{"id":"https://openalex.org/W2975683732","doi":"https://doi.org/10.1080/00207160.2019.1673892","title":"A perturbation-based approach for solving fractional-order Volterra\u2013Fredholm integro differential equations and its convergence analysis","display_name":"A perturbation-based approach for solving fractional-order Volterra\u2013Fredholm integro differential equations and its convergence analysis","publication_year":2019,"publication_date":"2019-09-28","ids":{"openalex":"https://openalex.org/W2975683732","doi":"https://doi.org/10.1080/00207160.2019.1673892","mag":"2975683732"},"language":"en","primary_location":{"id":"doi:10.1080/00207160.2019.1673892","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1673892","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://hdl.handle.net/10366/157027","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080855274","display_name":"Pratibhamoy Das","orcid":"https://orcid.org/0000-0001-5095-0360"},"institutions":[{"id":"https://openalex.org/I132153292","display_name":"Indian Institute of Technology Patna","ror":"https://ror.org/01ft5vz71","country_code":"IN","type":"education","lineage":["https://openalex.org/I132153292"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Pratibhamoy Das","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Technology, Patna, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Technology, Patna, India","institution_ids":["https://openalex.org/I132153292"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066185540","display_name":"Subrata Rana","orcid":"https://orcid.org/0000-0002-5431-0974"},"institutions":[{"id":"https://openalex.org/I132153292","display_name":"Indian Institute of Technology Patna","ror":"https://ror.org/01ft5vz71","country_code":"IN","type":"education","lineage":["https://openalex.org/I132153292"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Subrata Rana","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Technology, Patna, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Technology, Patna, India","institution_ids":["https://openalex.org/I132153292"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005180650","display_name":"Higinio Ramos","orcid":"https://orcid.org/0000-0003-2791-6230"},"institutions":[{"id":"https://openalex.org/I184999862","display_name":"Universidad de Salamanca","ror":"https://ror.org/02f40zc51","country_code":"ES","type":"education","lineage":["https://openalex.org/I184999862"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Higinio Ramos","raw_affiliation_strings":["Department of Applied Mathematics, University of Salamanca, Salamanca, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, University of Salamanca, Salamanca, Spain","institution_ids":["https://openalex.org/I184999862"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5005180650"],"corresponding_institution_ids":["https://openalex.org/I184999862"],"apc_list":null,"apc_paid":null,"fwci":4.6054,"has_fulltext":false,"cited_by_count":153,"citation_normalized_percentile":{"value":0.95787832,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"97","issue":"10","first_page":"1994","last_page":"2014"},"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.996999979019165,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9851999878883362,"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/mathematics","display_name":"Mathematics","score":0.7983551025390625},{"id":"https://openalex.org/keywords/volterra-equations","display_name":"Volterra equations","score":0.5629587173461914},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5335389375686646},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5319661498069763},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.5293247103691101},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5205137133598328},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.519964337348938},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5089552402496338},{"id":"https://openalex.org/keywords/volterra-integral-equation","display_name":"Volterra integral equation","score":0.4763091802597046},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.46883076429367065},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.4680739641189575},{"id":"https://openalex.org/keywords/fredholm-integral-equation","display_name":"Fredholm integral equation","score":0.4349508583545685},{"id":"https://openalex.org/keywords/integro-differential-equation","display_name":"Integro-differential equation","score":0.4279831051826477},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.2756039500236511},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0857844352722168},{"id":"https://openalex.org/keywords/first-order-partial-differential-equation","display_name":"First-order partial differential equation","score":0.07919126749038696}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7983551025390625},{"id":"https://openalex.org/C2983756050","wikidata":"https://www.wikidata.org/wiki/Q150052","display_name":"Volterra equations","level":3,"score":0.5629587173461914},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5335389375686646},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5319661498069763},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.5293247103691101},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5205137133598328},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.519964337348938},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5089552402496338},{"id":"https://openalex.org/C21965488","wikidata":"https://www.wikidata.org/wiki/Q2632613","display_name":"Volterra integral equation","level":3,"score":0.4763091802597046},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.46883076429367065},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.4680739641189575},{"id":"https://openalex.org/C89285879","wikidata":"https://www.wikidata.org/wiki/Q2991898","display_name":"Fredholm integral equation","level":3,"score":0.4349508583545685},{"id":"https://openalex.org/C70985411","wikidata":"https://www.wikidata.org/wiki/Q2380676","display_name":"Integro-differential equation","level":4,"score":0.4279831051826477},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.2756039500236511},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0857844352722168},{"id":"https://openalex.org/C64057670","wikidata":"https://www.wikidata.org/wiki/Q4381442","display_name":"First-order partial differential equation","level":3,"score":0.07919126749038696},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1080/00207160.2019.1673892","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1673892","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:gredos.usal.es:10366/157027","is_oa":true,"landing_page_url":"http://hdl.handle.net/10366/157027","pdf_url":null,"source":{"id":"https://openalex.org/S4377196290","display_name":"Gredos (University of Salamanca)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I184999862","host_organization_name":"Universidad de Salamanca","host_organization_lineage":["https://openalex.org/I184999862"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:gredos.usal.es:10366/157027","is_oa":true,"landing_page_url":"http://hdl.handle.net/10366/157027","pdf_url":null,"source":{"id":"https://openalex.org/S4377196290","display_name":"Gredos (University of Salamanca)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I184999862","host_organization_name":"Universidad de Salamanca","host_organization_lineage":["https://openalex.org/I184999862"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G524484662","display_name":null,"funder_award_id":"ECR/2017/001973","funder_id":"https://openalex.org/F4320334771","funder_display_name":"Science and Engineering Research Board"}],"funders":[{"id":"https://openalex.org/F4320334771","display_name":"Science and Engineering Research Board","ror":"https://ror.org/03ffdsr55"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W820976346","https://openalex.org/W1553338987","https://openalex.org/W1584611818","https://openalex.org/W1999651452","https://openalex.org/W2007111134","https://openalex.org/W2013395650","https://openalex.org/W2015256935","https://openalex.org/W2021629544","https://openalex.org/W2041079718","https://openalex.org/W2055701161","https://openalex.org/W2081754386","https://openalex.org/W2107207615","https://openalex.org/W2163017860","https://openalex.org/W2171377300","https://openalex.org/W2435011107","https://openalex.org/W2488968130","https://openalex.org/W2558664449","https://openalex.org/W2571132309","https://openalex.org/W2749568566","https://openalex.org/W2771262646","https://openalex.org/W2783832701","https://openalex.org/W2800550366","https://openalex.org/W2807325390","https://openalex.org/W2811445391","https://openalex.org/W2892204294","https://openalex.org/W4241819982"],"related_works":["https://openalex.org/W4249149186","https://openalex.org/W2082102072","https://openalex.org/W1541040149","https://openalex.org/W128596308","https://openalex.org/W2038297679","https://openalex.org/W2184249577","https://openalex.org/W1539637428","https://openalex.org/W2906624979","https://openalex.org/W3200459157","https://openalex.org/W3097583914"],"abstract_inverted_index":{"The":[0],"present":[1,29,65,95,172],"work":[2],"considers":[3],"the":[4,17,33,51,61,64,71,99,117,121,130,142,152,157,171,179],"approximation":[5,97,140],"of":[6,8,11,32,50,60,63,82,98,101,106,120,141],"solutions":[7,154],"a":[9,43,57,80,107,134],"type":[10],"fractional-order":[12,34],"Volterra\u2013Fredholm":[13],"integro-differential":[14],"equations,":[15],"where":[16],"fractional":[18,163],"derivative":[19],"is":[20,124,133,149,174],"introduced":[21],"in":[22],"Caputo":[23],"sense.":[24],"In":[25,166],"addition,":[26,167],"we":[27,41,68,94,115,168],"also":[28,54],"several":[30],"applications":[31],"differential":[35,164],"equations":[36,83],"and":[37,48,53,182],"integral":[38],"equations.":[39,165],"Here,":[40],"provide":[42],"sufficient":[44],"condition":[45],"for":[46,138,145,161],"existence":[47],"uniqueness":[49],"solution":[52,62,100],"obtain":[55],"an":[56,96],"priori":[58],"bound":[59],"problem.":[66],"Then,":[67],"discuss":[69,116],"about":[70],"higher-order":[72,146,162],"model":[73],"equation":[74],"which":[75],"can":[76],"be":[77],"written":[78],"as":[79],"system":[81],"whose":[84],"orders":[85],"are":[86],"less":[87,184],"than":[88],"or":[89],"equal":[90],"to":[91,156,178],"one.":[92],"Next,":[93],"this":[102],"problem":[103],"by":[104],"means":[105],"perturbation":[108],"approach":[109],"based":[110],"on":[111],"homotopy":[112],"analysis.":[113],"Also,":[114],"convergence":[118],"analysis":[119],"method.":[122],"It":[123,148],"observed":[125],"through":[126],"different":[127],"examples":[128],"that":[129,151,170],"adopted":[131],"strategy":[132],"very":[135],"effective":[136,176],"one":[137],"good":[139],"solution,":[143,159],"even":[144,160],"problems.":[147],"shown":[150],"approximate":[153],"converge":[155],"exact":[158],"show":[169],"method":[173,181],"highly":[175],"compared":[177],"existed":[180],"produces":[183],"error.":[185]},"counts_by_year":[{"year":2026,"cited_by_count":17},{"year":2025,"cited_by_count":46},{"year":2024,"cited_by_count":27},{"year":2023,"cited_by_count":18},{"year":2022,"cited_by_count":17},{"year":2021,"cited_by_count":19},{"year":2020,"cited_by_count":9}],"updated_date":"2026-08-15T07:11:24.734988","created_date":"2025-10-10T00:00:00"}