{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T05:58:51Z","timestamp":1778392731636,"version":"3.51.4"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, an a priori estimate for the corresponding\ndifferential problem is obtained by using the method of the energy\ninequalities. We construct a difference analog of the Caputo\nfractional derivative with generalized memory kernel (L1 formula).\nThe basic properties of this difference operator are investigated\nand on its basis some difference schemes generating approximations\nof the second and fourth order in space and the <jats:inline-formula id=\"j_cmam-2017-0035_ineq_9999_w2aab3b7b7b1b6b1aab1c16b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                 <m:mn>2<\/m:mn>\n                                 <m:mo>-<\/m:mo>\n                                 <m:mi>\u03b1<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0035_eq_mi232.png\"\/>\n                        <jats:tex-math>{(2-\\alpha)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-th\norder in time for the generalized time-fractional diffusion equation\nwith variable coefficients are considered. Stability of the\nsuggested schemes and also their convergence in the grid <jats:inline-formula id=\"j_cmam-2017-0035_ineq_9998_w2aab3b7b7b1b6b1aab1c16b1b3Aa\">\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-2017-0035_eq_mi326.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 are\nproved. The obtained results are supported by numerical\ncalculations carried out for some test problems.<\/jats:p>","DOI":"10.1515\/cmam-2017-0035","type":"journal-article","created":{"date-parts":[[2017,9,9]],"date-time":"2017-09-09T10:01:19Z","timestamp":1504951279000},"page":"647-660","source":"Crossref","is-referenced-by-count":20,"title":["A Time-Fractional Diffusion Equation with Generalized Memory Kernel in Differential and Difference Settings with Smooth Solutions"],"prefix":"10.1515","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0684-6667","authenticated-orcid":false,"given":"Anatoly A.","family":"Alikhanov","sequence":"first","affiliation":[{"name":"Institute of Applied Mathematics and Automation KBSC RAS , ul. Shortanova 89 A , Nalchik , 360000 , Russia"}]}],"member":"374","published-online":{"date-parts":[[2017,9,9]]},"reference":[{"key":"2023033116270840103_j_cmam-2017-0035_ref_001_w2aab3b7b7b1b6b1ab1b7b1Aa","doi-asserted-by":"crossref","unstructured":"G.-H.  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