New explicit and accelerated techniques for solving fractional order differential equations

Abstract

This paper proposes new direct and acceleration numerical methods for solving Fractional order Differential Equations (FDEs). For the Caputo differential operator with fractional order 0 < nu < 1, we rewrite the FDE as an equivalent integral form circumventing the derivative of the solution by the integral by parts. We obtain a discrete formulation directly from the integral form using the Lagrange interpolate polynomials. We provide truncation errors depending on the fractional order v for linear and quadratic interpolations, which are O(h(2-nu)) and O(h(3-nu)) respectively. In order to overcome a strong singular issue as nu approximate to 0 we propose the explicit predictor-corrector scheme with perturbation technique. In case of nu approximate to 1, the truncation errors are reduced by O(h) and O(h(2)). In order to accelerate the con- vergence rate, we decompose nu = nu/2 + nu/2, convert the FDE into the system of FDEs, and apply the predictor-corrector scheme. Numerical tests for linear, nonlinear, variable order and time-fractional sub-diffusion problems demonstrate that the proposed methods give a prominent performance. We also compare the numerical results with other high-order explicit and implicit schema. (C) 2020 Elsevier Inc. All rights reserved.