Fourier Series and Numerical Methods for Partial Differential Equations

by Bernatz, Richard

• ISBN13: 9780470617960
• ISBN10: 0470617969
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-07-20
• Publisher: Wiley

Summary

Unable to find a course book that provided all the topics needed for an introductory PDE course, the author pursued this book, which covers all of the essential topics.The needed foundation and theory is complimented with tangible applications in physics and other disciplines. Since many practical applications are non-linear, numerical solution techniques are required. Consequently, the book introduces this topic in a general way before providing the necessary details. As for an introduction to a specific method, the finite difference method is the natural place to begin. With this approach, readers more clearly understand the notations of order and convergence as well as explicit and implicit methodologies.Later, readers are introduced to the finite element method in such a way that it is seen as essentially a sub-space approximation technique. Finally, the finite analytic method is introduced, where readers are presented with the application of the Fourier Series methodology to linearized versions of non-linear PDEs. In terms of theory, material on linear PDEs reinforces the important concept of inner product spaces introduced in a linear algebra course, especially those of infinite dimension. Further, it introduces the concept of completeness, thereby introducing readers to Hilbert Spaces. Past experience with ordinary differential equations is called upon to understand the solution process for Sturm-Liouville boundary value ODE problems, which leads to an infinite-dimensional basis for an inner product space, and ultimately, a Fourier Series representation of the solution of an initial boundary value problems. Computer algebra resources such as Maple TM , Mathematicareg;, and MATLABreg; can be used to aid in understanding and applying the solution techniques to interesting problems. This can begin as soon as the theoretical work is in Sturm-Liouville problems and Fourier series is covered.Later on, it is used to apply numerical solution methods to various applications.

Author Biography

RICHARD A. BERNATZ, PhD, is Professor in the Department of Mathematics at Luther College. Dr. Bernatz is the author of numerous journal articles in his areas of research interest, which include climatology, mathematical models of watersheds, and computational fluid dynamics with applications in meteorology.

Table of Contents

Preface

Acknowledgments

Introduction

Terminology and Notation

Classification

Canonical Forms

Common PDEs

Cauchy-Kowalevski Theorem

Initial Boundary Value Problems

Solution Techniques

Separation of Variables

Exercises

Fourier Series

Vector Spaces

The Integral as an Inner Product

Principle of Superposition

General Fourier Series

Fourier Sine Series on (0, c)

Fourier Cosine Series on (0, c)

Fourier Series on (¡c; c)

Best Approximation

Bessel's Inequality

Piecewise Smooth Functions

Fourier Series Convergence

2c-Periodic Functions

Concluding Remarks

Exercises

Sturm-Liouville Problems

Basic Examples

Regular Sturm-Liouville Problems

Properties

Examples

Bessel's Equation

Legendre's Equation

Exercises

Heat Equation

Heat Equation in One Dimension

Boundary Conditions

Heat Equation in Two Dimensions

Heat Equation in Three Dimensions

Polar-Cylindrical Coordinates

Spherical Coordinates

Exercises

Heat Transfer in 1D

Homogeneous IBVP

Semi-homogeneous PDE

Non-homogeneous Boundary Conditions

Spherical Coordinate Example

Exercises

Heat Transfer in 2D and 3D

Homogeneous 2D IBVP

Semi-Homogeneous 2D IBVP

Non-Homogeneous 2D IBVP

2D BVP: Laplace & Poisson Equations

Non-homogeneous 2D Example

Time-Dependent BCs

Homogeneous 3D IBVP

Exercises

Wave Equation

Wave Equation in One Dimension

Wave Equation in Two Dimensions

Exercises

Numerical Methods: an Overview

Grid Generation

Numerical Methods

Consistency and Convergence

The Finite Difference Method

Discretization

Finite Difference Formulas

One-Dimensional Heat Equation

Crank-Nicolson Method

Error and Stability

Convergence in Practice

One-Dimensional Wave Equation

2D Heat Equation in Cartesian Coordinates

Two-Dimensional Wave Equation

2D Heat Equation in Polar Coordinates

Exercises

Finite Element Method

General Framework

1D Elliptical Example

2D Elliptical Example

Error Analysis

1D Parabolic Example

Exercises

Finite Analytic Method

1D Transport Equation

2D Transport Equation

Convergence and Accuracy

Exercises

FA One Dimensional Case

FA Two-Dimensional Case

References

Index

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