{"id":"https://openalex.org/W4200147366","doi":"https://doi.org/10.1155/2021/6316477","title":"Langevin Equations with Generalized Proportional Hadamard\u2013Caputo Fractional Derivative","display_name":"Langevin Equations with Generalized Proportional Hadamard\u2013Caputo Fractional Derivative","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W4200147366","doi":"https://doi.org/10.1155/2021/6316477","pmid":"https://pubmed.ncbi.nlm.nih.gov/34925494"},"language":"en","primary_location":{"id":"doi:10.1155/2021/6316477","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2021/6316477","pdf_url":"https://downloads.hindawi.com/journals/cin/2021/6316477.pdf","source":{"id":"https://openalex.org/S72372694","display_name":"Computational Intelligence and Neuroscience","issn_l":"1687-5265","issn":["1687-5265","1687-5273"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Intelligence and Neuroscience","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://downloads.hindawi.com/journals/cin/2021/6316477.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007865861","display_name":"Mohamed A. Barakat","orcid":"https://orcid.org/0000-0003-4036-6368"},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]},{"id":"https://openalex.org/I72264486","display_name":"University of Tabuk","ror":"https://ror.org/04yej8x59","country_code":"SA","type":"education","lineage":["https://openalex.org/I72264486"]}],"countries":["EG","SA"],"is_corresponding":true,"raw_author_name":"M. A. Barakat","raw_affiliation_strings":["Department of Computer Science, College of Al Wajh, University of Tabuk, Tabuk, Saudi Arabia","Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524, Egypt","Department of Computer Science, College of Al Wajh, University of Tabuk, Tabuk","Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524"],"raw_orcid":"https://orcid.org/0000-0003-4036-6368","affiliations":[{"raw_affiliation_string":"Department of Computer Science, College of Al Wajh, University of Tabuk, Tabuk, Saudi Arabia","institution_ids":["https://openalex.org/I72264486"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524, Egypt","institution_ids":["https://openalex.org/I184834183"]},{"raw_affiliation_string":"Department of Computer Science, College of Al Wajh, University of Tabuk, Tabuk","institution_ids":["https://openalex.org/I72264486"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051829976","display_name":"Ahmed H. Soliman","orcid":"https://orcid.org/0000-0001-5732-0847"},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Ahmed H. Soliman","raw_affiliation_strings":["Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524, Egypt","Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524"],"raw_orcid":"https://orcid.org/0000-0001-5732-0847","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524, Egypt","institution_ids":["https://openalex.org/I184834183"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048312355","display_name":"Abd\u2010Allah Hyder","orcid":"https://orcid.org/0000-0001-9273-9512"},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]},{"id":"https://openalex.org/I82952536","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30","country_code":"SA","type":"education","lineage":["https://openalex.org/I82952536"]}],"countries":["EG","SA"],"is_corresponding":false,"raw_author_name":"Abd-Allah Hyder","raw_affiliation_strings":["Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo, Egypt","King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, Abha 61413, Saudi Arabia","Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo","King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, Abha 61413"],"raw_orcid":"https://orcid.org/0000-0001-9273-9512","affiliations":[{"raw_affiliation_string":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo, Egypt","institution_ids":["https://openalex.org/I184834183"]},{"raw_affiliation_string":"King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, Abha 61413, Saudi Arabia","institution_ids":["https://openalex.org/I82952536"]},{"raw_affiliation_string":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo","institution_ids":["https://openalex.org/I184834183"]},{"raw_affiliation_string":"King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, Abha 61413","institution_ids":["https://openalex.org/I82952536"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5007865861"],"corresponding_institution_ids":["https://openalex.org/I184834183","https://openalex.org/I72264486"],"apc_list":{"value":2550,"currency":"USD","value_usd":2550},"apc_paid":{"value":2550,"currency":"USD","value_usd":2550},"fwci":1.0135,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.74698795,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"2021","issue":"1","first_page":"6316477","last_page":"6316477"},"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":0.9997000098228455,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9937000274658203,"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/hadamard-transform","display_name":"Hadamard transform","score":0.7920874357223511},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.781444787979126},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6801566481590271},{"id":"https://openalex.org/keywords/langevin-equation","display_name":"Langevin equation","score":0.537390410900116},{"id":"https://openalex.org/keywords/fixed-point-theorem","display_name":"Fixed-point theorem","score":0.5335885286331177},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5245183706283569},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4966452717781067},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.48554345965385437},{"id":"https://openalex.org/keywords/contraction-principle","display_name":"Contraction principle","score":0.4712778925895691},{"id":"https://openalex.org/keywords/banach-fixed-point-theorem","display_name":"Banach fixed-point theorem","score":0.4709136188030243},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.46533599495887756},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.42665189504623413},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.4255763292312622},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15991583466529846},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.15810784697532654},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12014874815940857}],"concepts":[{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.7920874357223511},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.781444787979126},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6801566481590271},{"id":"https://openalex.org/C2777577648","wikidata":"https://www.wikidata.org/wiki/Q584537","display_name":"Langevin equation","level":2,"score":0.537390410900116},{"id":"https://openalex.org/C45962547","wikidata":"https://www.wikidata.org/wiki/Q1422068","display_name":"Fixed-point theorem","level":2,"score":0.5335885286331177},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5245183706283569},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4966452717781067},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.48554345965385437},{"id":"https://openalex.org/C2775958659","wikidata":"https://www.wikidata.org/wiki/Q5165690","display_name":"Contraction principle","level":3,"score":0.4712778925895691},{"id":"https://openalex.org/C2780549194","wikidata":"https://www.wikidata.org/wiki/Q220680","display_name":"Banach fixed-point theorem","level":3,"score":0.4709136188030243},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.46533599495887756},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.42665189504623413},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.4255763292312622},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15991583466529846},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.15810784697532654},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12014874815940857},{"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2021/6316477","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2021/6316477","pdf_url":"https://downloads.hindawi.com/journals/cin/2021/6316477.pdf","source":{"id":"https://openalex.org/S72372694","display_name":"Computational Intelligence and Neuroscience","issn_l":"1687-5265","issn":["1687-5265","1687-5273"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Intelligence and Neuroscience","raw_type":"journal-article"},{"id":"pmid:34925494","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/34925494","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational intelligence and neuroscience","raw_type":null},{"id":"pmh:oai:doaj.org/article:5e5a0e67e21f4b9da97e1fe191438f93","is_oa":true,"landing_page_url":"https://doaj.org/article/5e5a0e67e21f4b9da97e1fe191438f93","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":"Computational Intelligence and Neuroscience, Vol 2021 (2021)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:8674044","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/8674044","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Comput Intell Neurosci","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1155/2021/6316477","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2021/6316477","pdf_url":"https://downloads.hindawi.com/journals/cin/2021/6316477.pdf","source":{"id":"https://openalex.org/S72372694","display_name":"Computational Intelligence and Neuroscience","issn_l":"1687-5265","issn":["1687-5265","1687-5273"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Intelligence and Neuroscience","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5830182537","display_name":null,"funder_award_id":"RGP.1/68/42","funder_id":"https://openalex.org/F4320324433","funder_display_name":"King Khalid University"}],"funders":[{"id":"https://openalex.org/F4320324433","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4200147366.pdf","grobid_xml":"https://content.openalex.org/works/W4200147366.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W2003874621","https://openalex.org/W2010114544","https://openalex.org/W2025169577","https://openalex.org/W2111217632","https://openalex.org/W2132013227","https://openalex.org/W2914933876","https://openalex.org/W2916801737","https://openalex.org/W2940412213","https://openalex.org/W2962847206","https://openalex.org/W2963186934","https://openalex.org/W2981874317","https://openalex.org/W3011783530","https://openalex.org/W3014859463","https://openalex.org/W3021461492","https://openalex.org/W3036813548","https://openalex.org/W3120092315","https://openalex.org/W3149735771","https://openalex.org/W4240465921"],"related_works":["https://openalex.org/W2980485881","https://openalex.org/W3154148476","https://openalex.org/W2520698300","https://openalex.org/W3172638807","https://openalex.org/W4224318062","https://openalex.org/W2270341331","https://openalex.org/W2402357946","https://openalex.org/W2131451500","https://openalex.org/W4319083558","https://openalex.org/W2793924561"],"abstract_inverted_index":{"We":[0],"look":[1],"at":[2],"fractional":[3],"Langevin":[4],"equations":[5],"(FLEs)":[6],"with":[7],"generalized":[8],"proportional":[9],"Hadamard-Caputo":[10],"derivative":[11],"of":[12,39,76],"different":[13],"orders.":[14],"Moreover,":[15],"nonlocal":[16],"integrals":[17],"and":[18,36,44,62],"nonperiodic":[19],"boundary":[20],"conditions":[21],"are":[22,42,57],"considered":[23],"in":[24],"this":[25],"paper.":[26],"For":[27],"the":[28,31,40,74,77],"proposed":[29],"equations,":[30],"Hyres-Ulam":[32],"(HU)":[33],"stability,":[34],"existence,":[35],"uniqueness":[37],"(EU)":[38],"solution":[41],"defined":[43],"investigated.":[45],"In":[46],"implementing":[47],"our":[48],"results,":[49],"we":[50],"rely":[51],"on":[52],"two":[53],"important":[54],"theories":[55],"that":[56],"Krasnoselskii":[58],"fixed":[59],"point":[60],"theorem":[61],"Banach":[63],"contraction":[64],"principle.":[65],"Also,":[66],"an":[67],"application":[68],"example":[69],"is":[70],"given":[71],"to":[72],"bolster":[73],"accuracy":[75],"acquired":[78],"results.":[79]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":7}],"updated_date":"2026-07-27T08:26:11.824852","created_date":"2025-10-10T00:00:00"}