{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T19:39:31Z","timestamp":1740166771946,"version":"3.37.3"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"crossref"}]},{"name":"NSFC of China","award":["11371263"],"award-info":[{"award-number":["11371263"]}]},{"name":"Program for New Century Excellent Talents in University of China"},{"name":"Russian Foundation for Basic Research","award":["13-01-00096-a, 15-01-00026-a"],"award-info":[{"award-number":["13-01-00096-a, 15-01-00026-a"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The semidiscretization methods for solving the Cauchy problem\n<jats:disp-formula id=\"eq1_w2aab2b8b7b1b7b1aab1c13b1b1Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0001_5c4f1c23616c2f5932bdc1ad4ca8b97c.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:msubsup>\n                                    <m:mi>\ud835\udc03<\/m:mi>\n                                    <m:mrow>\n                                       <m:mi>t<\/m:mi>\n                                    <\/m:mrow>\n                                    <m:mi>\u03b1<\/m:mi>\n                                 <\/m:msubsup>\n                                 <m:mi>u<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>A<\/m:mi>\n                              <m:mi>u<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>+<\/m:mo>\n                              <m:msup>\n                                 <m:mi>J<\/m:mi>\n                                 <m:mrow>\n                                    <m:mn>1<\/m:mn>\n                                    <m:mo>-<\/m:mo>\n                                    <m:mi>\u03b1<\/m:mi>\n                                 <\/m:mrow>\n                              <\/m:msup>\n                              <m:mi>f<\/m:mi>\n                              <m:mfenced separators=\"\" open=\"(\" close=\")\">\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mi>u<\/m:mi>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mfenced>\n                              <m:mo>,<\/m:mo>\n                              <m:mspace width=\"1.em\"\/>\n                              <m:mi>t<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>[<\/m:mo>\n                                 <m:mn>0<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mi>T<\/m:mi>\n                                 <m:mo>]<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>,<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mi>\u03b1<\/m:mi>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>,<\/m:mo>\n                              <m:mspace width=\"2.em\"\/>\n                              <m:mi>u<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mn>0<\/m:mn>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:msup>\n                                 <m:mi>u<\/m:mi>\n                                 <m:mn>0<\/m:mn>\n                              <\/m:msup>\n                              <m:mo>,<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$(\\mathbf {D}_{t}^{\\alpha }u)(t) = A u(t) + J^{1-\\alpha } f\\big (t,u(t)\\big ), \\quad t \\in [0,T], 0 &amp;lt; \\alpha &amp;lt;1,\\qquad u(0) = u^0,$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:disp-formula>\nwith operator <jats:italic>A<\/jats:italic>, which generates an analytic and compact resolution family <jats:inline-formula id=\"eq2_w2aab2b8b7b1b7b1aab1c13b1b5Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0001_d8c8e79353729680237c61121d2a09c6.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mrow>\n                                 <m:mo>{<\/m:mo>\n                                 <m:msub>\n                                    <m:mi>S<\/m:mi>\n                                    <m:mi>\u03b1<\/m:mi>\n                                 <\/m:msub>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mi>t<\/m:mi>\n                                    <m:mo>,<\/m:mo>\n                                    <m:mi>A<\/m:mi>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                                 <m:mo>}<\/m:mo>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>\u2265<\/m:mo>\n                                 <m:mn>0<\/m:mn>\n                              <\/m:mrow>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>${\\lbrace S_{\\alpha }(t,A)\\rbrace _{t\\ge 0}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, in a Banach space <jats:italic>E<\/jats:italic> are presented. It is proved that the compact convergence of resolvents implies the convergence of semidiscrete approximations to an exact solution. We give an analysis of a general approximation scheme, which includes finite differences and projective methods.<\/jats:p>","DOI":"10.1515\/cmam-2015-0001","type":"journal-article","created":{"date-parts":[[2015,2,4]],"date-time":"2015-02-04T12:51:12Z","timestamp":1423054272000},"page":"203-212","source":"Crossref","is-referenced-by-count":7,"title":["Approximation of Semilinear Fractional Cauchy Problem"],"prefix":"10.1515","volume":"15","author":[{"given":"Ru","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Miao","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey","family":"Piskarev","sequence":"additional","affiliation":[{"name":"Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119991, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2015,2,3]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/15\/2\/article-p203.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0001\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0001\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T22:39:54Z","timestamp":1680302394000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0001\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,3,10]]},"published-print":{"date-parts":[[2015,4,1]]}},"alternative-id":["10.1515\/cmam-2015-0001"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2015-0001","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2015,2,3]]}}}