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Higher order numerical methods for solving fractional differential equations

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Abstract

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0<α<1. The order of convergence of the numerical method is O(h 3−α). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α>0. The order of convergence of the numerical method is O(h 3) for α≥1 and O(h 1+2α) for 0<α≤1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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Acknowledgements

The authors wish to thank Kai Diethelm for reading the first version of this paper and making useful suggestions. The authors also wish to thank the anonymous reviewers of this paper for their careful reading and useful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Chester, Chester, CH1 4BJ, UK

    Yubin Yan, Kamal Pal & Neville J. Ford

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  1. Yubin Yan
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  2. Kamal Pal
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  3. Neville J. Ford
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Correspondence to Yubin Yan.

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Communicated by Jan Hesthaven.

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Yan, Y., Pal, K. & Ford, N.J. Higher order numerical methods for solving fractional differential equations. Bit Numer Math 54, 555–584 (2014). https://doi.org/10.1007/s10543-013-0443-3

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  • DOI: https://doi.org/10.1007/s10543-013-0443-3

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