

A.2 Function Spaces 231

A.3 The Fourier Transform 238

A.4 Problems 240

B Orientation on Numerical Linear Algebra 245

B.1 Direct Methods 245

B.2 Iterative Methods. Relaxation, Overrelaxation, and Acceleration 246

B.3 Alternating Direction Methods 248

B.4 Preconditioned Conjugate Gradient Methods 249

B.5 Multigrid and Domain Decomposition Methods 250

Bibliography 253

Index 257


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Partial Differential Equations with Numerical Methods, The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elli, Partial Differential Equations with Numerical Methods

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Partial Differential Equations with Numerical Methods, The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elli, Partial Differential Equations with Numerical Methods

Partial Differential Equations with Numerical Methods

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Partial Differential Equations with Numerical Methods, The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elli, Partial Differential Equations with Numerical Methods

Partial Differential Equations with Numerical Methods

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