New explicit and accelerated techniques for solving fractional order differential equations

Hyunju Kim, Keon Ho Kim, Seyeon Lee and Bongsoo Jang

Applied Mathematics and Computation, 2020, vol. 379, issue C

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 < ν < 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 ν for linear and quadratic interpolations, which are O(h2−ν) and O(h3−ν) respectively. In order to overcome a strong singular issue as ν ≈ 0 we propose the explicit predictor-corrector scheme with perturbation technique. In case of ν ≈ 1, the truncation errors are reduced by O(h) and O(h2). In order to accelerate the convergence rate, we decompose ν=ν/2+ν/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.

Keywords: Caputo fractional derivative; Fractional differential equations; Predictor-corrector methods; Explicit scheme; High-order method (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:379:y:2020:i:c:s0096300320301971

DOI: 10.1016/j.amc.2020.125228

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