{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T18:21:47Z","timestamp":1778696507997,"version":"3.51.4"},"reference-count":39,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we propose an improvement of the classical compact finite difference (CFD) method by using a proper\northogonal decomposition (POD) technique for time-fractional diffusion equations\nin one- and two-dimensional space.\nA reduced CFD method is constructed with lower dimensions such that it maintains the accuracy and\ndecreases the computational time in comparison with classical CFD method.\nSince the solution of time-fractional diffusion equation typically has a weak singularity near the\ninitial time <jats:inline-formula id=\"j_cmam-2020-0158_ineq_9999\">\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-2020-0158_eq_0370.png\"\/>\n                        <jats:tex-math>{t=0}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, the classical L1 scheme on uniform meshes\nmay obtain a scheme with low accuracy. So, we use the L1 scheme on graded meshes for time discretization.\nMoreover, we provide the error estimation between the reduced CFD method\nbased on POD and classical CFD solutions. Some numerical examples show the effectiveness and accuracy of the proposed method.<\/jats:p>","DOI":"10.1515\/cmam-2020-0158","type":"journal-article","created":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T21:54:05Z","timestamp":1613858045000},"page":"827-840","source":"Crossref","is-referenced-by-count":2,"title":["A Low-Dimensional Compact Finite Difference Method on Graded Meshes for Time-Fractional Diffusion Equations"],"prefix":"10.1515","volume":"21","author":[{"given":"Rezvan","family":"Ghaffari","sequence":"first","affiliation":[{"name":"Faculty of Mathematics , K.\u2009N. Toosi University of Technology , Tehran , Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0437-2616","authenticated-orcid":false,"given":"Farideh","family":"Ghoreishi","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics , K.\u2009N. Toosi University of Technology , Tehran , Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,2,20]]},"reference":[{"key":"2023033111295733869_j_cmam-2020-0158_ref_001","doi-asserted-by":"crossref","unstructured":"M.  Abbaszadeh and A.  Mohebbi,\nA fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term,\nComput. Math. Appl. 66 (2013), no. 8, 1345\u20131359.","DOI":"10.1016\/j.camwa.2013.08.010"},{"key":"2023033111295733869_j_cmam-2020-0158_ref_002","doi-asserted-by":"crossref","unstructured":"J.  An, Z.  Luo, H.  Li and P.  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