Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations

Abstract

In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). Besides, the proposed GFPDEs represent a great generalization of a significant type of FPDEs and their applications, which contain many previous reports as a special case. Moreover, the proposed scheme uses shifted Chebyshev sixth-kind (SCSK) polynomials with spectral collocation approach. The fractional differential derivatives are expressed in terms of the Caputo's definition. Furthermore, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. In addition, the classical fourth-order Runge-Kotta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple and effective numerical method.

Keywords

• collocation method
• chebyshev sixth-kind
• generalized space fractional partial differential equations
• finite difference method
• caputo's fractional derivatives