Higher order numerical methods for solving fractional differential equations

Hdl Handle:
http://hdl.handle.net/10034/580035
Title:
Higher order numerical methods for solving fractional differential equations
Authors:
Yan, Yubin; Pal, Kamal; Ford, Neville J.
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.
Affiliation:
University of Chester
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
Publication Date:
5-Oct-2013
URI:
http://hdl.handle.net/10034/580035
DOI:
10.1007/s10543-013-0443-3
Additional Links:
http://link.springer.com/10.1007/s10543-013-0443-3
Type:
Article
Language:
en
Description:
The final publication is available at Springer via http://dx.doi.org/10.1007/s10543-013-0443-3
ISSN:
1572-9125
Appears in Collections:
Mathematics

Full metadata record

DC FieldValue Language
dc.contributor.authorYan, Yubinen
dc.contributor.authorPal, Kamalen
dc.contributor.authorFord, Neville J.en
dc.date.accessioned2015-10-21T18:08:56Zen
dc.date.available2015-10-21T18:08:56Zen
dc.date.issued2013-10-05en
dc.identifier.citationYan, 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/s10543-013-0443-3en
dc.identifier.issn1572-9125en
dc.identifier.doi10.1007/s10543-013-0443-3en
dc.identifier.urihttp://hdl.handle.net/10034/580035en
dc.descriptionThe final publication is available at Springer via http://dx.doi.org/10.1007/s10543-013-0443-3en
dc.description.abstractIn 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.en
dc.language.isoenen
dc.publisherSpringeren
dc.relation.urlhttp://link.springer.com/10.1007/s10543-013-0443-3en
dc.rightsArchived with thanks to BIT Numerical Mathematicsen
dc.rightsAn error occurred on the license name.*
dc.rights.uriAn error occurred getting the license - uri.*
dc.subjectFractional differential equationen
dc.subjectFinite difference methoden
dc.subjectCaputo derivativeen
dc.subjectError estimatesen
dc.titleHigher order numerical methods for solving fractional differential equationsen
dc.typeArticleen
dc.contributor.departmentUniversity of Chesteren
dc.identifier.journalBIT Numerical Mathematicsen
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