New explicit and accelerated techniques for solving fractional order differential equations
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- New explicit and accelerated techniques for solving fractional order differential equations
- Kim, Hyunju; Kim, Keon Ho; Lee, Seyeon; Jang, Bongsoo
- Issue Date
- Elsevier BV
- APPLIED MATHEMATICS AND COMPUTATION , v.379, no.125228, pp.125228
- 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 and 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 and . In order to accelerate the convergence rate, we decompose 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.
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