{"id":"https://openalex.org/W4214848306","doi":"https://doi.org/10.3390/sym9010013","title":"A (1 + 2)-Dimensional Simplified Keller\u2013Segel Model: Lie Symmetry and Exact Solutions. II","display_name":"A (1 + 2)-Dimensional Simplified Keller\u2013Segel Model: Lie Symmetry and Exact Solutions. II","publication_year":2017,"publication_date":"2017-01-20","ids":{"openalex":"https://openalex.org/W4214848306","doi":"https://doi.org/10.3390/sym9010013"},"language":"en","primary_location":{"id":"doi:10.3390/sym9010013","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym9010013","pdf_url":"https://www.mdpi.com/2073-8994/9/1/13/pdf?version=1484918619","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/2073-8994/9/1/13/pdf?version=1484918619","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051428126","display_name":"Roman Cherniha","orcid":"https://orcid.org/0000-0002-1733-5240"},"institutions":[{"id":"https://openalex.org/I149851306","display_name":"National Academy of Sciences of Ukraine","ror":"https://ror.org/00je4t102","country_code":"UA","type":"government","lineage":["https://openalex.org/I149851306"]},{"id":"https://openalex.org/I190444043","display_name":"Institute of Mathematics","ror":"https://ror.org/04q3mp532","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I190444043"]}],"countries":["UA"],"is_corresponding":true,"raw_author_name":"Roman Cherniha","raw_affiliation_strings":["Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, Kyiv 01004, Ukraine","Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka Street, Kyiv 01004, Ukraine"],"raw_orcid":"https://orcid.org/0000-0002-1733-5240","affiliations":[{"raw_affiliation_string":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, Kyiv 01004, Ukraine","institution_ids":["https://openalex.org/I190444043","https://openalex.org/I149851306"]},{"raw_affiliation_string":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka Street, Kyiv 01004, Ukraine","institution_ids":["https://openalex.org/I190444043","https://openalex.org/I149851306"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5020674633","display_name":"Maksym Didovych","orcid":null},"institutions":[{"id":"https://openalex.org/I149851306","display_name":"National Academy of Sciences of Ukraine","ror":"https://ror.org/00je4t102","country_code":"UA","type":"government","lineage":["https://openalex.org/I149851306"]},{"id":"https://openalex.org/I190444043","display_name":"Institute of Mathematics","ror":"https://ror.org/04q3mp532","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I190444043"]}],"countries":["UA"],"is_corresponding":false,"raw_author_name":"Maksym Didovych","raw_affiliation_strings":["Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, Kyiv 01004, Ukraine","Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka Street, Kyiv 01004, Ukraine"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, Kyiv 01004, Ukraine","institution_ids":["https://openalex.org/I190444043","https://openalex.org/I149851306"]},{"raw_affiliation_string":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka Street, Kyiv 01004, Ukraine","institution_ids":["https://openalex.org/I190444043","https://openalex.org/I149851306"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5051428126"],"corresponding_institution_ids":["https://openalex.org/I149851306","https://openalex.org/I190444043"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.3218,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.61451049,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"9","issue":"1","first_page":"13","last_page":"13"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T11829","display_name":"Mathematical Biology Tumor Growth","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10621","display_name":"Gene Regulatory Network Analysis","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10492","display_name":"Microtubule and mitosis dynamics","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/1307","display_name":"Cell Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/homogeneous-space","display_name":"Homogeneous space","score":0.6972332000732422},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6098200678825378},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.5878677368164062},{"id":"https://openalex.org/keywords/lie-group","display_name":"Lie group","score":0.5564186573028564},{"id":"https://openalex.org/keywords/neumann-boundary-condition","display_name":"Neumann boundary condition","score":0.5393689870834351},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.5360714793205261},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.5350386500358582},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.47936493158340454},{"id":"https://openalex.org/keywords/cauchy-distribution","display_name":"Cauchy distribution","score":0.47399652004241943},{"id":"https://openalex.org/keywords/lie-algebra","display_name":"Lie algebra","score":0.4449985921382904},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.44365832209587097},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3754926323890686},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.35117071866989136},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.18274062871932983},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10909175872802734}],"concepts":[{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.6972332000732422},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6098200678825378},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.5878677368164062},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.5564186573028564},{"id":"https://openalex.org/C163681178","wikidata":"https://www.wikidata.org/wiki/Q1149279","display_name":"Neumann boundary condition","level":3,"score":0.5393689870834351},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.5360714793205261},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5350386500358582},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.47936493158340454},{"id":"https://openalex.org/C49344536","wikidata":"https://www.wikidata.org/wiki/Q726441","display_name":"Cauchy distribution","level":2,"score":0.47399652004241943},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.4449985921382904},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.44365832209587097},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3754926323890686},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35117071866989136},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.18274062871932983},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10909175872802734}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym9010013","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym9010013","pdf_url":"https://www.mdpi.com/2073-8994/9/1/13/pdf?version=1484918619","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:1677a6f8b7b843009ce3ee82f2bdfdc9","is_oa":true,"landing_page_url":"https://doaj.org/article/1677a6f8b7b843009ce3ee82f2bdfdc9","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 9, Iss 1, p 13 (2017)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/9/1/13/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym9010013","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry; Volume 9; Issue 1; Pages: 13","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym9010013","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym9010013","pdf_url":"https://www.mdpi.com/2073-8994/9/1/13/pdf?version=1484918619","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":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1496087027","https://openalex.org/W1506176481","https://openalex.org/W1534983138","https://openalex.org/W1808720348","https://openalex.org/W1978941072","https://openalex.org/W1989853286","https://openalex.org/W2028496490","https://openalex.org/W2042892258","https://openalex.org/W2045384647","https://openalex.org/W2109369213","https://openalex.org/W2119563001","https://openalex.org/W2285346760","https://openalex.org/W2563650214","https://openalex.org/W2797256054","https://openalex.org/W3022230513","https://openalex.org/W4240986989","https://openalex.org/W4292170110","https://openalex.org/W6631848652","https://openalex.org/W6695652942"],"related_works":["https://openalex.org/W4300711749","https://openalex.org/W2246967737","https://openalex.org/W2080482296","https://openalex.org/W3186968207","https://openalex.org/W3016004364","https://openalex.org/W2089433228","https://openalex.org/W2000026364","https://openalex.org/W2963288691","https://openalex.org/W4298401637","https://openalex.org/W2940979654"],"abstract_inverted_index":{"A":[0,40],"simplified":[1,98,137],"Keller\u2013Segel":[2,21,99,138],"model":[3],"is":[4,14,32,54,140,158,180],"studied":[5],"by":[6],"means":[7],"of":[8,44,64,70,76,128,145,152,166,172,192],"Lie":[9,38,41,57,125,143],"symmetry":[10,42,126,144],"based":[11],"approaches.":[12],"It":[13],"shown":[15],"that":[16,69,89],"a":[17,121,178,187,193,199],"(1":[18,71,78,95,130],"+":[19,72,79,96,131],"2)-dimensional":[20,132],"type":[22],"system,":[23],"together":[24],"with":[25,34],"the":[26,45,50,65,90,94,129,136,153,156,173,174,183],"correctly-specified":[27],"boundary":[28],"and/or":[29],"initial":[30,51],"conditions,":[31],"invariant":[33],"respect":[35],"to":[36,68,119],"infinite-dimensional":[37],"algebras.":[39],"classification":[43,127],"Cauchy":[46,66,91],"problem":[47,67,92,134,157,176,197],"depending":[48],"on":[49,150,177,198],"profile":[52],"form":[53],"presented.":[55],"The":[56,124],"symmetries":[58,184],"obtained":[59],"are":[60,82],"used":[61],"for":[62,93,135],"reduction":[63],"1)-dimensional.":[73],"Exact":[74],"solutions":[75],"some":[77],"1)-dimensional":[80,97],"problems":[81,147],"constructed.":[83],"In":[84],"particular,":[85],"we":[86],"have":[87],"proved":[88],"system":[100,139],"can":[101],"be":[102],"linearized":[103],"and":[104],"solved":[105],"in":[106,117],"an":[107,189],"explicit":[108],"form.":[109],"Moreover,":[110],"additional":[111],"biologically":[112],"motivated":[113],"restrictions":[114],"were":[115,169],"established":[116],"order":[118],"obtain":[120],"unique":[122],"solution.":[123],"Neumann":[133,175,196],"derived.":[141],"Because":[142],"boundary-value":[146],"depends":[148],"essentially":[149],"geometry":[151],"domain,":[154],"which":[155],"formulated":[159],"for,":[160],"all":[161],"realistic":[162],"(from":[163],"applicability":[164],"point":[165],"view)":[167],"domains":[168],"examined.":[170],"Reduction":[171],"strip":[179],"derived":[181],"using":[182],"obtained.":[185],"As":[186],"result,":[188],"exact":[190],"solution":[191],"nonlinear":[194],"two-dimensional":[195],"finite":[200],"interval":[201],"was":[202],"found.":[203]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}