APPLIED MATHEMATICS AND COMPUTATION , v.379, pp.125228
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.