An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions

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dc.contributor.author Chen, Hu
dc.contributor.author Holland, Finbarr
dc.contributor.author Stynes, Martin
dc.date.accessioned 2019-02-12T10:25:16Z
dc.date.available 2019-02-12T10:25:16Z
dc.date.issued 2019-01-15
dc.identifier.citation Chen, H., Holland, F. and Stynes, M. (2019) 'An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions', Applied Numerical Mathematics, 139, pp. 52-61. doi:10.1016/j.apnum.2019.01.004 en
dc.identifier.volume 139 en
dc.identifier.startpage 52 en
dc.identifier.endpage 61 en
dc.identifier.issn 0168-9274
dc.identifier.issn 1873-5460
dc.identifier.uri http://hdl.handle.net/10468/7479
dc.identifier.doi 10.1016/j.apnum.2019.01.004
dc.description.abstract A convergence analysis is given for the Grünwald–Letnikov discretisation of a Riemann–Liouville fractional initial-value problem on a uniform mesh tm=mτ with m=0,1,…,M. For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t=0. Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the Grünwald–Letnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the Grünwald–Letnikov scheme is O(τtmα−1) at each mesh point tm; hence the scheme is globally only O(τα) accurate, but it is O(τ) accurate for mesh points tm that are bounded away from t=0. Numerical results for a test example show that these theoretical results are sharp. en
dc.description.sponsorship China Postdoctoral Science Foundation (Grant 2018M631316); National Natural Science Foundation of China (Grants 11801026; 91430216; NSAF-U1530401)
dc.format.mimetype application/pdf en
dc.language.iso en en
dc.publisher Elsevier B.V. en
dc.rights © 2019, IMACS. Published by Elsevier B.V. All rights reserved. This manuscript version is made available under the CC BY-NC-ND 4.0 license. en
dc.rights.uri https://creativecommons.org/licenses/by-nc-nd/4.0/ en
dc.subject Weak singularity en
dc.subject Convergence analysis en
dc.subject Riemann–Liouville derivative en
dc.subject Grünwald–Letnikov scheme en
dc.title An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions en
dc.type Article (peer-reviewed) en
dc.internal.authorcontactother Finbarr Holland, Pensioners, University College Cork, Cork, Ireland. +353-21-490-3000 Email: f.holland@ucc.ie en
dc.internal.availability Full text available en
dc.check.info Access to this article is restricted until 24 months after publication by request of the publisher. en
dc.check.date 2021-01-15
dc.date.updated 2019-02-12T10:06:07Z
dc.description.version Accepted Version en
dc.internal.rssid 473313212
dc.contributor.funder China Postdoctoral Science Foundation
dc.contributor.funder National Natural Science Foundation of China
dc.description.status Peer reviewed en
dc.identifier.journaltitle Applied Numerical Mathematics en
dc.internal.copyrightchecked Yes en
dc.internal.licenseacceptance Yes en
dc.internal.IRISemailaddress f.holland@ucc.ie en


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© 2019, IMACS. Published by Elsevier B.V. All rights reserved. This manuscript version is made available under the CC BY-NC-ND 4.0 license. Except where otherwise noted, this item's license is described as © 2019, IMACS. Published by Elsevier B.V. All rights reserved. This manuscript version is made available under the CC BY-NC-ND 4.0 license.
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