{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T11:21:01Z","timestamp":1773487261642,"version":"3.50.1"},"reference-count":11,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801332"],"award-info":[{"award-number":["11801332"]}],"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":["11971259"],"award-info":[{"award-number":["11971259"]}],"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-U1930402"],"award-info":[{"award-number":["NSAF-U1930402"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>An initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered.\nIts spatial domain is <jats:inline-formula id=\"j_cmam-2019-0042_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mn>0<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mi>d<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0042_eq_0101.png\"\/>\n                        <jats:tex-math>{(0,1)^{d}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> for some <jats:inline-formula id=\"j_cmam-2019-0042_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>d<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">{<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>2<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>3<\/m:mn>\n                                 <m:mo stretchy=\"false\">}<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0042_eq_0221.png\"\/>\n                        <jats:tex-math>{d\\in\\{1,2,3\\}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nThis problem is a generalisation of the problem considered by Stynes, O\u2019Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp.\u20091057\u20131079, where <jats:inline-formula id=\"j_cmam-2019-0042_ineq_9997\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>d<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mn>1<\/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-2019-0042_eq_0219.png\"\/>\n                        <jats:tex-math>{d=1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and only one fractional time derivative was present.\nA priori bounds on the derivatives of the unknown solution are derived.\nA finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time.\nStability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way.\nNumerical results supporting our theoretical results are provided.<\/jats:p>","DOI":"10.1515\/cmam-2019-0042","type":"journal-article","created":{"date-parts":[[2020,1,15]],"date-time":"2020-01-15T09:03:09Z","timestamp":1579078989000},"page":"815-825","source":"Crossref","is-referenced-by-count":49,"title":["Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3554-7885","authenticated-orcid":false,"given":"Chaobao","family":"Huang","sequence":"first","affiliation":[{"name":"School of Mathematics and Quantitative Economics , Shandong University of Finance and Economics , Jinan 250014 , P. R. China"}]},{"given":"Xiaohui","family":"Liu","sequence":"additional","affiliation":[{"name":"Applied and Computational Mathematics Division , Beijing Computational Science Research Center , Beijing 100193 , P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5327-6068","authenticated-orcid":false,"given":"Xiangyun","family":"Meng","sequence":"additional","affiliation":[{"name":"School of Science , Beijing Jiaotong University , Beijing 100044 , P. R. China"}]},{"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 100193 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2020,1,15]]},"reference":[{"key":"2023033110450894424_j_cmam-2019-0042_ref_001","unstructured":"R.  Courant and D.  Hilbert,\nMethods of Mathematical Physics. Vol. II: Partial Differential Equations,\nInterscience, New York, 1962."},{"key":"2023033110450894424_j_cmam-2019-0042_ref_002","doi-asserted-by":"crossref","unstructured":"D.  Henry,\nGeometric Theory of Semilinear Parabolic Equations,\nLecture Notes in Math. 840,\nSpringer, Berlin, 1981.","DOI":"10.1007\/BFb0089647"},{"key":"2023033110450894424_j_cmam-2019-0042_ref_003","doi-asserted-by":"crossref","unstructured":"C.  Huang and M.  Stynes,\nSuperconvergence of a finite element method for the multi-term time-fractional diffusion problem,\npreprint (2019), https:\/\/www.researchgate.net\/publication\/336104219.","DOI":"10.1007\/s10915-019-01115-w"},{"key":"2023033110450894424_j_cmam-2019-0042_ref_004","doi-asserted-by":"crossref","unstructured":"B.  Jin, R.  Lazarov, Y.  Liu and Z.  Zhou,\nThe Galerkin finite element method for a multi-term time-fractional diffusion equation,\nJ. Comput. Phys. 281 (2015), 825\u2013843.","DOI":"10.1016\/j.jcp.2014.10.051"},{"key":"2023033110450894424_j_cmam-2019-0042_ref_005","doi-asserted-by":"crossref","unstructured":"N.  Kopteva,\nError analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions,\nMath. Comp. 88 (2019), no. 319, 2135\u20132155.","DOI":"10.1090\/mcom\/3410"},{"key":"2023033110450894424_j_cmam-2019-0042_ref_006","doi-asserted-by":"crossref","unstructured":"Z.  Li, Y.  Liu and M.  Yamamoto,\nInitial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients,\nAppl. Math. Comput. 257 (2015), 381\u2013397.","DOI":"10.1016\/j.amc.2014.11.073"},{"key":"2023033110450894424_j_cmam-2019-0042_ref_007","unstructured":"X.  Liu and M.  Stynes,\nError analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem,\npreprint (2019), https:\/\/www.researchgate.net\/publication\/331465694."},{"key":"2023033110450894424_j_cmam-2019-0042_ref_008","doi-asserted-by":"crossref","unstructured":"Y.  Luchko,\nInitial-boundary problems for the generalized multi-term time-fractional diffusion equation,\nJ. Math. Anal. Appl. 374 (2011), no. 2, 538\u2013548.","DOI":"10.1016\/j.jmaa.2010.08.048"},{"key":"2023033110450894424_j_cmam-2019-0042_ref_009","doi-asserted-by":"crossref","unstructured":"Y.  Luchko,\nInitial-boundary-value problems for the one-dimensional time-fractional diffusion equation,\nFract. Calc. Appl. Anal. 15 (2012), no. 1, 141\u2013160.","DOI":"10.2478\/s13540-012-0010-7"},{"key":"2023033110450894424_j_cmam-2019-0042_ref_010","unstructured":"I.  Podlubny,\nFractional Differential Equations,\nMath. Sci. Eng. 198,\nAcademic Press, San Diego, 1999."},{"key":"2023033110450894424_j_cmam-2019-0042_ref_011","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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