{"id":"https://openalex.org/W2120847465","doi":"https://doi.org/10.3390/e15093931","title":"Solutions of Some Nonlinear Diffusion Equations and Generalized Entropy Framework","display_name":"Solutions of Some Nonlinear Diffusion Equations and Generalized Entropy Framework","publication_year":2013,"publication_date":"2013-09-18","ids":{"openalex":"https://openalex.org/W2120847465","doi":"https://doi.org/10.3390/e15093931","mag":"2120847465"},"language":"en","primary_location":{"id":"doi:10.3390/e15093931","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15093931","pdf_url":"https://www.mdpi.com/1099-4300/15/9/3931/pdf?version=1424785094","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/15/9/3931/pdf?version=1424785094","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060481487","display_name":"E. K. Lenzi","orcid":"https://orcid.org/0000-0003-3853-1790"},"institutions":[{"id":"https://openalex.org/I123443094","display_name":"Universidade Estadual de Maring\u00e1","ror":"https://ror.org/04bqqa360","country_code":"BR","type":"education","lineage":["https://openalex.org/I123443094"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Ervin Lenzi","raw_affiliation_strings":["Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil","institution_ids":["https://openalex.org/I123443094"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5011457702","display_name":"M. A. F. dos Santos","orcid":"https://orcid.org/0000-0001-8735-0617"},"institutions":[{"id":"https://openalex.org/I123443094","display_name":"Universidade Estadual de Maring\u00e1","ror":"https://ror.org/04bqqa360","country_code":"BR","type":"education","lineage":["https://openalex.org/I123443094"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Maike Dos Santos","raw_affiliation_strings":["Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil","institution_ids":["https://openalex.org/I123443094"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089352653","display_name":"Fl\u00e1vio S. Michels","orcid":"https://orcid.org/0000-0002-2952-3230"},"institutions":[{"id":"https://openalex.org/I123443094","display_name":"Universidade Estadual de Maring\u00e1","ror":"https://ror.org/04bqqa360","country_code":"BR","type":"education","lineage":["https://openalex.org/I123443094"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Flavio Michels","raw_affiliation_strings":["Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil","institution_ids":["https://openalex.org/I123443094"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043332827","display_name":"R. S. Mendes","orcid":"https://orcid.org/0009-0002-5216-0578"},"institutions":[{"id":"https://openalex.org/I123443094","display_name":"Universidade Estadual de Maring\u00e1","ror":"https://ror.org/04bqqa360","country_code":"BR","type":"education","lineage":["https://openalex.org/I123443094"]}],"countries":["BR"],"is_corresponding":true,"raw_author_name":"Renio Mendes","raw_affiliation_strings":["Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil","institution_ids":["https://openalex.org/I123443094"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006098704","display_name":"L. R. Evangelista","orcid":"https://orcid.org/0000-0001-7202-1386"},"institutions":[{"id":"https://openalex.org/I123443094","display_name":"Universidade Estadual de Maring\u00e1","ror":"https://ror.org/04bqqa360","country_code":"BR","type":"education","lineage":["https://openalex.org/I123443094"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Luiz Evangelista","raw_affiliation_strings":["Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil","institution_ids":["https://openalex.org/I123443094"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5043332827"],"corresponding_institution_ids":["https://openalex.org/I123443094"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.13517794,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"15","issue":"9","first_page":"3931","last_page":"3940"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9976999759674072,"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/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9567000269889832,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6477842926979065},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6028648614883423},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5945937633514404},{"id":"https://openalex.org/keywords/tsallis-entropy","display_name":"Tsallis entropy","score":0.5924644470214844},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5647779107093811},{"id":"https://openalex.org/keywords/tsallis-statistics","display_name":"Tsallis statistics","score":0.5093227624893188},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4982433319091797},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.4909132122993469},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.48267868161201477},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.4618467688560486},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4431007504463196},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.25842413306236267},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.15111243724822998},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07309037446975708},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06216949224472046}],"concepts":[{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6477842926979065},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6028648614883423},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5945937633514404},{"id":"https://openalex.org/C117521176","wikidata":"https://www.wikidata.org/wiki/Q7849341","display_name":"Tsallis entropy","level":3,"score":0.5924644470214844},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5647779107093811},{"id":"https://openalex.org/C2780056601","wikidata":"https://www.wikidata.org/wiki/Q7849339","display_name":"Tsallis statistics","level":2,"score":0.5093227624893188},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4982433319091797},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.4909132122993469},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.48267868161201477},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.4618467688560486},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4431007504463196},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.25842413306236267},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.15111243724822998},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07309037446975708},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06216949224472046},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/e15093931","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15093931","pdf_url":"https://www.mdpi.com/1099-4300/15/9/3931/pdf?version=1424785094","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:62552d60886444299b1eae9c491f5ce2","is_oa":true,"landing_page_url":"https://doaj.org/article/62552d60886444299b1eae9c491f5ce2","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":"Entropy, Vol 15, Iss 9, Pp 3931-3940 (2013)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/15/9/3931/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e15093931","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":"Entropy; Volume 15; Issue 9; Pages: 3931-3940","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e15093931","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15093931","pdf_url":"https://www.mdpi.com/1099-4300/15/9/3931/pdf?version=1424785094","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2120847465.pdf","grobid_xml":"https://content.openalex.org/works/W2120847465.grobid-xml"},"referenced_works_count":54,"referenced_works":["https://openalex.org/W145038029","https://openalex.org/W433794350","https://openalex.org/W1568783337","https://openalex.org/W1587522179","https://openalex.org/W1672570498","https://openalex.org/W1873431535","https://openalex.org/W1964988733","https://openalex.org/W1967155852","https://openalex.org/W1973910972","https://openalex.org/W1978824346","https://openalex.org/W1982910148","https://openalex.org/W1992670807","https://openalex.org/W2004369070","https://openalex.org/W2007200841","https://openalex.org/W2009387827","https://openalex.org/W2011487726","https://openalex.org/W2011952030","https://openalex.org/W2014672328","https://openalex.org/W2024244955","https://openalex.org/W2032180849","https://openalex.org/W2038047494","https://openalex.org/W2040726100","https://openalex.org/W2046567809","https://openalex.org/W2051638496","https://openalex.org/W2053116694","https://openalex.org/W2057908063","https://openalex.org/W2068267828","https://openalex.org/W2068738591","https://openalex.org/W2068946487","https://openalex.org/W2070467675","https://openalex.org/W2078477559","https://openalex.org/W2079987131","https://openalex.org/W2081925860","https://openalex.org/W2084055475","https://openalex.org/W2086183205","https://openalex.org/W2089852843","https://openalex.org/W2095582234","https://openalex.org/W2097509761","https://openalex.org/W2099733338","https://openalex.org/W2111271983","https://openalex.org/W2115195242","https://openalex.org/W2155587292","https://openalex.org/W2170439816","https://openalex.org/W2239362388","https://openalex.org/W2563554623","https://openalex.org/W3004847442","https://openalex.org/W3021966610","https://openalex.org/W3040881605","https://openalex.org/W3101264823","https://openalex.org/W3103885682","https://openalex.org/W4244695582","https://openalex.org/W4255777804","https://openalex.org/W4299635615","https://openalex.org/W6731068750"],"related_works":["https://openalex.org/W2154579008","https://openalex.org/W3159939403","https://openalex.org/W2379111240","https://openalex.org/W2908383647","https://openalex.org/W2021252433","https://openalex.org/W1516425925","https://openalex.org/W3098194525","https://openalex.org/W1989185299","https://openalex.org/W2361046054","https://openalex.org/W3101893557"],"abstract_inverted_index":{"We":[0],"investigate":[1],"solutions":[2,22,60],"of":[3,15,39,51,79],"a":[4,27,76],"generalized":[5],"diffusion":[6],"equation":[7],"that":[8],"contains":[9],"nonlinear":[10],"terms":[11,38],"in":[12,37,44,55],"the":[13,40,45,49,52,56,66,72],"presence":[14],"external":[16],"forces":[17],"and":[18,33,84],"reaction":[19],"terms.":[20],"The":[21],"found":[23],"here":[24],"can":[25,34,61,87],"have":[26],"compact":[28],"or":[29],"long":[30],"tail":[31],"behavior":[32],"be":[35,63,88],"expressed":[36],"q-exponential":[41],"functions":[42],"present":[43],"Tsallis":[46],"framework.":[47],"In":[48,69],"case":[50],"long-tailed":[53],"behavior,":[54],"asymptotic":[57],"limit,":[58],"these":[59],"also":[62],"connected":[64],"with":[65],"L\u00b4evy":[67],"distributions.":[68],"addition,":[70],"from":[71],"results":[73],"presented":[74],"here,":[75],"rich":[77],"class":[78],"diffusive":[80],"processes,":[81],"including":[82],"normal":[83],"anomalous":[85],"ones,":[86],"obtained.":[89]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2016-06-24T00:00:00"}