{"id":"https://openalex.org/W4388540589","doi":"https://doi.org/10.1155/2023/7210126","title":"Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator","display_name":"Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator","publication_year":2023,"publication_date":"2023-11-09","ids":{"openalex":"https://openalex.org/W4388540589","doi":"https://doi.org/10.1155/2023/7210126"},"language":"en","primary_location":{"id":"doi:10.1155/2023/7210126","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2023/7210126","pdf_url":"https://downloads.hindawi.com/journals/complexity/2023/7210126.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"],"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":"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/2023/7210126.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045227516","display_name":"Kamran Kamran","orcid":"https://orcid.org/0000-0001-8217-3308"},"institutions":[{"id":"https://openalex.org/I206573129","display_name":"Islamia College University","ror":"https://ror.org/02p2c1595","country_code":"PK","type":"education","lineage":["https://openalex.org/I206573129"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Kamran","raw_affiliation_strings":["Department of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I206573129"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5093232238","display_name":"Aisha Subhan","orcid":"https://orcid.org/0009-0007-8419-4402"},"institutions":[{"id":"https://openalex.org/I206573129","display_name":"Islamia College University","ror":"https://ror.org/02p2c1595","country_code":"PK","type":"education","lineage":["https://openalex.org/I206573129"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Aisha Subhan","raw_affiliation_strings":["Department of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I206573129"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073895350","display_name":"Kamal Shah","orcid":"https://orcid.org/0000-0002-8851-4844"},"institutions":[{"id":"https://openalex.org/I142024983","display_name":"Prince Sultan University","ror":"https://ror.org/053mqrf26","country_code":"SA","type":"education","lineage":["https://openalex.org/I142024983"]},{"id":"https://openalex.org/I56306041","display_name":"Lebanese American University","ror":"https://ror.org/00hqkan37","country_code":"LB","type":"education","lineage":["https://openalex.org/I56306041"]}],"countries":["LB","SA"],"is_corresponding":false,"raw_author_name":"Kamal Shah","raw_affiliation_strings":["Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon","Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon","institution_ids":["https://openalex.org/I56306041"]},{"raw_affiliation_string":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia","institution_ids":["https://openalex.org/I142024983"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029433594","display_name":"Suhad Subhi Aiadi","orcid":"https://orcid.org/0000-0002-0018-9873"},"institutions":[{"id":"https://openalex.org/I142024983","display_name":"Prince Sultan University","ror":"https://ror.org/053mqrf26","country_code":"SA","type":"education","lineage":["https://openalex.org/I142024983"]},{"id":"https://openalex.org/I206573129","display_name":"Islamia College University","ror":"https://ror.org/02p2c1595","country_code":"PK","type":"education","lineage":["https://openalex.org/I206573129"]}],"countries":["PK","SA"],"is_corresponding":false,"raw_author_name":"Suhad Subhi Aiadi","raw_affiliation_strings":["Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia","Department of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia","institution_ids":["https://openalex.org/I142024983"]},{"raw_affiliation_string":"Department of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I206573129"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021626938","display_name":"Nabil Mlaiki","orcid":"https://orcid.org/0000-0002-7986-886X"},"institutions":[{"id":"https://openalex.org/I142024983","display_name":"Prince Sultan University","ror":"https://ror.org/053mqrf26","country_code":"SA","type":"education","lineage":["https://openalex.org/I142024983"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Nabil Mlaiki","raw_affiliation_strings":["Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia","institution_ids":["https://openalex.org/I142024983"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006465720","display_name":"Fahad Alotaibi","orcid":"https://orcid.org/0000-0001-9885-1775"},"institutions":[{"id":"https://openalex.org/I185163786","display_name":"King Abdulaziz University","ror":"https://ror.org/02ma4wv74","country_code":"SA","type":"education","lineage":["https://openalex.org/I185163786"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Fahad M. Alotaibi","raw_affiliation_strings":["Department of Information Systems, Faculty of Computing and Information Technology (FCIT), King Abdulaziz University, Jeddah 34025, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Information Systems, Faculty of Computing and Information Technology (FCIT), King Abdulaziz University, Jeddah 34025, Saudi Arabia","institution_ids":["https://openalex.org/I185163786"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5021626938"],"corresponding_institution_ids":["https://openalex.org/I142024983"],"apc_list":{"value":2300,"currency":"USD","value_usd":2300},"apc_paid":{"value":2300,"currency":"USD","value_usd":2300},"fwci":0.6484,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.65709156,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"2023","issue":null,"first_page":"1","last_page":"22"},"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9980999827384949,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9979000091552734,"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/mathematics","display_name":"Mathematics","score":0.7244008183479309},{"id":"https://openalex.org/keywords/laplace-transform","display_name":"Laplace transform","score":0.6129967570304871},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5753289461135864},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.570968747138977},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.5606379508972168},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.523919403553009},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4831625819206238},{"id":"https://openalex.org/keywords/laplaces-equation","display_name":"Laplace's equation","score":0.4719212055206299},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.44173464179039},{"id":"https://openalex.org/keywords/differential-operator","display_name":"Differential operator","score":0.424445778131485},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3691573739051819}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7244008183479309},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.6129967570304871},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5753289461135864},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.570968747138977},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.5606379508972168},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.523919403553009},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4831625819206238},{"id":"https://openalex.org/C70615421","wikidata":"https://www.wikidata.org/wiki/Q339444","display_name":"Laplace's equation","level":3,"score":0.4719212055206299},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.44173464179039},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.424445778131485},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3691573739051819},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1155/2023/7210126","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2023/7210126","pdf_url":"https://downloads.hindawi.com/journals/complexity/2023/7210126.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"],"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":"Complexity","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:e04f2d80f92e4e4b9901787aea767dc1","is_oa":true,"landing_page_url":"https://doaj.org/article/e04f2d80f92e4e4b9901787aea767dc1","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Complexity, Vol 2023 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2023/7210126","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2023/7210126","pdf_url":"https://downloads.hindawi.com/journals/complexity/2023/7210126.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"],"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":"Complexity","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322322","display_name":"King Abdulaziz University","ror":"https://ror.org/02ma4wv74"},{"id":"https://openalex.org/F4320327702","display_name":"Prince Sultan University","ror":"https://ror.org/053mqrf26"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4388540589.pdf"},"referenced_works_count":56,"referenced_works":["https://openalex.org/W1922149384","https://openalex.org/W1965557233","https://openalex.org/W1965914676","https://openalex.org/W1966688129","https://openalex.org/W1969446688","https://openalex.org/W1982261380","https://openalex.org/W1986135809","https://openalex.org/W1995453571","https://openalex.org/W2005669138","https://openalex.org/W2006205876","https://openalex.org/W2007072278","https://openalex.org/W2034340583","https://openalex.org/W2057342974","https://openalex.org/W2064244612","https://openalex.org/W2083372910","https://openalex.org/W2083828525","https://openalex.org/W2086783890","https://openalex.org/W2135094424","https://openalex.org/W2145796457","https://openalex.org/W2164971571","https://openalex.org/W2337776243","https://openalex.org/W2506586217","https://openalex.org/W2518477744","https://openalex.org/W2540514250","https://openalex.org/W2544152878","https://openalex.org/W2578400478","https://openalex.org/W2606385482","https://openalex.org/W2609978083","https://openalex.org/W2790041383","https://openalex.org/W2791375747","https://openalex.org/W2797499680","https://openalex.org/W2809354586","https://openalex.org/W2890721927","https://openalex.org/W2943950752","https://openalex.org/W2944787851","https://openalex.org/W2963540395","https://openalex.org/W2969811770","https://openalex.org/W2970328856","https://openalex.org/W2971212185","https://openalex.org/W2982881409","https://openalex.org/W2995310942","https://openalex.org/W3004354476","https://openalex.org/W3015243030","https://openalex.org/W3023220629","https://openalex.org/W3035774213","https://openalex.org/W3043590186","https://openalex.org/W3048379687","https://openalex.org/W3110914724","https://openalex.org/W3129420110","https://openalex.org/W3161806095","https://openalex.org/W3211332687","https://openalex.org/W4200111179","https://openalex.org/W4200170779","https://openalex.org/W4220842969","https://openalex.org/W4309915471","https://openalex.org/W4319227002"],"related_works":["https://openalex.org/W2013332237","https://openalex.org/W2295517574","https://openalex.org/W4375959275","https://openalex.org/W4382753044","https://openalex.org/W3024206168","https://openalex.org/W4317627639","https://openalex.org/W3005643231","https://openalex.org/W2930126963","https://openalex.org/W2041486430","https://openalex.org/W963546583"],"abstract_inverted_index":{"In":[0,47],"this":[1],"paper,":[2],"a":[3,27],"class":[4],"of":[5,22,41,140,158],"integrodifferential":[6,52,63,73,81,92],"equations":[7],"with":[8,26,54,65],"the":[9,17,23,38,51,55,66,71,79,84,91,95,100,108,114,118,122,141,159],"Caputo":[10,85,96],"fractal-fractional":[11,28,56],"derivative":[12],"is":[13,34,59,75],"considered.":[14],"We":[15],"study":[16],"exact":[18],"and":[19,44,134],"numerical":[20,135,138,156],"solutions":[21],"said":[24],"problem":[25],"differential":[29,57,68,86,97],"operator.":[30,69,87],"The":[31],"abovementioned":[32,160],"operator":[33,58,98],"arising":[35],"widely":[36],"in":[37,107],"mathematical":[39],"modeling":[40],"various":[42],"physical":[43],"biological":[45],"problems.":[46],"our":[48,168],"scheme,":[49],"first,":[50],"equation":[53,64,74,82,93,106],"converted":[60,77],"to":[61,78,103,146,166],"an":[62,104],"Riemann\u2013Liouville":[67],"Additionally,":[70],"obtained":[72,115],"then":[76],"equivalent":[80],"involving":[83],"Then,":[88],"we":[89,112,126,144,162],"convert":[90,113],"under":[94],"using":[99],"Laplace":[101,109,119,142],"transform":[102],"algebraic":[105],"space.":[110],"Finally,":[111],"solution":[116],"from":[117],"space":[120],"into":[121],"real":[123],"domain.":[124],"Moreover,":[125],"utilize":[127],"two":[128],"techniques":[129],"which":[130],"include":[131],"analytic":[132],"inversion":[133,139],"inversion.":[136],"For":[137],"transforms,":[143],"have":[145],"evaluate":[147],"five":[148],"methods.":[149],"Extensive":[150],"results":[151],"are":[152],"presented.":[153],"Furthermore,":[154],"for":[155],"illustration":[157],"methods,":[161],"consider":[163],"three":[164],"problems":[165],"demonstrate":[167],"results.":[169]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2023-11-10T00:00:00"}
