{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T02:23:48Z","timestamp":1773800628004,"version":"3.50.1"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014,7,1]]},"abstract":"<jats:title>Abstract.<\/jats:title>\n               <jats:p>We construct an usual linearized difference scheme for initial boundary value problems (IBVP) for one-dimensional quasilinear parabolic equations with generalized solutions.\nThe uniform parabolicity condition <jats:inline-formula id=\"eq1_w2aab2b8b5b1b7b1aab1c13b1b1Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0008_62f7e3a4017d73df0ea35733fb8aae53.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>0<\/m:mn>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:msub>\n                                 <m:mi>k<\/m:mi>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:msub>\n                              <m:mo>\u2264<\/m:mo>\n                              <m:mi>k<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>u<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>\u2264<\/m:mo>\n                              <m:msub>\n                                 <m:mi>k<\/m:mi>\n                                 <m:mn>2<\/m:mn>\n                              <\/m:msub>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$0&amp;lt;k_1\\le k(u) \\le k_2$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is assumed to be\nfulfilled for the sign alternating solution <jats:inline-formula id=\"eq2_w2aab2b8b5b1b7b1aab1c13b1b3Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0008_e73151735d23a53e294b3b897ade6811.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>u<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>x<\/m:mi>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mover accent=\"true\">\n                                 <m:mi>D<\/m:mi>\n                                 <m:mo>\u00af<\/m:mo>\n                              <\/m:mover>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>u<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$u(x,t) \\in \\bar{D}(u)$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\nonly in the domain of exact solution values (unbounded\nnon-linearity). On the basis of new corollaries of the\nmaximum principle, we establish not only two-sided estimates for the approximate\nsolution <jats:italic>y<\/jats:italic> but its belonging to the domain of exact solution\nvalues. We assume that the solution is continuous\nand its first derivative <jats:inline-formula id=\"eq3_w2aab2b8b5b1b7b1aab1c13b1b7Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0008_df821859383d5dfb407623b91e6f7d04.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mfrac>\n                              <m:mrow>\n                                 <m:mi>\u2202<\/m:mi>\n                                 <m:mi>u<\/m:mi>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mi>\u2202<\/m:mi>\n                                 <m:mi>x<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:mfrac>\n                        <\/m:math>\n                        <jats:tex-math>$\\frac{\\partial u}{\\partial x}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> has\ndiscontinuity of the first kind in the neighborhood of the finite\nnumber of discontinuity lines. An existence of time derivative in\nany sense is not assumed. We prove convergence of the approximate solution to\nthe generalized solution of the differential problem in the grid norm <jats:italic>L<\/jats:italic>\n                  <jats:sub>2<\/jats:sub>.<\/jats:p>","DOI":"10.1515\/cmam-2014-0008","type":"journal-article","created":{"date-parts":[[2014,5,21]],"date-time":"2014-05-21T12:49:59Z","timestamp":1400676599000},"page":"361-371","source":"Crossref","is-referenced-by-count":21,"title":["On Convergence of Difference Schemes for IBVP\nfor Quasilinear Parabolic Equations with Generalized Solutions"],"prefix":"10.1515","volume":"14","author":[{"given":"Piotr","family":"Matus","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20-950 Lublin, Poland; and Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2014,4,26]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/14\/3\/article-p361.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0008\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0008\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:08:45Z","timestamp":1680296925000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0008\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,4,26]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,4,24]]},"published-print":{"date-parts":[[2014,7,1]]}},"alternative-id":["10.1515\/cmam-2014-0008"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2014-0008","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,4,26]]}}}