Chebyshev Delta Shaped and Chebyshev Pseudo-Spectral Methods for Solutions of Differential Equations

Abstract

In this paper we introduce a new Chebyshev delta-shaped function (CDSF) and establish its relationship with Chebyshev polynomials in interpolation problems. We first prove that CDSF is indeed form a basis for a Haar space. We then derive the conditions for the selection of suitable collocation points. Next, we introduce and develop Chebyshev delta-shaped pseudo-spectral method. Error bounds on discrete L2-norm and Sobolev norm (Hp) are presented for the Chebyshev pseudo-spectral method. Tests to find approximate solutions for the Poisson, Poisson-Boltzmann equations and Stokes second problem and comparisons of the predictions using the following methods are presented: 1. Chebyshev pseudo-spectral method, 2. Cosine-sine delta-shaped pseudo-spectral method, and 3. Cosine-sine pseudo-spectral method. Excellent convergent and stable results are obtained by using our newly defined Chebyshev delta-shaped basis functions and this is documented for the first time.