{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T15:35:29Z","timestamp":1775835329793,"version":"3.50.1"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>s<\/m:mi>\n                              <m:mo>+<\/m:mo>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0231_eq_0203.png\"\/>\n                        <jats:tex-math>{s^{+}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is better than that at <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mn>0<\/m:mn>\n                              <m:mo>+<\/m:mo>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0231_eq_0107.png\"\/>\n                        <jats:tex-math>{0^{+}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, where <jats:italic>s<\/jats:italic> is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9997\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>L<\/m:mi>\n                              <m:mo>\u2062<\/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-2022-0231_eq_0133.png\"\/>\n                        <jats:tex-math>{L1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9996\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03b1<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">[<\/m:mo>\n                                 <m:mfrac>\n                                    <m:mn>2<\/m:mn>\n                                    <m:mn>3<\/m:mn>\n                                 <\/m:mfrac>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>1<\/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-2022-0231_eq_0151.png\"\/>\n                        <jats:tex-math>{\\alpha\\in[\\frac{2}{3},1)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, \u03b1 is the order of fractional derivative. Furthermore, an improved fitted <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9995\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>L<\/m:mi>\n                              <m:mo>\u2062<\/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-2022-0231_eq_0133.png\"\/>\n                        <jats:tex-math>{L1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> method is proposed and the region of optimal convergence order is larger. For the case <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9994\">\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>&gt;<\/m:mo>\n                              <m:mi>s<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0231_eq_0206.png\"\/>\n                        <jats:tex-math>{t&gt;s}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, stability and <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9993\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>min<\/m:mi>\n                              <m:mo>\u2061<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">{<\/m:mo>\n                                 <m:mrow>\n                                    <m:mn>2<\/m:mn>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>\u03b1<\/m:mi>\n                                 <\/m:mrow>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>1<\/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-2022-0231_eq_0166.png\"\/>\n                        <jats:tex-math>{\\min\\{2\\alpha,1\\}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> order convergence of the fitted <jats:inline-formula id=\"j_cmam-2022-0231_ineq_9992\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>L<\/m:mi>\n                              <m:mo>\u2062<\/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-2022-0231_eq_0133.png\"\/>\n                        <jats:tex-math>{L1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.<\/jats:p>","DOI":"10.1515\/cmam-2022-0231","type":"journal-article","created":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T13:50:51Z","timestamp":1677505851000},"page":"591-601","source":"Crossref","is-referenced-by-count":15,"title":["The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted <i>L<\/i>1 Method"],"prefix":"10.1515","volume":"23","author":[{"given":"Dakang","family":"Cen","sequence":"first","affiliation":[{"name":"Department of Mathematics , University of Macau , Macao , P. R. China"}]},{"given":"Seakweng","family":"Vong","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Macau , Macao , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2023,2,28]]},"reference":[{"key":"2023070413382846042_j_cmam-2022-0231_ref_001","doi-asserted-by":"crossref","unstructured":"A.  Bellen and M.  Zennaro,\nNumerical Methods for Delay Differential Equations,\nNumer. Math. Sci. Comput.,\nThe Clarendon, Oxford, 2003.","DOI":"10.1093\/acprof:oso\/9780198506546.001.0001"},{"key":"2023070413382846042_j_cmam-2022-0231_ref_002","doi-asserted-by":"crossref","unstructured":"R.  Bellman and K. L.  Cooke,\nDifferential-Difference Equations,\nAcademic Press, New York, 1963.","DOI":"10.1063\/1.3050672"},{"key":"2023070413382846042_j_cmam-2022-0231_ref_003","doi-asserted-by":"crossref","unstructured":"J. L.  Gracia, E.  O\u2019Riordan and M.  Stynes,\nA fitted scheme for a Caputo initial-boundary value problem,\nJ. Sci. Comput. 76 (2018), no. 1, 583\u2013609.","DOI":"10.1007\/s10915-017-0631-4"},{"key":"2023070413382846042_j_cmam-2022-0231_ref_004","doi-asserted-by":"crossref","unstructured":"J. L.  Gracia, E.  O\u2019Riordan and M.  Stynes,\nConvergence in positive time for a finite difference method applied to a fractional convection-diffusion problem,\nComput. Methods Appl. Math. 18 (2018), no. 1, 33\u201342.","DOI":"10.1515\/cmam-2017-0019"},{"key":"2023070413382846042_j_cmam-2022-0231_ref_005","doi-asserted-by":"crossref","unstructured":"H.-L.  Liao, W.  McLean and J.  Zhang,\nA discrete Gr\u00f6nwall inequality with applications to numerical schemes for subdiffusion problems,\nSIAM J. Numer. Anal. 57 (2019), no. 1, 218\u2013237.","DOI":"10.1137\/16M1175742"},{"key":"2023070413382846042_j_cmam-2022-0231_ref_006","doi-asserted-by":"crossref","unstructured":"M. L.  Morgado, N. J.  Ford and P. M.  Lima,\nAnalysis and numerical methods for fractional differential equations with delay,\nJ. Comput. Appl. Math. 252 (2013), 159\u2013168.","DOI":"10.1016\/j.cam.2012.06.034"},{"key":"2023070413382846042_j_cmam-2022-0231_ref_007","unstructured":"I.  Podlubny,\nFractional Differential Equations,\nMath. Sci. Eng. 198,\nAcademic Press, San Diego, 1999."},{"key":"2023070413382846042_j_cmam-2022-0231_ref_008","doi-asserted-by":"crossref","unstructured":"M.  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Comput. 92 (2022), no. 3, Paper No. 98.","DOI":"10.1007\/s10915-022-01948-y"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0231\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0231\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,4]],"date-time":"2023-07-04T14:17:59Z","timestamp":1688480279000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0231\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,28]]},"references-count":10,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,12,7]]},"published-print":{"date-parts":[[2023,7,1]]}},"alternative-id":["10.1515\/cmam-2022-0231"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0231","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,28]]}}}