Global Spectral Collocation Method with Fourier Transform to Solve Differential Equations

Abstract

Numerical analysis is the area of mathematics that creates, analyzes, and implements algorithms for solving numerically the problems from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. Till now, numerous numerical methods have been introduced. Spectral method is one of those techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform (FFT). This study presents some of the fundamental ideas of spectral method. Orthogonal basis are used to establish computational algorithm. The accuracy and efficiency of proposed model are discussed. This manuscript estimates for the error between the exact and approximated discrete solutions. This paper shows that, grid points for polynomial spectral methods should lie approximately in a minimal energy configuration associated with inverse linear repulsion between points. The wave equation, linear and non-linear boundary value problems are solved using spectral method on the platform of MATLAB language.

GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 28-42