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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

IMPACT FACTOR 2018: 1.218
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CiteScore 2018: 1.42

SCImago Journal Rank (SJR) 2018: 0.947
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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


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

About the article

Received: 2015-04-02

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

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