The goal of this book is to teach spectral methods for solving boundary value, eigenvalue, and time-dependent problems. Although the title speaks only of Chebyshev polynomials and trigonometric functions, the book also discusses Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions. These notes evolved from a course I have taught the past five years to an audience drawn from half a dozen different disciplines at the University of Michigan: aerospace engineering, meteorology, physical oceanography, mechanical engineering, naval architecture, and nuclear engineering. With such a diverse audience, this book is not focused on a particular discipline, but rather upon solving differential equations in general. The style is not lemma-theorem-Sobolev space, but algorithms guidelines-rules-of-thumb. Although the course is aimed at graduate students, the required background is limited. It helps if the reader has taken an elementary course in computer methods and also has been exposed to Fourier series and complex variables at the undergraduate level. However, even this background is not absolutely necessary. Chapters 2 to 5 are a self contained treatment of basic convergence and interpolation theory.

Inhalt

1. Introduction.- 2. Convergence Theory.- 3. Galerkin's Method & Inner Products.- 4. Interpolation, Collocation & All That.- 5. Cardinal Functions.- 6. Pseudospectral Methods for Boundary Value Problems.- 7. Symmetry & Parity.- 8. Explicit Time-Integration Methods.- 9. Practical Matters.- 10. Fractional Steps Time Integration: Splitting and Its Cousins.- 11. Case Studies of Time Integration.- 12. Iterative Methods for Solving Matrix Equations.- 13. The Many Uses of Coordinate Transformation.- 14. Methods for Unbounded Intervals.- 15. Spherical Coordinates.- 16. Special Tricks.- 17. Analytical Applications and Symbolic Manipulation.- 18. The Tau-Method.- 19. Domain Decomposition Methods.- Appendix A. A Bestiary of Basis Functions.- 0. Trigonometric Basis Functions: Fourier Series.- 4. Gegenbauer Polynomials.- 5. Laguerre Functions.- 6. Hermite Functions.- Table A-1. Flow Chart on Choice of Basis Functions.- Fig. A-1. Regions of Convergence of Basis Sets in the Complex Plane.- Appendix B. Matrix Methods.- 1. Gaussian Elimination & LU Decomposition.- 2. Block-Banded Elimination: the Lindzen-Kuo Algorithm.- 3. Block and Bordered Matrices: the Fadeev-Fadeeva Factorization.- 4. Global Methods for Linear Eigenvalue Problems: The QR algorithm & the Pseudospectral Method.- Table B-1. Operation Counts for Banded Matrices.- Appendix C. The Newton-Kantorovich Method for Nonlinear Boundary and Eigenvalue Problems 1. Introduction.- 2. Examples.- 3. Eigenvalue Problems.- 4. Summary.- Appendix D. The Continuation Method.- 1. Introduction.- 2. Examples.- 3. Initialization Strategies.- 4. Limit Points.- 5. Bifurcation Points.- 6. Pseudoarclength Continuation.- Appendix E. Mapping Transformations.- Table E-1 [General Mapping].- Table E-2 [y = cos(x)].- Table E-3 [y =arccos(x)].- Table E-4 [y = L cot(x)].- Table E-8. [y = L arctanh(x)].- 2. Derivative Boundary Conditions.- Appendix F. Cardinal Functions.- 1. Introduction.- 2. General Fourier Series: Endpoint Grid.- 3. Fourier Cosine Series: Endpoint Grid.- 4. Fourier Sine Series: Endpoint Grid.- 5. Sinc(x): Whittaker Cardinal Functions.- 6. Chebyshev Polynomials: Extrema & Endpoints Grid.- 7. Chebyshev Polynomials: Interior Grid.- 8. Legendre Polynomials: Extrema & Endpoints Grid.- 9. Cosine Cardinal Functions on the Interior [Rectangle Rule or Roots] Grid.- 10. Sine Cardinal Functions on the Interior [Rectangle Rule or Roots] Grid.- Appendix G. Minimization of the Square of the Residual (Least Squares) for Solving Differential Equations via Nonlinear Degrees of Freedom.- 1. Introduction.- 2. Newton's Method.- 3. Linear Least-Squares Fitting and the Neglect of the Second Derivative.- 4. Evaluating the Second Derivatives for the Hessian Matrix.- 5. Steepest Descent.- 6. Convexity, Positive Definiteness, and Conditions for a Minimum.- 7. Approximations that Depend Nonlinearly on the Free Parameters.- 8. Nonlinear Approximation to the KdV Soliton: A Worked Example.- Errata.