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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. Product details Format Paperback pages Dimensions x x Other books in this series. Turbulent Reactive Flows R. Add to basket. Table of contents 1. Series Expansions. Choice of Basis Functions. Comparison with Finite Element Methods. Boundary Conditions. Accuracy and Efficiency. Convergence Theory. Fourier Series.
Orders of Convergence. An Upper Bound on the Truncation Error. Asymptotic Calculation of Coefficients: Power Series. Asymptotic Calculation of Coefficients: Fourier Series. Model Functions for Fourier Series. Convergence Theory for Chebyshev Polynomials.
Mean Weighted Residual Methods. General Properties of Basis Functions: Completeness. Galerkin's Method. Galerkin's Method: Case Studies.
Galerkin's Method Today. Polynomial Interpolation. Pseudospectral: Galerkin's Method via Gaussian Quadrature. The Envelope of the Interpolation Error. Cardinal Functions. Whittaker Cardinal or "Sinc" Functions. Cardinal Functions for Trigonometric Interpolation. Cardinal Functions for Orthogonal Polynomials.
Transformations and Interpolation. Pseudospectral Methods for Boundary Value Problems. Choice of Basis Set. The Cardinal Function Basis. The Interpolation Grid. Special Problems of Higher Dimensions: Indexing. Matrix Methods. Other Discrete Symmetries.
Continuous Symmetries. Symmetry Classes for Trigonometric Functions. Explicit Time-Integration Methods. Differential Equations with Variable Coefficients. Linear and Nonlinear. Practical Matters. Partial Summation in Two or More Dimensions. The Fast Fourier Transform. Boundary Layers. Endpoint versus Interior Singularities. Diffusion Equation: Analytical and Numerical Background. Operator Theory of Time-Stepping Methods. Splitting the Navier-Stokes Equation. Case Studies of Time Integration.
Three-Dimensional Periodic Turbulence: Brachet. Stellar Convection in Annular Shell: Glatzmaier. Complex Geometry and Variable Coefficients: Zang. Iterative Methods for Solving Matrix Equations. Chebyshev Acceleration. Pre-conditioning: Finite Difference. Multigrid: An Overview. Stationary One-Step Iterative Methods. The Many Uses of Coordinate Transformation. Programming Chebyshev Methods. The General Theory of Coordinate Transformations. Mapping and Infinite and Semi-Infinite Intervals.
Adaptive Methods. Methods for Unbounded Intervals. Whittaker Cardinal or "Sinc" Expansions. Hermite functions. Behavioral versus Numerical Boundary Conditions. Rational Chebyshev Functions on y? Spherical Coordinates. Icosahedral Grids and the Radiolaria.
The Parity Factor: Sphere versus Torus. The Pole Problem. Spherical Harmonics: An Overview. Spherical Harmonics and Physics. Transformation of the Horizontal Velocities. Cylindrical, Toroidal and Polar Coordinates. An Illustration of Triangular Truncation.
Chebyshev and Fourier Spectral Methods
It seems that you're in Germany. We have a dedicated site for Germany. 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.
Chebyshev & Fourier Spectral Methods
Account Options Sign in. My library Help Advanced Book Search. View eBook. John P. Courier Corporation , Dec 3, - Mathematics - pages.