Fourier Series and Numerical Methods for Partial Differential Equations

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 MapleTM, Mathematica®, and MATLAB® 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.

Spis treści:
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 One Dimension.
4.2 Boundary Conditions.
4.3 Heat Equation in Two Dimensions.
4.4 Heat Equation in Three Dimensions.
4.5 Polar–Cylindrical Coordinates.
4.6 Spherical Coordinates.
Exercises.
5 Heat Transfer in 1D.
5.1 Homogeneous IBVP.
5.2 Semi–homogeneous PDE.
5.3 Non–homogeneous Boundary Conditions.
5.4 Spherical Coordinate Example.
Exercises.
6 Heat Transfer in 2D and 3D.
6.1 Homogeneous 2D IBVP.
6.2 Semi–Homogeneous 2D IBVP.
6.3 Non–Homogeneous 2D IBVP.
6.4 2D BVP: Laplace & Poisson Equations.
6.5 Non–homogeneous 2D Example.
6.6 Time–Dependent BCs.
6.7 Homogeneou s 3D IBVP.
Exercises.
7 Wave Equation.
7.1 Wave Equation in One Dimension.
7.2 Wave Equation in Two Dimensions.
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 One–Dimensional Heat Equation.
9.4 Crank–Nicolson Method.
9.5 Error and Stability.
9.6 Convergence in Practice.
9.7 One–Dimensional 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 One Dimensional Case.
Appendix B: FA Two–Dimensional Case.
References.
Index.