{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,5]],"date-time":"2025-04-05T09:09:09Z","timestamp":1743844149907},"reference-count":20,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T00:00:00Z","timestamp":1680220800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"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>We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain.\nWe approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping.\nFor the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msubsup>\n                                 <m:mi>L<\/m:mi>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                              <\/m:msubsup>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:msubsup>\n                                    <m:mi>L<\/m:mi>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                                 <\/m:msubsup>\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-2022-0217_ineq_0001.png\" \/>\n                        <jats:tex-math>L^{\\infty}_{t}(L^{\\infty}_{x})<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msubsup>\n                                 <m:mi>L<\/m:mi>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                              <\/m:msubsup>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:msubsup>\n                                    <m:mi>H<\/m:mi>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mn>1<\/m:mn>\n                                 <\/m:msubsup>\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-2022-0217_ineq_0002.png\" \/>\n                        <jats:tex-math>L^{\\infty}_{t}(H^{1}_{x})<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norm, allowing a mild mesh condition to be satisfied.\nFinally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments.\nIt is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.<\/jats:p>","DOI":"10.1515\/cmam-2022-0217","type":"journal-article","created":{"date-parts":[[2023,3,30]],"date-time":"2023-03-30T17:22:38Z","timestamp":1680196958000},"page":"695-713","source":"Crossref","is-referenced-by-count":2,"title":["Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity"],"prefix":"10.1515","volume":"23","author":[{"given":"Panagiotis","family":"Paraschis","sequence":"first","affiliation":[{"name":"School of Applied Mathematical and Physical Sciences , National Technical University of Athens , GR-157 80 Zografou , Greece"}]},{"given":"Georgios E.","family":"Zouraris","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics , Division of Applied Mathematics: Differential Equations and Numerical Analysis , University of Crete , GR-700 13 Voutes Campus , Heraklion , Crete , Greece"}]}],"member":"374","published-online":{"date-parts":[[2023,3,31]]},"reference":[{"key":"2023070413382866451_j_cmam-2022-0217_ref_001","doi-asserted-by":"crossref","unstructured":"M. Alfaro and R. Carles,\nSuperexponential growth or decay in the heat equation with a logarithmic nonlinearity,\nDyn. Partial Differ. Equ. 14 (2017), no. 4, 343\u2013358.","DOI":"10.4310\/DPDE.2017.v14.n4.a2"},{"key":"2023070413382866451_j_cmam-2022-0217_ref_002","doi-asserted-by":"crossref","unstructured":"W. Bao, R. Carles, C. Su and Q. Tang,\nError estimates of a regularized finite difference method for the logarithmic Schr\u00f6dinger equation,\nSIAM J. Numer. Anal. 57 (2019), no. 2, 657\u2013680.","DOI":"10.1137\/18M1177445"},{"key":"2023070413382866451_j_cmam-2022-0217_ref_003","doi-asserted-by":"crossref","unstructured":"W. Bao, R. Carles, C. Su and Q. Tang,\nRegularized numerical methods for the logarithmic Schr\u00f6dinger equation,\nNumer. Math. 143 (2019), no. 2, 461\u2013487.","DOI":"10.1007\/s00211-019-01058-2"},{"key":"2023070413382866451_j_cmam-2022-0217_ref_004","doi-asserted-by":"crossref","unstructured":"W. Bao, R. Carles, C. Su and Q. Tang,\nError estimates of local energy regularization for the logarithmic Schr\u00f6dinger equation,\nMath. Models Methods Appl. Sci. 32 (2022), no. 1, 101\u2013136.","DOI":"10.1142\/S0218202522500038"},{"key":"2023070413382866451_j_cmam-2022-0217_ref_005","doi-asserted-by":"crossref","unstructured":"H. Chen, P. Luo and G. Liu,\nGlobal solution and blow-up of a semilinear heat equation with logarithmic nonlinearity,\nJ. Math. Anal. Appl. 422 (2015), no. 1, 84\u201398.","DOI":"10.1016\/j.jmaa.2014.08.030"},{"key":"2023070413382866451_j_cmam-2022-0217_ref_006","doi-asserted-by":"crossref","unstructured":"B. Cheng and Z. Guo,\nRegularized splitting spectral method for space-fractional logarithmic Schr\u00f6dinger equation,\nAppl. Numer. Math. 167 (2021), 330\u2013355.","DOI":"10.1016\/j.apnum.2021.05.003"},{"key":"2023070413382866451_j_cmam-2022-0217_ref_007","doi-asserted-by":"crossref","unstructured":"R. C. Dalang, D. Khoshnevisan and T. 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