An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Neville J. Ford; Kamal Pal; Yubin Yan

doi:10.1515/cmam-2015-0022

Published by De Gruyter August 20, 2015

An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Neville J. Ford , Kamal Pal and Yubin Yan

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2015-0022

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Abstract

We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order O(Δx3-α), 1<α<2. A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O(Δt+Δxmin(3-α,β)), 1<α<2, β>0, where Δt,Δx denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

Keywords: Finite Difference Methods; Error Estimates; Riemann–Liouville Derivative; Space-Fractional Partial Differential Equations

MSC: 65M12; 65M06; 65M70; 35S10

Received: 2015-4-2

Revised: 2015-6-18

Accepted: 2015-8-9

Published Online: 2015-8-20

Published in Print: 2015-10-1

An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Abstract

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