Numerical Approximations Using Chebyshev Polynomial Expansions
نویسندگان
چکیده
The aim of this work is to find numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N . The solutions are exact at these points, apart from round-off computer errors and the convergence of other numerical methods used in connection to solving the linear system of equations. Applications to initial-value problems in time-dependent quantum field theory, and second order boundary-value problems in fluid dynamics are presented.
منابع مشابه
Rigorous uniform approximation of D-finite functions using Chebyshev expansions
A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-kn...
متن کاملNumerical approximations using Chebyshev polynomial expansions: El-gendi’s method revisited
Abstract We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi’s method). The solutions are exact at these points, apart from round-off computer errors and the convergen...
متن کاملA Hybrid Fourier-Chebyshev Method for Partial Differential Equations
We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use...
متن کاملTau method approximation of a generalized Epstein-Hubbell elliptic-type integral
We consider the evaluation of a recent generalization of the Epstein-Hubbell elliptic-type integral using the tau method approximation with a Chebyshev polynomial basis. This also leads to an approximation of Lauricella’s hypergeometric function of three variables. Numerical results are given for polynomial approximations of degree 6.
متن کاملAccurate solution of the Orr-Sommerfeld stability equation
The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshevpolynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772.22. It is explained why expansions in Chebyshev polynomials are better s...
متن کامل