{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T12:43:31Z","timestamp":1774529011277,"version":"3.50.1"},"reference-count":28,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["22-21-00363"],"award-info":[{"award-number":["22-21-00363"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171177"],"award-info":[{"award-number":["12171177"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In the current work, we build a difference analog of the Caputo fractional derivative with generalized memory kernel (<jats:sub>\ud835\udf07<\/jats:sub>L2-1<jats:sub>\ud835\udf0e<\/jats:sub> formula).\nThe fundamental features of this difference operator are studied, and on its ground, some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out.\nWe have proved stability and convergence of the given schemes in the grid <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0089_ineq_0001.png\"\/>\n                        <jats:tex-math>L_{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norm with the rate equal to the order of the approximation error.\nThe achieved results are supported by the numerical computations performed for some test problems.<\/jats:p>","DOI":"10.1515\/cmam-2022-0089","type":"journal-article","created":{"date-parts":[[2023,5,22]],"date-time":"2023-05-22T17:10:09Z","timestamp":1684775409000},"page":"101-117","source":"Crossref","is-referenced-by-count":8,"title":["A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5727-7540","authenticated-orcid":false,"given":"Aslanbek","family":"Khibiev","sequence":"first","affiliation":[{"name":"North-Caucasus Center for Mathematical Research , North-Caucasus Federal University , Stavropol , Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0684-6667","authenticated-orcid":false,"given":"Anatoly","family":"Alikhanov","sequence":"additional","affiliation":[{"name":"North-Caucasus Center for Mathematical Research , North-Caucasus Federal University , Stavropol , Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chengming","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , Huazhong University of Science and Technology , Wuhan , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,5,23]]},"reference":[{"key":"2024010712054004454_j_cmam-2022-0089_ref_001","doi-asserted-by":"crossref","unstructured":"O. P. Agrawal,\nSome generalized fractional calculus operators and their applications in integral equations,\nFract. Calc. Appl. Anal. 15 (2012), no. 4, 700\u2013711.","DOI":"10.2478\/s13540-012-0047-7"},{"key":"2024010712054004454_j_cmam-2022-0089_ref_002","doi-asserted-by":"crossref","unstructured":"A. A. Alikhanov,\nA priori estimates for solutions of boundary value problems for equations of fractional order,\nDiffer. Equ. 46 (2010), 660\u2013666.","DOI":"10.1134\/S0012266110050058"},{"key":"2024010712054004454_j_cmam-2022-0089_ref_003","doi-asserted-by":"crossref","unstructured":"A. A. Alikhanov,\nA new difference scheme for the time fractional diffusion equation,\nJ. Comput. 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