# Fourier Series and Numerical Methods for Partial Differential Equations

ISBN: 978-0-470-61796-0
318 pages
July 2010

## Description

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.

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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 Fourier Series.

2.1 Vector Spaces.

2.2 The Integral as an Inner Product.

2.3 Principle of Superposition.

2.4 General Fourier Series.

2.5 Fourier Sine Series on (0, c).

2.6 Fourier Cosine Series on (0, c).

2.7 Fourier Series on (–c; c).

2.8 Best Approximation.

2.9 Bessel's Inequality.

2.10 Piecewise Smooth Functions.

2.11 Fourier Series Convergence.

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.

3.4 Examples.

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 1D.

5.1 Homogeneous IBVP.

5.2 Semihomogeneous PDE.

5.3 Nonhomogeneous Boundary Conditions.

5.4 Spherical Coordinate Example.

Exercises.

6 Heat Transfer in 2D and 3D.

6.1 Homogeneous 2D IBVP.

6.2 Semihomogeneous 2D IBVP.

6.3 Nonhomogeneous 2D IBVP.

6.4 2D BVP: Laplace and Poisson Equations.

6.5 Nonhomogeneous 2D Example.

6.6 Time-Dependent BCs.

6.7 Homogeneous 3D IBVP.

Exercises.

7 Wave Equation.

7.1 Wave Equation in 1D.

7.2 Wave Equation in 2D.

Exercises.

8 Numerical Methods: an Overview.

8.1 Grid Generation.

8.2 Numerical Methods.

8.3 Consistency and Convergence.

9 The Finite Difference Method.

9.1 Discretization.

9.2 Finite Difference Formulas.

9.3 1D Heat Equation.

9.4 Crank–Nicolson Method.

9.5 Error and Stability.

9.6 Convergence in Practice.

9.7 1D Wave Equation.

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.

10.3 2D Elliptical Example.

10.4 Error Analysis.

10.5 1D Parabolic Example.

Exercises.

11 Finite Analytic Method.

11.1 1D Transport Equation.

11.2 2D Transport Equation.

11.3 Convergence and Accuracy.

Exercises.

Appendix A: FA 1D Case.

Appendix B: FA 2D Case.

References.

Index.

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## Author Information

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.
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## Reviews

"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." (Mathematical Reviews, 2011)

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