{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T01:29:16Z","timestamp":1777512556993,"version":"3.51.4"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91430216"],"award-info":[{"award-number":["91430216"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["NSAF U1530401"],"award-info":[{"award-number":["NSAF U1530401"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["MTM2016-75139-R"],"award-info":[{"award-number":["MTM2016-75139-R"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A standard finite difference method on a uniform mesh is used to solve a time-fractional convection-diffusion initial-boundary value problem. Such problems typically exhibit a mild singularity at the initial time <jats:inline-formula id=\"j_cmam-2017-0019_ineq_9999_w2aab3b7d277b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>t<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0019_eq_mi249.png\"\/>\n                        <jats:tex-math>{t=0}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. It is proved that the rate of convergence of the maximum nodal error on any subdomain that is bounded away from <jats:inline-formula id=\"j_cmam-2017-0019_ineq_9998_w2aab3b7d277b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>t<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0019_eq_mi249.png\"\/>\n                        <jats:tex-math>{t=0}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is higher than the rate obtained when the maximum nodal error is measured over the entire space-time domain. Numerical results are provided to illustrate the theoretical error bounds.<\/jats:p>","DOI":"10.1515\/cmam-2017-0019","type":"journal-article","created":{"date-parts":[[2017,7,6]],"date-time":"2017-07-06T10:01:41Z","timestamp":1499335301000},"page":"33-42","source":"Crossref","is-referenced-by-count":60,"title":["Convergence in Positive Time for a Finite Difference Method Applied to a Fractional Convection-Diffusion Problem"],"prefix":"10.1515","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2538-9027","authenticated-orcid":false,"given":"Jos\u00e9 Luis","family":"Gracia","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics , University of Zaragoza , 50018 Zaragoza , Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eugene","family":"O\u2019Riordan","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , Dublin City University , Glasnevin , Dublin 9 , Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2085-7354","authenticated-orcid":false,"given":"Martin","family":"Stynes","sequence":"additional","affiliation":[{"name":"Applied and Computational Mathematics Division , Beijing Computational Science Research Center , Beijing , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,7,6]]},"reference":[{"key":"2023033115122839896_j_cmam-2017-0019_ref_001_w2aab3b7d277b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"K.  Diethelm,\nThe Analysis of Fractional Differential Equations,\nLecture Notes in Math. 2004,\nSpringer, Berlin, 2010.","DOI":"10.1007\/978-3-642-14574-2"},{"key":"2023033115122839896_j_cmam-2017-0019_ref_002_w2aab3b7d277b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"P. A.  Farrell, A. F.  Hegarty, J. J. H.  Miller, E.  O\u2019Riordan and G. I.  Shishkin,\nRobust Computational Techniques for Boundary Layers,\nAppl. Math. Math. Comput. 16,\nChapman & Hall\/CRC, Boca Raton, 2000.","DOI":"10.1201\/9781482285727"},{"key":"2023033115122839896_j_cmam-2017-0019_ref_003_w2aab3b7d277b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"J. L.  Gracia, E.  O\u2019Riordan and M.  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McLean,\nPiecewise-linear, discontinuous Galerkin method for a fractional diffusion equation,\nNumer. Algorithms 56 (2011), no. 2, 159\u2013184.","DOI":"10.1007\/s11075-010-9379-8"},{"key":"2023033115122839896_j_cmam-2017-0019_ref_009_w2aab3b7d277b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"M.  Stynes,\nToo much regularity may force too much uniqueness,\nFract. Calc. Appl. Anal. 19 (2016), no. 6, 1554\u20131562.","DOI":"10.1515\/fca-2016-0080"},{"key":"2023033115122839896_j_cmam-2017-0019_ref_010_w2aab3b7d277b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"M.  Stynes, E.  O\u2019Riordan and J. L.  Gracia,\nError analysis of a finite difference method on graded meshes for a time-fractional diffusion equation,\nSIAM J. Numer. 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