{"id":"https://openalex.org/W4281550816","doi":"https://doi.org/10.1155/2022/4475491","title":"Euler\u2019s Numerical Method on Fractional DSEK Model under ABC Derivative","display_name":"Euler\u2019s Numerical Method on Fractional DSEK Model under ABC Derivative","publication_year":2022,"publication_date":"2022-01-01","ids":{"openalex":"https://openalex.org/W4281550816","doi":"https://doi.org/10.1155/2022/4475491"},"language":"en","primary_location":{"id":"doi:10.1155/2022/4475491","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2022/4475491","pdf_url":"https://downloads.hindawi.com/journals/complexity/2022/4475491.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/complexity/2022/4475491.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5061499682","display_name":"Fareeha Sami Khan","orcid":"https://orcid.org/0000-0002-4766-028X"},"institutions":[{"id":"https://openalex.org/I40597779","display_name":"Federal Urdu University","ror":"https://ror.org/02b52th27","country_code":"PK","type":"education","lineage":["https://openalex.org/I40597779"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Fareeha Sami Khan","raw_affiliation_strings":["Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300, Pakistan","1, Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300, Pakistan","institution_ids":["https://openalex.org/I40597779"]},{"raw_affiliation_string":"1, Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300","institution_ids":["https://openalex.org/I40597779"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074574938","display_name":"Muhammad Khalid","orcid":"https://orcid.org/0000-0002-5233-6505"},"institutions":[{"id":"https://openalex.org/I40597779","display_name":"Federal Urdu University","ror":"https://ror.org/02b52th27","country_code":"PK","type":"education","lineage":["https://openalex.org/I40597779"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"M. Khalid","raw_affiliation_strings":["Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300, Pakistan","1, Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300, Pakistan","institution_ids":["https://openalex.org/I40597779"]},{"raw_affiliation_string":"1, Department of Mathematical Sciences, Federal Urdu University of Arts, Science and Technology, University Road, Karachi 75300","institution_ids":["https://openalex.org/I40597779"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034511816","display_name":"Omar Bazighifan","orcid":"https://orcid.org/0000-0002-7251-9608"},"institutions":[{"id":"https://openalex.org/I319841309","display_name":"Hadhramout University","ror":"https://ror.org/02kv0px94","country_code":"YE","type":"education","lineage":["https://openalex.org/I319841309"]},{"id":"https://openalex.org/I4405263500","display_name":"Seiyun University","ror":"https://ror.org/01xp8dp03","country_code":"YE","type":"education","lineage":["https://openalex.org/I4405263500"]}],"countries":["YE"],"is_corresponding":true,"raw_author_name":"Omar Bazighifan","raw_affiliation_strings":["Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen","Department of Mathematics, Faculty of Science, Hadhramaut University, Hadhramaut, Al Mukalla 50512, Yemen","2, Department of Mathematics, Faculty of Science, Hadhramaut University, Hadhramaut, Al Mukalla 50512","Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512"],"raw_orcid":"https://orcid.org/0000-0002-7251-9608","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen","institution_ids":["https://openalex.org/I4405263500"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Hadhramaut University, Hadhramaut, Al Mukalla 50512, Yemen","institution_ids":["https://openalex.org/I319841309"]},{"raw_affiliation_string":"2, Department of Mathematics, Faculty of Science, Hadhramaut University, Hadhramaut, Al Mukalla 50512","institution_ids":["https://openalex.org/I319841309"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512","institution_ids":["https://openalex.org/I4405263500"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035334111","display_name":"A. El-Mesady","orcid":"https://orcid.org/0000-0003-2009-3905"},"institutions":[{"id":"https://openalex.org/I63601056","display_name":"Menoufia University","ror":"https://ror.org/05sjrb944","country_code":"EG","type":"education","lineage":["https://openalex.org/I63601056"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"A. El-Mesady","raw_affiliation_strings":["Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt","Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952"],"raw_orcid":"https://orcid.org/0000-0003-2009-3905","affiliations":[{"raw_affiliation_string":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt","institution_ids":["https://openalex.org/I63601056"]},{"raw_affiliation_string":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952","institution_ids":["https://openalex.org/I63601056"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5034511816"],"corresponding_institution_ids":["https://openalex.org/I319841309","https://openalex.org/I4405263500"],"apc_list":{"value":2760,"currency":"USD","value_usd":2760},"apc_paid":{"value":2760,"currency":"USD","value_usd":2760},"fwci":1.4143,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.78621039,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"2022","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","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/T10288","display_name":"Fractional Differential Equations Solutions","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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9987999796867371,"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9878000020980835,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.8962323069572449},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6880478858947754},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.6557970643043518},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5098994970321655},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.5010201930999756},{"id":"https://openalex.org/keywords/fixed-point-theorem","display_name":"Fixed-point theorem","score":0.48901495337486267},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4854796230792999},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.46810340881347656},{"id":"https://openalex.org/keywords/contraction-mapping","display_name":"Contraction mapping","score":0.43946048617362976},{"id":"https://openalex.org/keywords/contraction","display_name":"Contraction (grammar)","score":0.4298381209373474},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.42543715238571167},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4087095260620117},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1539348065853119}],"concepts":[{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.8962323069572449},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6880478858947754},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6557970643043518},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5098994970321655},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.5010201930999756},{"id":"https://openalex.org/C45962547","wikidata":"https://www.wikidata.org/wiki/Q1422068","display_name":"Fixed-point theorem","level":2,"score":0.48901495337486267},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4854796230792999},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.46810340881347656},{"id":"https://openalex.org/C55524764","wikidata":"https://www.wikidata.org/wiki/Q515173","display_name":"Contraction mapping","level":3,"score":0.43946048617362976},{"id":"https://openalex.org/C163415756","wikidata":"https://www.wikidata.org/wiki/Q126473","display_name":"Contraction (grammar)","level":2,"score":0.4298381209373474},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.42543715238571167},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4087095260620117},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1539348065853119},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C126322002","wikidata":"https://www.wikidata.org/wiki/Q11180","display_name":"Internal medicine","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1155/2022/4475491","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2022/4475491","pdf_url":"https://downloads.hindawi.com/journals/complexity/2022/4475491.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:805be4b3dda1456cb5d936b80eea22c0","is_oa":true,"landing_page_url":"https://doaj.org/article/805be4b3dda1456cb5d936b80eea22c0","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":"Complexity, Vol 2022 (2022)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2022/4475491","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2022/4475491","pdf_url":"https://downloads.hindawi.com/journals/complexity/2022/4475491.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4281550816.pdf","grobid_xml":"https://content.openalex.org/works/W4281550816.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1533006072","https://openalex.org/W1972551677","https://openalex.org/W1981981778","https://openalex.org/W1987711797","https://openalex.org/W1997848545","https://openalex.org/W2003874621","https://openalex.org/W2056542410","https://openalex.org/W2063679835","https://openalex.org/W2127836593","https://openalex.org/W2145401214","https://openalex.org/W2187913422","https://openalex.org/W2251268089","https://openalex.org/W2382056600","https://openalex.org/W2609419633","https://openalex.org/W2766701865","https://openalex.org/W2792479230","https://openalex.org/W2808745722","https://openalex.org/W2892606629","https://openalex.org/W2922747723","https://openalex.org/W2963376087","https://openalex.org/W2963641381","https://openalex.org/W2963855174","https://openalex.org/W2964166956","https://openalex.org/W3037393044","https://openalex.org/W3120189783","https://openalex.org/W4200227480","https://openalex.org/W4224057846","https://openalex.org/W4231537439","https://openalex.org/W4255430550","https://openalex.org/W4383739001","https://openalex.org/W6979976599"],"related_works":["https://openalex.org/W43907689","https://openalex.org/W175278553","https://openalex.org/W4391096285","https://openalex.org/W1992609729","https://openalex.org/W4384932940","https://openalex.org/W2302784006","https://openalex.org/W4389349827","https://openalex.org/W1989319665","https://openalex.org/W2141092213","https://openalex.org/W329565398"],"abstract_inverted_index":{"In":[0],"this":[1,50],"paper,":[2],"DSEK":[3,46,79,104],"model":[4],"with":[5,96],"fractional":[6,24,45,78,86,98,103,121,129],"derivatives":[7],"of":[8,21,41,138],"the":[9,22,37,44,77,112,117,156,162],"Atangana\u2010Baleanu":[10],"Caputo":[11],"(ABC)":[12],"is":[13,73,90],"proposed.":[14],"This":[15],"paper":[16],"gives":[17],"a":[18],"brief":[19],"overview":[20],"ABC":[23,97],"derivative":[25,99],"and":[26,39,60,67,100,144],"its":[27],"attributes.":[28],"Fixed":[29],"point":[30],"theory":[31],"has":[32],"been":[33,133],"used":[34],"to":[35,49,75,110,120,161],"establish":[36],"uniqueness":[38],"existence":[40],"solutions":[42,126],"for":[43,92,115,135],"model.":[47,105],"According":[48],"theory,":[51],"we":[52],"will":[53,152],"define":[54],"two":[55],"operators":[56],"based":[57],"on":[58,102],"Lipschitzian":[59],"prove":[61,76],"that":[62,158],"they":[63],"are":[64],"contraction":[65],"mapping":[66],"relatively":[68],"compact.":[69],"Ulam\u2010Hyers":[70],"stability":[71,81],"theorem":[72],"implemented":[74,101],"model\u2019s":[80],"in":[82],"Banach":[83],"space.":[84],"Also,":[85],"Euler\u2019s":[87,130],"numerical":[88,125],"method":[89,114,131],"derived":[91],"initial":[93],"value":[94],"problems":[95],"The":[106,124],"symmetric":[107],"properties":[108],"contribute":[109],"determining":[111],"appropriate":[113],"finding":[116],"correct":[118],"solution":[119],"differential":[122],"equations.":[123],"generated":[127],"using":[128],"have":[132],"plotted":[134],"different":[136,145],"values":[137],"\u03b1":[139,141],"where":[140],"\u2208":[142],"(0,1]":[143],"step":[146,163],"sizes":[147],"h":[148,165],".":[149,166],"Result":[150],"discussion":[151],"be":[153],"given,":[154],"describing":[155],"changes":[157],"occur":[159],"due":[160],"size":[164]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}