{"id":"https://openalex.org/W4283728672","doi":"https://doi.org/10.3390/sym14071343","title":"Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis","display_name":"Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis","publication_year":2022,"publication_date":"2022-06-29","ids":{"openalex":"https://openalex.org/W4283728672","doi":"https://doi.org/10.3390/sym14071343"},"language":"en","primary_location":{"id":"doi:10.3390/sym14071343","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14071343","pdf_url":"https://www.mdpi.com/2073-8994/14/7/1343/pdf?version=1656490602","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/2073-8994/14/7/1343/pdf?version=1656490602","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079171037","display_name":"Samsul Ariffin Abdul Karim","orcid":"https://orcid.org/0000-0001-6518-6705"},"institutions":[{"id":"https://openalex.org/I161371597","display_name":"Universiti of Malaysia Sabah","ror":"https://ror.org/040v70252","country_code":"MY","type":"education","lineage":["https://openalex.org/I161371597"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Samsul Ariffin Abdul Karim","raw_affiliation_strings":["Software Engineering Programme, Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu 88400, Malaysia"],"raw_orcid":"https://orcid.org/0000-0001-6518-6705","affiliations":[{"raw_affiliation_string":"Software Engineering Programme, Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu 88400, Malaysia","institution_ids":["https://openalex.org/I161371597"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070644190","display_name":"Faheem Khan","orcid":"https://orcid.org/0000-0003-1520-4690"},"institutions":[{"id":"https://openalex.org/I174731842","display_name":"Islamia University of Bahawalpur","ror":"https://ror.org/002rc4w13","country_code":"PK","type":"education","lineage":["https://openalex.org/I174731842"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Faheem Khan","raw_affiliation_strings":["Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan"],"raw_orcid":"https://orcid.org/0000-0003-1520-4690","affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan","institution_ids":["https://openalex.org/I174731842"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100713256","display_name":"Muhammad Abdul Basit","orcid":"https://orcid.org/0009-0005-7805-5938"},"institutions":[{"id":"https://openalex.org/I160968435","display_name":"University of Sargodha","ror":"https://ror.org/0086rpr26","country_code":"PK","type":"education","lineage":["https://openalex.org/I160968435"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Basit","raw_affiliation_strings":["Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan","institution_ids":["https://openalex.org/I160968435"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5079171037"],"corresponding_institution_ids":["https://openalex.org/I161371597"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.3758,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.78242788,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"14","issue":"7","first_page":"1343","last_page":"1343"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9994999766349792,"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":0.9994999766349792,"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":0.9976999759674072,"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/T12508","display_name":"Functional Equations Stability Results","score":0.9908999800682068,"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":0.7397469282150269},{"id":"https://openalex.org/keywords/bernstein-polynomial","display_name":"Bernstein polynomial","score":0.688498854637146},{"id":"https://openalex.org/keywords/volterra-integral-equation","display_name":"Volterra integral equation","score":0.6868434548377991},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.6342654824256897},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6226231455802917},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.5937486886978149},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5594162940979004},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.5575444102287292},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.47283703088760376},{"id":"https://openalex.org/keywords/numerical-stability","display_name":"Numerical stability","score":0.4460245668888092},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.4445369839668274},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4347638487815857},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.41381973028182983},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.41144710779190063},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.11977481842041016},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11610734462738037},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.06324046850204468},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.061193883419036865}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7397469282150269},{"id":"https://openalex.org/C34179328","wikidata":"https://www.wikidata.org/wiki/Q826841","display_name":"Bernstein polynomial","level":2,"score":0.688498854637146},{"id":"https://openalex.org/C21965488","wikidata":"https://www.wikidata.org/wiki/Q2632613","display_name":"Volterra integral equation","level":3,"score":0.6868434548377991},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.6342654824256897},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6226231455802917},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.5937486886978149},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5594162940979004},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.5575444102287292},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47283703088760376},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.4460245668888092},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4445369839668274},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4347638487815857},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.41381973028182983},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.41144710779190063},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.11977481842041016},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11610734462738037},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.06324046850204468},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.061193883419036865},{"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/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14071343","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14071343","pdf_url":"https://www.mdpi.com/2073-8994/14/7/1343/pdf?version=1656490602","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:43a77dc68a434e358b15aaddb3c686af","is_oa":true,"landing_page_url":"https://doaj.org/article/43a77dc68a434e358b15aaddb3c686af","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":"Symmetry, Vol 14, Iss 7, p 1343 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/7/1343/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14071343","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; Volume 14; Issue 7; Pages: 1343","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14071343","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14071343","pdf_url":"https://www.mdpi.com/2073-8994/14/7/1343/pdf?version=1656490602","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":[{"id":"https://openalex.org/F4320323704","display_name":"Universiti Malaysia Sabah","ror":"https://ror.org/040v70252"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4283728672.pdf","grobid_xml":"https://content.openalex.org/works/W4283728672.grobid-xml"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W1948238714","https://openalex.org/W1972915717","https://openalex.org/W1976700083","https://openalex.org/W1984230568","https://openalex.org/W1992586393","https://openalex.org/W1996079888","https://openalex.org/W2019739960","https://openalex.org/W2020498805","https://openalex.org/W2028011742","https://openalex.org/W2044856674","https://openalex.org/W2048739721","https://openalex.org/W2061505012","https://openalex.org/W2065170087","https://openalex.org/W2076449788","https://openalex.org/W2080517481","https://openalex.org/W2083294239","https://openalex.org/W2144181355","https://openalex.org/W2341016936","https://openalex.org/W2536379864","https://openalex.org/W2751974595","https://openalex.org/W2896814518","https://openalex.org/W2984533924","https://openalex.org/W3112632525","https://openalex.org/W3198857156","https://openalex.org/W6640870746","https://openalex.org/W6644678197","https://openalex.org/W6703942286"],"related_works":["https://openalex.org/W2006021087","https://openalex.org/W4388522829","https://openalex.org/W2519730419","https://openalex.org/W2282343183","https://openalex.org/W1996416189","https://openalex.org/W2182117216","https://openalex.org/W1635536058","https://openalex.org/W39027593","https://openalex.org/W2021440751","https://openalex.org/W3174665301"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3],"new":[4],"numerical":[5,33,48,118],"technique":[6,77],"is":[7,62,124],"introduced":[8],"to":[9,30,42,105,126],"find":[10,31],"the":[11,14,32,47,51,66,70,79,85,107,110,121],"solution":[12,49,119],"of":[13,16,27,35,50,53,82,87,109],"system":[15,52,81,86],"Volterra":[17,54],"integral":[18,38,55],"equations":[19,39,56,83,89],"based":[20],"on":[21,57],"symmetric":[22],"Bernstein":[23,28,72],"polynomials.":[24],"The":[25,75,113],"use":[26],"polynomials":[29],"solutions":[34],"differential":[36],"and":[37,117],"increased":[40],"due":[41],"its":[43,128],"fast":[44,129],"convergence.":[45,130],"Here,":[46],"any":[58,96],"finite":[59],"interval":[60],"[m,n]":[61],"obtained":[63],"by":[64,94],"replacing":[65],"unknown":[67],"functions":[68],"with":[69],"generalized":[71],"basis":[73],"functions.":[74],"proposed":[76],"converts":[78],"given":[80,111],"into":[84],"algebraic":[88],"which":[90],"can":[91],"be":[92],"solved":[93],"using":[95],"standard":[97],"rule.":[98],"Further,":[99],"Hyers\u2013Ulam":[100],"stability":[101,108],"criteria":[102],"are":[103],"used":[104],"check":[106],"technique.":[112],"comparison":[114],"between":[115],"exact":[116],"for":[120],"distinct":[122],"nodes":[123],"demonstrated":[125],"show":[127]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}
