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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 15, Issue 4

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

Neville J. Ford
/ Kamal Pal
/ Yubin Yan
Published Online: 2015-08-20 | DOI: https://doi.org/10.1515/cmam-2015-0022

## 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 $O\left(\Delta {x}^{3-\alpha }\right)$, $1<\alpha <2$. 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 $O\left(\Delta t+\Delta {x}^{min\left(3-\alpha ,\beta \right)}\right)$, $1<\alpha <2$, $\beta >0$, where $\Delta t,\Delta x$ 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.

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

Revised: 2015-06-18

Accepted: 2015-08-09

Published Online: 2015-08-20

Published in Print: 2015-10-01

Citation Information: Computational Methods in Applied Mathematics, Volume 15, Issue 4, Pages 497–514, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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