High-order numerical schemes based on B-spline for solving a time-fractional Fokker–Planck equation

Abstract

The authors of Jiang (2014), Vong and Wang (2014) and Roul et al. (2022) proposed lower-orders computational techniques for solving a time-fractional Fokker–Planck (TFFP) equation. This paper deals with the design of two high-order computational schemes for the TFFP equation. The first scheme is based on a combination of $L2-1_{\sigma }$ scheme and standard quintic B-spline collocation method, while the second one is based on a combination of $L2-1_{\sigma }$ scheme and a new technique, namely improvised quintic B-spline collocation method. Convergence of the suggested method is analyzed. An illustrative example is provided to demonstrate the applicability and efficiency of the proposed method. The convergence orders of first and second methods are $O(\Delta t^2+\Delta x^4)$, $O(\Delta t^2+\Delta x^6),$ respectively, where $\Delta t$ and $\Delta x$ are the step-sizes in time and space domain, respectively. We compare the computed results with those obtained by the finite difference method (FDM), compact FDM and quartic B-spline collocation method to justify the advantage of proposed schemes.