Fourier Series and Numerical Methods for Partial Differential Equations

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.