Chebyshev expansion method for solving second and fourth-order elliptic equations

Abstract

In this paper we introduce a new spectral method based on Chebyshev polynomials for solving second and fourth-order elliptic equations. Moreover the suggested method is applicable for a wide area of differential equations. An explicit formula for the Chebyshev polynomials in terms of arbitrary order of their derivatives is presented. Also a formula for the successive integration of Chebyshev polynomials in terms of Chebyshev polynomials is proved. Numerical results indicate that the suggested method is significantly more accurate than that based on the Chebyshev–Tau method. The present results are in satisfactory agreement with the exact solutions.