{"id":"https://openalex.org/W3033435644","doi":"https://doi.org/10.3390/sym12060921","title":"On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term","display_name":"On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term","publication_year":2020,"publication_date":"2020-06-02","ids":{"openalex":"https://openalex.org/W3033435644","doi":"https://doi.org/10.3390/sym12060921","mag":"3033435644"},"language":"en","primary_location":{"id":"doi:10.3390/sym12060921","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12060921","pdf_url":"https://www.mdpi.com/2073-8994/12/6/921/pdf","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/12/6/921/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041522452","display_name":"\u0410. \u041b. \u041a\u0430\u0437\u0430\u043a\u043e\u0432","orcid":"https://orcid.org/0000-0002-3047-1650"},"institutions":[{"id":"https://openalex.org/I4210148533","display_name":"Institute for System Dynamics and Control Theory","ror":"https://ror.org/04wteqw51","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I1313323035","https://openalex.org/I4210127387","https://openalex.org/I4210148533","https://openalex.org/I4210164537"]},{"id":"https://openalex.org/I4210127387","display_name":"Siberian Branch of the Russian Academy of Sciences","ror":"https://ror.org/02frkq021","country_code":"RU","type":"funder","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210127387"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Alexander Kazakov","raw_affiliation_strings":["Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, 134 Lermontov St., Irkutsk 664033, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, 134 Lermontov St., Irkutsk 664033, Russia","institution_ids":["https://openalex.org/I4210148533","https://openalex.org/I4210127387"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053571975","display_name":"\u041b. \u0424. \u0421\u043f\u0435\u0432\u0430\u043a","orcid":"https://orcid.org/0000-0003-2957-6962"},"institutions":[{"id":"https://openalex.org/I4210119369","display_name":"Ural Branch of the Russian Academy of Sciences","ror":"https://ror.org/02s4h3z39","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210119369"]},{"id":"https://openalex.org/I4210134084","display_name":"Institute of Engineering Science","ror":"https://ror.org/03zeyg902","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210134084","https://openalex.org/I4210164537"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Lev Spevak","raw_affiliation_strings":["Institute of Engineering Science, Ural Branch of the Russian Academy of Sciences, 34 Komsomolskaya St., Ekaterinburg 620049, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Engineering Science, Ural Branch of the Russian Academy of Sciences, 34 Komsomolskaya St., Ekaterinburg 620049, Russia","institution_ids":["https://openalex.org/I4210119369","https://openalex.org/I4210134084"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017226998","display_name":"\u041e. \u0410. \u041d\u0435\u0444\u0435\u0434\u043e\u0432\u0430","orcid":"https://orcid.org/0000-0003-1163-6699"},"institutions":[{"id":"https://openalex.org/I4210134084","display_name":"Institute of Engineering Science","ror":"https://ror.org/03zeyg902","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210134084","https://openalex.org/I4210164537"]},{"id":"https://openalex.org/I4210119369","display_name":"Ural Branch of the Russian Academy of Sciences","ror":"https://ror.org/02s4h3z39","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210119369"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Olga Nefedova","raw_affiliation_strings":["Institute of Engineering Science, Ural Branch of the Russian Academy of Sciences, 34 Komsomolskaya St., Ekaterinburg 620049, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Engineering Science, Ural Branch of the Russian Academy of Sciences, 34 Komsomolskaya St., Ekaterinburg 620049, Russia","institution_ids":["https://openalex.org/I4210119369","https://openalex.org/I4210134084"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030697249","display_name":"Anna Lempert","orcid":"https://orcid.org/0000-0001-9562-7903"},"institutions":[{"id":"https://openalex.org/I4210148533","display_name":"Institute for System Dynamics and Control Theory","ror":"https://ror.org/04wteqw51","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I1313323035","https://openalex.org/I4210127387","https://openalex.org/I4210148533","https://openalex.org/I4210164537"]},{"id":"https://openalex.org/I4210127387","display_name":"Siberian Branch of the Russian Academy of Sciences","ror":"https://ror.org/02frkq021","country_code":"RU","type":"funder","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210127387"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Anna Lempert","raw_affiliation_strings":["Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, 134 Lermontov St., Irkutsk 664033, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, 134 Lermontov St., Irkutsk 664033, Russia","institution_ids":["https://openalex.org/I4210148533","https://openalex.org/I4210127387"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5030697249"],"corresponding_institution_ids":["https://openalex.org/I4210127387","https://openalex.org/I4210148533"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":3.547,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.93123408,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"12","issue":"6","first_page":"921","last_page":"921"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9933000206947327,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12710","display_name":"Engineering and Agricultural Innovations","score":0.9861000180244446,"subfield":{"id":"https://openalex.org/subfields/1104","display_name":"Aquatic Science"},"field":{"id":"https://openalex.org/fields/11","display_name":"Agricultural and Biological Sciences"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6563307046890259},{"id":"https://openalex.org/keywords/heat-equation","display_name":"Heat equation","score":0.6313526034355164},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.5826985836029053},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5638784766197205},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5407500267028809},{"id":"https://openalex.org/keywords/thermal-conduction","display_name":"Thermal conduction","score":0.4856163561344147},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.4760812819004059},{"id":"https://openalex.org/keywords/reciprocity","display_name":"Reciprocity (cultural anthropology)","score":0.4523509442806244},{"id":"https://openalex.org/keywords/power-series","display_name":"Power series","score":0.4438365399837494},{"id":"https://openalex.org/keywords/stefan-problem","display_name":"Stefan problem","score":0.42151787877082825},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.42141205072402954},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.42095476388931274},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.35707414150238037},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.2777109146118164},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1262328326702118}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6563307046890259},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.6313526034355164},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.5826985836029053},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5638784766197205},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5407500267028809},{"id":"https://openalex.org/C172100665","wikidata":"https://www.wikidata.org/wiki/Q7465774","display_name":"Thermal conduction","level":2,"score":0.4856163561344147},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4760812819004059},{"id":"https://openalex.org/C169903001","wikidata":"https://www.wikidata.org/wiki/Q3264987","display_name":"Reciprocity (cultural anthropology)","level":2,"score":0.4523509442806244},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.4438365399837494},{"id":"https://openalex.org/C2778021227","wikidata":"https://www.wikidata.org/wiki/Q3775789","display_name":"Stefan problem","level":3,"score":0.42151787877082825},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.42141205072402954},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.42095476388931274},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.35707414150238037},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.2777109146118164},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1262328326702118},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym12060921","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12060921","pdf_url":"https://www.mdpi.com/2073-8994/12/6/921/pdf","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:54e616cd2e0b432ca1b9ddcaa17020ba","is_oa":true,"landing_page_url":"https://doaj.org/article/54e616cd2e0b432ca1b9ddcaa17020ba","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 12, Iss 6, p 921 (2020)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym12060921","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12060921","pdf_url":"https://www.mdpi.com/2073-8994/12/6/921/pdf","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7","score":0.5299999713897705}],"awards":[{"id":"https://openalex.org/G7695368781","display_name":null,"funder_award_id":"20-07-00407","funder_id":"https://openalex.org/F4320321079","funder_display_name":"Russian Foundation for Basic Research"}],"funders":[{"id":"https://openalex.org/F4320321079","display_name":"Russian Foundation for Basic Research","ror":"https://ror.org/02mh1ke95"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3033435644.pdf","grobid_xml":"https://content.openalex.org/works/W3033435644.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W33353517","https://openalex.org/W615448849","https://openalex.org/W1204786174","https://openalex.org/W1855169650","https://openalex.org/W1965739805","https://openalex.org/W2003223889","https://openalex.org/W2004774349","https://openalex.org/W2014385945","https://openalex.org/W2037839126","https://openalex.org/W2061720377","https://openalex.org/W2074797641","https://openalex.org/W2080051145","https://openalex.org/W2081334928","https://openalex.org/W2082768525","https://openalex.org/W2083090771","https://openalex.org/W2092250306","https://openalex.org/W2095769364","https://openalex.org/W2123967333","https://openalex.org/W2158092963","https://openalex.org/W2267809190","https://openalex.org/W2488273306","https://openalex.org/W2490615762","https://openalex.org/W2804845204","https://openalex.org/W2809496349","https://openalex.org/W2809756657","https://openalex.org/W2891923107","https://openalex.org/W2922779598","https://openalex.org/W2937071557","https://openalex.org/W2951198851","https://openalex.org/W4205717468","https://openalex.org/W4206832447","https://openalex.org/W6669145261","https://openalex.org/W6678559251","https://openalex.org/W6761231428"],"related_works":["https://openalex.org/W1985897728","https://openalex.org/W3016158432","https://openalex.org/W2353039109","https://openalex.org/W2375700760","https://openalex.org/W4315588385","https://openalex.org/W2061564118","https://openalex.org/W3124638673","https://openalex.org/W2094793377","https://openalex.org/W1516308869","https://openalex.org/W2087162510"],"abstract_inverted_index":{"The":[0,56,70,109,145,156],"paper":[1],"deals":[2],"with":[3,13,44,78],"two-dimensional":[4],"boundary-value":[5],"problems":[6],"for":[7,119],"the":[8,19,25,28,33,50,65,93,98,106,112,115,135,138,152,161,165],"degenerate":[9],"nonlinear":[10],"parabolic":[11],"equation":[12],"a":[14,39,45,75,84],"source":[15],"term,":[16],"which":[17],"describes":[18],"process":[20],"of":[21,27,32,52,67,111,114,122,134,140,164],"heat":[22,34,40],"conduction":[23],"in":[24,49,64,137],"case":[26,51,139],"power-law":[29],"temperature":[30],"dependence":[31],"conductivity":[35],"coefficient.":[36],"We":[37],"consider":[38],"wave":[41],"propagation":[42],"problem":[43,107,136],"specified":[46],"zero":[47],"front":[48],"two":[53],"spatial":[54],"variables.":[55],"solution":[57,71],"existence":[58],"and":[59],"uniqueness":[60],"theorem":[61],"is":[62,72,102,126,143],"proved":[63],"class":[66],"analytic":[68],"functions.":[69],"constructed":[73,146],"as":[74],"power":[76],"series":[77],"coefficients":[79],"to":[80,104,130,150],"be":[81],"calculated":[82],"by":[83],"proposed":[85],"constructive":[86],"recurrent":[87],"procedure.":[88],"An":[89,128],"algorithm":[90],"based":[91],"on":[92],"boundary":[94],"element":[95],"method":[96,101,118],"using":[97],"dual":[99,116],"reciprocity":[100,117],"developed":[103,153],"solve":[105],"numerically.":[108],"efficiency":[110,163],"application":[113],"various":[120],"systems":[121],"radial":[123],"basis":[124],"functions":[125],"analyzed.":[127],"approach":[129],"constructing":[131],"invariant":[132],"solutions":[133,147],"central":[141],"symmetry":[142],"proposed.":[144],"are":[148],"used":[149],"verify":[151],"numerical":[154],"algorithm.":[155,166],"test":[157],"calculations":[158],"have":[159],"shown":[160],"high":[162]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":4}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}