Intro to Partial Differential Equations: A Computational Approach - Springer HC

Introduction to Partial Differential Equations A Computational Approach by Aslak Tveito and Ragnar Winther Texts in Applied Mathematics 29

Publisher: Springer /
Year: 2004 ./
ISBN: 354022551X /
Description: Hardcover without dust jacket, Fine condition /
Condition: no marks, no underlining, no highlighting, like new condition /
Pages: 392 pages

Introduction to Partial Differential Equations: A Computational Approach

Texts in Applied Mathematics #29

This book teaches the basic methods of partial differential equations and introduces related important ideas associated with the analysis of numerical methods for those partial differential equations. Standard topics such as separation of variables, Fourier analysis, maximum principles and energy estimates are included. Numerical methods are introduced in parallel to the classical theory. The numerical experiments are used to illustrate properties of differential equations and theory for finite difference approximations is developed.
Numerical methods are included in the book to show the significance of computations in partial differential equations and to illustrate the strong interaction between mathematical theory and the development of numerical methods. Great care has been taken throughout the book to seek a sound balance between the analytical and numerical techniques.

The authors present the material at an easy pace with well-organized exercises ranging from the straightforward to the challenging. In addition, special projects are included, containing step by step hints and instructions, to help guide students in the correct way of approaching partial differential equations.
The text would be suitable for advanced undergraduate and graduate courses in mathematics and engineering. Necessary prerequisites for this text are basic calculus and linear algebra. Some elementary knowledge of ordinary differential equations is also preferable.

CONTENTS

1 . Setting the Scene

1.1 What Is a Differential Equation? .
1.1.1 Concepts .
1.2 The Solution and Its Properties .
1.2.1 An Ordinary\' Differential Equation
1.3 A Numerical Method .
1.4 Cauchy Problems
1.4.1 First-Order Homogeneous Equations . ,
1.4.2 First-Order Nonhomogeneous Equations
1.4.3 The Wave Equation .
1.4.4 The Heat Equation
1.5 - Exercises
1.6 Projects

2 Two-Point Boundary Value Problems

2.1 Poisson\'s Equation in One Dimension : .
2.1.1 Green\'s Function
2.1.2, Smoothness Of the Solution
2.1.3 A Maximum-Principle .
2.2 A Finite Difference Approximation
2.2.1 Taylor Series .
2.2.2 A System of Algebraic Equations
2.2.3 Gaussian Elimination for TridiagOrial Linear System\'s
2.2.4 Diagonal Dominant Matrices
2.2.5 Positive Definite Matrices
2.3 Continuous and Discrete Solutions
2.3.1 Difference and Differential Equations
2.3.2 Symmetry .
2.3.3 Uniqueness
2.3.4 A Maximum Principle for the Discrete Problem .
2.3.5 Convergence of the Discrete Solutions
2.4 Eigenvalue Problems
2.4.1 The Continuous Eigenvalue Problem
2.4.2 The Discrete Eigenvalue Problem .
2.5 Exercises
2.6 Projects

3 The Heat Equation

3.1 A Brief Overview .
3.2 Separation of Variables .
3.3 The Principle of Superposition
3.4 Fourier Coefficients
3.5 Other Boundary Conditions
3.6 The Neumann Problem
3.6.1 The Eigenvalue Problem .
3.6.2 Particular Solutions
3 6 3 A Formal Solution
3.7 Energy Arguments
3.8 Differentiation of Integrals
3.9 Exercises
3.10 Projects

4 Finite Difference Schemes for the Heat Equation

4.1 Ari Explicit Scheme .
4.2 Fourier Analysis of the Numerical Solution
4.2.1 Particular Solutions .
4.2.2 Comparison of the Analytical and Discrete Solution,
4.2.3 Stability Considerations
4.2.4 The Accuracy of the Approximation.,
4.2.5 Summary of the Comparison .
4.3 Von Neumann\'s Stability Analysis .
4.3.1 Particular Solutions: Continuous and Discrete
4.3.2 Examples .
4.3.3 A Nonlinear Problem .
4.4 An Implicit Scheme .
4.4.1 Stability Analysis . :
4.5 Numerical Stability brEnergy Arguments
4.6 Exercises

5 The Wave Equation

5.1 Separation of Variables
5.2 Uniqueness and Energy Arguments
5.3 . A Finite Difference Approximation
5.3.1 . Stability Analysis
5.4 Exercises .

6 :Maximum Principles

6.1 A Two-Point Boundary Value Problem
6.2 The Linear Heat Equation .
6.2.1 The Continuous Case
6.2.2 Uniqueness and Stability
6.2.3 The Explicit Finite Difference Scheme .
6.2.4. The Implicit Finite Difference. Scheme The Nonlinear Heat Equation,
6.3.1 The. Continuous Case,
6.3.2 An Explicit Finite Difference Scheme
6.4 Harmonic Functions .
6.4.1 Maximum Principles for Harmonic FunctionE.
6.5 Discrete Harmonic Functions
6.6 Exercises

7 Poisson\'s Equation in Two,Space,Dimensions

7.1 Rectangular Domains
7.2 Polar Coordinates
7.2.1 . The. Disc, .
7.2.2 A Wedge
7.2.3 A Corner Singularity :
7.3 Applications of the. Divergence ;Theorem: .
7.4 . The,Meari,Value.Property for,Harmonic Functions
7.5 A Finite Difference Approximation
7.5.1 . The Five:-Point Stencil .
7.5.2 An Error.Estimate
7.6 Gaussian Elimination for General Systems
7.6.1 Upper Triangular Systems
7.6.2 . General Systems
7.6.3 Banded Systems
7.6.4. Positive Definite. Systems .
7.7 Exercises

8 -Orthogonality and General Fourier Series

8.1 The Full Fourier Series
8.1.1 Even and Odd. Functions
8.1.2 Differentiation.of Fourier Series
8.1.3 The Complex Form:.
8.1.4 Changing the Scale
8.2 . Boundary Value. Problems and Orthogonal Functions . :
8.2.1 Other Boundary Conditions
8.2.2 Sturm-Liouville Problems
8.3 „ The Mean Square Distance
8.4 General Fourier Series
8.5 A Poincare Inequality
8.6 Exercises

9 Convergence of Fourier Series

9.1 . Different Notions of Convergence \' . :
9.2 . Pointwise Convergence .
9.3 Uniform Convergence
9.4 Mean Square Convergence . :
9.5 Smoothness and Decay of Fourier Coefficients .
9.6 . Exercises

10 .The Heat Equation Revisited

10.1 Compatibility Conditions
10.2 Fourier\'s Method: A Mathematical Justification
10.2.1 The Smoothing Property,
10.2.2 The Differential Equation .
10.2.3 The Initial Condition
10.2.4 Smooth and Compatible Initial Functions
10.3 Convergence of Finite Difference Solutions
10.4 Exercises .

11 Reaction-Diffusion Equations

11.1 The Logistic Model of. Population Growth
11.1.1 A\'Numerical Method for the Logistic Model
11.2 Fisher\'s Equation
11.3 A Finite Difference Scheme for Fisher\'s Equation .
11.4 An Invariant Region .
11.5. The Asymptotic Solution .
11.6 Energy Arguments
11.6.1. An Invariant Region .
11.6.2, Convergence Towards Equilibrium .
11.6.3 Decay of Derivatives :
11.7 Blowup of Solutions
11.8 Exercises
11.9 Projects

12 Applications of the Fourier Transform

12.1 The Fourier Transform .
12.2 Properties of the. Fourier Transform .
12.3 The Inversion Formula
12.4 The Convolution .
12.5 Partial Differential Equations
12.5.1 The Heat Equation
12.5.2 Laplace\'s Equation in a Half-Plane .
12.6 Exercises .

References
Index

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