Ebook
Fourier Series and Numerical Methods for Partial Differential EquationsISBN: 9780470651377
336 pages
July 2010

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 SturmLiouville 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 timedependent 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 nonlinear PDEs
Throughout the book, the author incorporates his own classtested material, ensuring an accessible and easytofollow 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 multidimensional 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 upperundergraduate 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.
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 2cPeriodic 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 PolarCylindrical 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 TimeDependent 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 TwoDimensional 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.