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Higher order numerical methods for solving fractional differential equations

Yan, Yubin
; 
Pal, Kamal
; 
Ford, Neville J.
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University of Chester
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2013-10-05
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Mathematics
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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.
Citation
Yan, Y., Pal, K., & Ford, N. J. (2014). Higher order numerical methods for solving fractional differential equations. BIT Numerical Mathematics, 54(2), 555-584. doi:10.1007/s10543-013-0443-3
Publisher
Springer
Journal
BIT Numerical Mathematics
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http://hdl.handle.net/10034/580035
DOI
10.1007/s10543-013-0443-3
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Article
Language
en
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The final publication is available at Springer via http://dx.doi.org/10.1007/s10543-013-0443-3
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1572-9125
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http://link.springer.com/10.1007/s10543-013-0443-3