{"id":"https://openalex.org/W4383535827","doi":"https://doi.org/10.3390/sym15071374","title":"Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach","display_name":"Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach","publication_year":2023,"publication_date":"2023-07-06","ids":{"openalex":"https://openalex.org/W4383535827","doi":"https://doi.org/10.3390/sym15071374"},"language":"en","primary_location":{"id":"doi:10.3390/sym15071374","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15071374","pdf_url":"https://www.mdpi.com/2073-8994/15/7/1374/pdf?version=1688639846","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/2073-8994/15/7/1374/pdf?version=1688639846","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069957230","display_name":"Muhammad Nadeem","orcid":"https://orcid.org/0000-0002-9349-4729"},"institutions":[{"id":"https://openalex.org/I4210105365","display_name":"Qujing Normal University","ror":"https://ror.org/02ad7ap24","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210105365"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Muhammad Nadeem","raw_affiliation_strings":["School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China"],"raw_orcid":"https://orcid.org/0000-0002-9349-4729","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China","institution_ids":["https://openalex.org/I4210105365"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101963275","display_name":"Asad Islam","orcid":"https://orcid.org/0000-0002-4192-7211"},"institutions":[{"id":"https://openalex.org/I899713450","display_name":"Air University","ror":"https://ror.org/03yfe9v83","country_code":"PK","type":"education","lineage":["https://openalex.org/I899713450"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Asad Islam","raw_affiliation_strings":["Department of Mechanical and Aerospace Engineering, Air University, Islamabad 44000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical and Aerospace Engineering, Air University, Islamabad 44000, Pakistan","institution_ids":["https://openalex.org/I899713450"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028739207","display_name":"Shazia Karim","orcid":"https://orcid.org/0000-0003-2401-5437"},"institutions":[{"id":"https://openalex.org/I142732210","display_name":"University of Engineering and Technology Lahore","ror":"https://ror.org/0051w2v06","country_code":"PK","type":"education","lineage":["https://openalex.org/I142732210"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Shazia Karim","raw_affiliation_strings":["Department of Basic Sciences, UET Lahore, FSD Campus, Faisalabad 54800, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, UET Lahore, FSD Campus, Faisalabad 54800, Pakistan","institution_ids":["https://openalex.org/I142732210"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016978313","display_name":"Sorin Mure\u015fan","orcid":"https://orcid.org/0000-0002-1882-3748"},"institutions":[{"id":"https://openalex.org/I162871001","display_name":"University of Oradea","ror":"https://ror.org/00wzhv093","country_code":"RO","type":"education","lineage":["https://openalex.org/I162871001"]}],"countries":["RO"],"is_corresponding":false,"raw_author_name":"Sorin Mure\u015fan","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania","institution_ids":["https://openalex.org/I162871001"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075452466","display_name":"Loredana Florentina Iambor","orcid":"https://orcid.org/0000-0001-8845-3095"},"institutions":[{"id":"https://openalex.org/I162871001","display_name":"University of Oradea","ror":"https://ror.org/00wzhv093","country_code":"RO","type":"education","lineage":["https://openalex.org/I162871001"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Loredana Florentina Iambor","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"],"raw_orcid":"https://orcid.org/0000-0001-8845-3095","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania","institution_ids":["https://openalex.org/I162871001"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5069957230","https://openalex.org/A5075452466"],"corresponding_institution_ids":["https://openalex.org/I162871001","https://openalex.org/I4210105365"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.9176,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.84587032,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"15","issue":"7","first_page":"1374","last_page":"1374"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9901000261306763,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9728999733924866,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6572226881980896},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6225910782814026},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6107047200202942},{"id":"https://openalex.org/keywords/porous-medium","display_name":"Porous medium","score":0.5629643201828003},{"id":"https://openalex.org/keywords/homotopy-analysis-method","display_name":"Homotopy analysis method","score":0.5343244671821594},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5061203837394714},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4917011857032776},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.48591557145118713},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.482991099357605},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.4749997854232788},{"id":"https://openalex.org/keywords/exact-solutions-in-general-relativity","display_name":"Exact solutions in general relativity","score":0.4426722526550293},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4413835406303406},{"id":"https://openalex.org/keywords/heat-transfer","display_name":"Heat transfer","score":0.4163738787174225},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3766211271286011},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.34017667174339294},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1831987202167511},{"id":"https://openalex.org/keywords/porosity","display_name":"Porosity","score":0.1612379252910614},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14870673418045044},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.0994977355003357}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6572226881980896},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6225910782814026},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6107047200202942},{"id":"https://openalex.org/C105569014","wikidata":"https://www.wikidata.org/wiki/Q3271208","display_name":"Porous medium","level":3,"score":0.5629643201828003},{"id":"https://openalex.org/C173636693","wikidata":"https://www.wikidata.org/wiki/Q17030668","display_name":"Homotopy analysis method","level":3,"score":0.5343244671821594},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5061203837394714},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4917011857032776},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.48591557145118713},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.482991099357605},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.4749997854232788},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.4426722526550293},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4413835406303406},{"id":"https://openalex.org/C50517652","wikidata":"https://www.wikidata.org/wiki/Q179635","display_name":"Heat transfer","level":2,"score":0.4163738787174225},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3766211271286011},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.34017667174339294},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1831987202167511},{"id":"https://openalex.org/C6648577","wikidata":"https://www.wikidata.org/wiki/Q622669","display_name":"Porosity","level":2,"score":0.1612379252910614},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14870673418045044},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.0994977355003357},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C187320778","wikidata":"https://www.wikidata.org/wiki/Q1349130","display_name":"Geotechnical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15071374","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15071374","pdf_url":"https://www.mdpi.com/2073-8994/15/7/1374/pdf?version=1688639846","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:4aaed79b02414b5398e50ec10c902d6e","is_oa":false,"landing_page_url":"https://doaj.org/article/4aaed79b02414b5398e50ec10c902d6e","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 7, p 1374 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/7/1374/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15071374","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","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15071374","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15071374","pdf_url":"https://www.mdpi.com/2073-8994/15/7/1374/pdf?version=1688639846","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":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4383535827.pdf"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1976686289","https://openalex.org/W1979366256","https://openalex.org/W1985375302","https://openalex.org/W2067238867","https://openalex.org/W2084823492","https://openalex.org/W2756974843","https://openalex.org/W2801670413","https://openalex.org/W2890250940","https://openalex.org/W2903953228","https://openalex.org/W2910124875","https://openalex.org/W2914216937","https://openalex.org/W2961482806","https://openalex.org/W2970219924","https://openalex.org/W2981140784","https://openalex.org/W2999946823","https://openalex.org/W3037097151","https://openalex.org/W3049160659","https://openalex.org/W3107497183","https://openalex.org/W3126199721","https://openalex.org/W3201185194","https://openalex.org/W4200161442","https://openalex.org/W4225263068","https://openalex.org/W4283274223","https://openalex.org/W4307684799","https://openalex.org/W4309341961","https://openalex.org/W4320037506","https://openalex.org/W4320477463","https://openalex.org/W4321211405","https://openalex.org/W4379525541","https://openalex.org/W6644253384","https://openalex.org/W6744966566"],"related_works":["https://openalex.org/W2391752819","https://openalex.org/W2362256999","https://openalex.org/W2470744056","https://openalex.org/W2007245702","https://openalex.org/W2313539504","https://openalex.org/W2048873866","https://openalex.org/W1584105255","https://openalex.org/W2016523436","https://openalex.org/W2052573303","https://openalex.org/W2066436349"],"abstract_inverted_index":{"In":[0],"nature,":[1],"symmetry":[2,8],"is":[3],"all":[4],"around":[5],"us.":[6],"The":[7,55,96],"framework":[9],"represents":[10],"integer":[11],"partial":[12],"differential":[13],"equations":[14],"and":[15,36,51,64,108,127],"their":[16],"fractional":[17,59],"order":[18],"in":[19,91,103],"the":[20,37,44,58,66,82,85,89,105,111,121,130,139],"sense":[21],"of":[22,47,84,93,123,132],"Caputo":[23],"derivatives.":[24],"This":[25,74],"article":[26],"suggests":[27],"a":[28],"semi-analytical":[29],"approach":[30],"based":[31],"on":[32],"Aboodh":[33],"transform":[34],"(AT)":[35],"homotopy":[38],"perturbation":[39],"scheme":[40,126,134],"(HPS)":[41],"for":[42],"achieving":[43],"approximate":[45],"solution":[46],"time-fractional":[48],"porous":[49],"media":[50],"heat":[52],"transfer":[53],"equations.":[54],"AT":[56],"converts":[57],"problems":[60],"into":[61],"simple":[62],"ones":[63],"obtains":[65],"recurrence":[67,76],"relation":[68,77],"without":[69],"any":[70],"discretization":[71],"or":[72],"assumption.":[73],"nonlinear":[75],"can":[78],"be":[79],"decomposed":[80],"via":[81],"use":[83],"HPS":[86],"to":[87,119],"obtain":[88],"iterations":[90,107],"terms":[92],"series":[94],"solutions.":[95],"initial":[97],"conditions":[98],"play":[99],"an":[100],"important":[101],"role":[102],"determining":[104],"successive":[106],"yields":[109],"towards":[110],"exact":[112,140],"solution.":[113],"We":[114],"provide":[115],"some":[116],"numerical":[117],"applications":[118],"analyze":[120],"accuracy":[122],"this":[124],"proposed":[125],"show":[128],"that":[129],"performance":[131],"our":[133],"has":[135],"strong":[136],"agreement":[137],"with":[138],"results.":[141]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}
