Fourier Series and Numerical Methods for Partial Differential Equations

✍ Scribed by Richard Bernatz

• Publisher: Wiley
• Year: 2010
• Tongue: English
• Leaves: 337
• Edition: 1
• Category: Library

✦ Synopsis

The importance of partial differential equations (PDEs) in modeling phenomena in engineering as well as in the physical, natural, and social sciences is well known by students and practitioners in these fields. Striking a balance between theory and applications, Fourier Series and Numerical Methods for Partial Differential Equations presents an introduction to the analytical and numerical methods that are essential for working with partial differential equations. Combining methodologies from calculus, introductory linear algebra, and ordinary differential equations (ODEs), the book strengthens and extends readers' knowledge of the power of linear spaces and linear transformations for purposes of understanding and solving a wide range of PDEs.

The book begins with an introduction to the general terminology and topics related to PDEs, including the notion of initial and boundary value problems and also various solution techniques. Subsequent chapters explore: 

• The solution process for Sturm-Liouville boundary value ODE problems and a Fourier series representation of the solution of initial boundary value problems in PDEs
• The concept of completeness, which introduces readers to Hilbert spaces 
• The application of Laplace transforms and Duhamel's theorem to solve time-dependent boundary conditions
•  The finite element method, using finite dimensional subspaces
•  The finite analytic method with applications of the Fourier series methodology to linear version of non-linear PDEs

 Throughout the book, the author incorporates his own class-tested material, ensuring an accessible and easy-to-follow presentation that helps readers connect presented objectives with relevant applications to their own work. Maple is used throughout to solve many exercises, and a related Web site features Maple worksheets for readers to use when working with the book's one- and multi-dimensional problems.

Fourier Series and Numerical Methods for Partial Differential Equations is an ideal book for courses on applied mathematics and partial differential equations at the upper-undergraduate and graduate levels. It is also a reliable resource for researchers and practitioners in the fields of mathematics, science, and engineering who work with mathematical modeling of physical phenomena, including diffusion and wave aspects.

✦ Table of Contents

FOURIER SERIES AND NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
CONTENTS
Preface
Acknowledgments
1 Introduction
1.1 Terminology and Notation
1.2 Classification
1.3 Canonical Forms
1.4 Common PDEs
1.5 Cauchy–Kowalevski Theorem
1.6 Initial Boundary Value Problems
1.7 Solution Techniques
1.8 Separation of Variables
Exercises
2 Fouier Series
2.1 Vector Spaces
2.1.1 Subspaces
2.1.2 Basis and Dimension
2.1.3 Inner Products
2.2 The Integral as an Inner Product
2.2.1 Piecewise Continuous Functions
2.2.2 Inner Product on Cp(a, b)
2.3 Principle of Superposition
2.3.1 Finite Case
2.3.2 Infinite Case
2.3.3 Hilbert Spaces
2.4 General Fourier Series
2.5 Fourier Sine Series on (0, c)
2.5.1 Odd, Periodic Extensions
2.6 Fourier Cosine Series on (0, c)
2.6.1 Even, Periodic Extensions
2.7 Fourier Series on (–c,c)
2.7.1 2c-Periodic Extensions
2.8 Best Approximation
2.9 Bessel's Inequality
2.10 Piecewise Smooth Functions
2.11 Fourier Series Convergence
2.11.1 Alternate Form
2.11.2 Riemann–Lebesgue Lemma
2.11.3 A Dirichlet Kernel Lemma
2.11.4 A Fourier Theorem
2.12 2c-Periodic Functions
2.13 Concluding Remarks
Exercises
3 Sturm–Liouville Problems
3.1 Basic Examples
3.2 Regular Sturm–Liouville Problems
3.3 Properties
3.3.1 Eigenfunction Orthogonality
3.3.2 Real Eigenvalues
3.3.3 Eigenfunction Uniqueness
3.3.4 Non-negative Eigenvalues
3.4 Examples
3.4.1 Neumann Boundary Conditions [0, c]
3.4.2 Robin and Neumann BCs
3.4.3 Periodic Boundary Conditions
3.5 Bessel's Equation
3.6 Legendre's Equation
Exercises
4 Heat Equation
4.1 Heat Equation in 1D
4.2 Boundary Conditions
4.3 Heat Equation in 2D
4.4 Heat Equation in 3D
4.5 Polar-Cylindrical Coordinates
4.6 Spherical Coordinates
Exercises
5 Heat Transfer in ID
5.1 Homogeneous IBVP
5.1.1 Example: Insulated Ends
5.2 Semihomogeneous PDE
5.2.1 Variation of Parameters
5.2.2 Example: Semihomogeneous IBVP
5.3 Nonhomogeneous Boundary Conditions
5.3.1 Example: Nonhomogeneous Boundary Condition
5.3.2 Example: Time-Dependent Boundary Condition
5.3.3 Laplace Transforms
5.3.4 Duhamel's Theorem
5.4 Spherical Coordinate Example
Exercises
6 Heat Transfer in 2D and 3D
6.1 Homogeneous 2D IBVP
6.1.1 Example: Homogeneous IBVP
6.2 Semihomogeneous 2D IBVP
6.2.1 Example: Internal Source or Sink
6.3 Nonhomogeneous 2D IBVP
6.4 2D BVP: Laplace and Poisson Equations
6.4.1 Dirichlet Problems
6.4.2 Dirichlet Example
6.4.3 Neumann Problems
6.4.4 Neumann Example
6.4.5 Dirichlet, Neumann BC Example
6.4.6 Poisson Problems
6.5 Nonhomogeneous 2D Example
6.6 Time-Dependent BCs
6.7 Homogeneous 3D IBVP
Exercises
7 Wave Equation
7.1 Wave Equation in ID
7.1.1 d'Alembert's Solution
7.1.2 Homogeneous IBVP: Series Solution
7.1.3 Semihomogeneous IBVP
7.1.4 Nonhomogeneous IBVP
7.1.5 Homogeneous IBVP in Polar Coordinates
7.2 Wave Equation in 2D
7.2.1 2D Homogeneous Solution
Exercises
8 Numerical Methods: an Overview
8.1 Grid Generation
8.1.1 Adaptive Grids
8.1.2 Multilevel Methods
8.2 Numerical Methods
8.2.1 Finite Difference Method
8.2.2 Finite Element Method
8.2.3 Finite Analytic Method
8.3 Consistency and Convergence
9 The Finite Difference Method
9.1 Discretization
9.2 Finite Difference Formulas
9.2.1 First Partíais
9.2.2 Second Partíais
9.3 ID Heat Equation
9.3.1 Explicit Formulation
9.3.2 Implicit Formulation
9.4 Crank–Nicolson Method
9.5 Error and Stability
9.5.1 Error Types
9.5.2 Stability
9.6 Convergence in Practice
9.7 1D Wave Equation
9.7.1 Implicit Formulation
9.7.2 Initial Conditions
9.8 2D Heat Equation in Cartesian Coordinates
9.9 Two-Dimensional Wave Equation
9.10 2D Heat Equation in Polar Coordinates
Exercises
10 Finite Element Method
10.1 General Framework
10.2 1D Elliptical Example
10.2.1 Reformulations
10.2.2 Equivalence in Forms
10.2.3 Finite Element Solution
10.3 2D Elliptical Example
10.3.1 Weak Formulation
10.3.2 Finite Element Approximation
10.4 Error Analysis
10.5 1D Parabolic Example
10.5.1 Weak Formulation
10.5.2 Method of Lines
10.5.3 Backward Euler's Method
Exercises
11 Finite Analytic Method
11.1 1D Transport Equation
11.1.1 Finite Analytic Solution
11.1.2 FA and FD Coefficient Comparison
11.1.3 Hybrid Finite Analytic Solution
11.2 2D Transport Equation
11.2.1 FA Solution on Uniform Grids
11.2.2 The Poisson Equation
11.3 Convergence and Accuracy
Exercises
Appendix A: FA 1D Case
Appendix B: FA 2D Case
B.l The Case ø = 1
B.2 The Case ø = –Bx + Ay
References
Index