{"id":"https://openalex.org/W4287218597","doi":"https://doi.org/10.3390/sym14081514","title":"An Effective Projection Method for Solving a Coupled System of Fractional-Order Bagley\u2013Torvik Equations via Fractional Shifted Legendre Polynomials","display_name":"An Effective Projection Method for Solving a Coupled System of Fractional-Order Bagley\u2013Torvik Equations via Fractional Shifted Legendre Polynomials","publication_year":2022,"publication_date":"2022-07-25","ids":{"openalex":"https://openalex.org/W4287218597","doi":"https://doi.org/10.3390/sym14081514"},"language":"en","primary_location":{"id":"doi:10.3390/sym14081514","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14081514","pdf_url":"https://www.mdpi.com/2073-8994/14/8/1514/pdf?version=1658844096","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/2073-8994/14/8/1514/pdf?version=1658844096","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060461833","display_name":"Saeed Althubiti","orcid":"https://orcid.org/0000-0001-9434-7037"},"institutions":[{"id":"https://openalex.org/I179331831","display_name":"Taif University","ror":"https://ror.org/014g1a453","country_code":"SA","type":"education","lineage":["https://openalex.org/I179331831"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Saeed Althubiti","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia","institution_ids":["https://openalex.org/I179331831"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007422577","display_name":"Abdelaziz Mennouni","orcid":"https://orcid.org/0000-0002-0791-5866"},"institutions":[{"id":"https://openalex.org/I162489102","display_name":"University of Batna 1","ror":"https://ror.org/04hrbe508","country_code":"DZ","type":"education","lineage":["https://openalex.org/I162489102"]}],"countries":["DZ"],"is_corresponding":true,"raw_author_name":"Abdelaziz Mennouni","raw_affiliation_strings":["Department of Mathematics, LTM, University of Batna 2, Mostefa Ben Boula\u00efd, Fesdis, Batna 05078, Algeria"],"raw_orcid":"https://orcid.org/0000-0002-0791-5866","affiliations":[{"raw_affiliation_string":"Department of Mathematics, LTM, University of Batna 2, Mostefa Ben Boula\u00efd, Fesdis, Batna 05078, Algeria","institution_ids":["https://openalex.org/I162489102"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5007422577"],"corresponding_institution_ids":["https://openalex.org/I162489102"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.4127,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.54570655,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"14","issue":"8","first_page":"1514","last_page":"1514"},"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":0.9973000288009644,"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9939000010490417,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/conformable-matrix","display_name":"Conformable matrix","score":0.6844022274017334},{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.6548395156860352},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.595421552658081},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5786842107772827},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.5680224895477295},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5647811889648438},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4811321794986725},{"id":"https://openalex.org/keywords/fractional-programming","display_name":"Fractional programming","score":0.46336740255355835},{"id":"https://openalex.org/keywords/work","display_name":"Work (physics)","score":0.42382222414016724},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3536544442176819},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.33609843254089355},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11600568890571594},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.09172388911247253},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.07996267080307007}],"concepts":[{"id":"https://openalex.org/C86072612","wikidata":"https://www.wikidata.org/wiki/Q5160239","display_name":"Conformable matrix","level":2,"score":0.6844022274017334},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.6548395156860352},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.595421552658081},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5786842107772827},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5680224895477295},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5647811889648438},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4811321794986725},{"id":"https://openalex.org/C169069869","wikidata":"https://www.wikidata.org/wiki/Q5477546","display_name":"Fractional programming","level":4,"score":0.46336740255355835},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.42382222414016724},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3536544442176819},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33609843254089355},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11600568890571594},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.09172388911247253},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.07996267080307007},{"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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C115527620","wikidata":"https://www.wikidata.org/wiki/Q769909","display_name":"Nonlinear programming","level":3,"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14081514","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14081514","pdf_url":"https://www.mdpi.com/2073-8994/14/8/1514/pdf?version=1658844096","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:76836383be3645acbf10bb7abffb6457","is_oa":true,"landing_page_url":"https://doaj.org/article/76836383be3645acbf10bb7abffb6457","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 8, p 1514 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/8/1514/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14081514","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","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14081514","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14081514","pdf_url":"https://www.mdpi.com/2073-8994/14/8/1514/pdf?version=1658844096","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":[{"id":"https://openalex.org/G710107805","display_name":null,"funder_award_id":"TURSP-2020/320","funder_id":"https://openalex.org/F4320323722","funder_display_name":"Taif University"},{"id":"https://openalex.org/G7299194679","display_name":null,"funder_award_id":"TURSP-2020","funder_id":"https://openalex.org/F4320323722","funder_display_name":"Taif University"}],"funders":[{"id":"https://openalex.org/F4320323722","display_name":"Taif University","ror":"https://ror.org/014g1a453"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4287218597.pdf","grobid_xml":"https://content.openalex.org/works/W4287218597.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1559244587","https://openalex.org/W2033827050","https://openalex.org/W2058292416","https://openalex.org/W2060477697","https://openalex.org/W2083559699","https://openalex.org/W2224569190","https://openalex.org/W2537664671","https://openalex.org/W2789913125","https://openalex.org/W2889078787","https://openalex.org/W3099140416","https://openalex.org/W4226463608","https://openalex.org/W6658727143"],"related_works":["https://openalex.org/W3158217983","https://openalex.org/W4248103533","https://openalex.org/W3046588275","https://openalex.org/W3085259849","https://openalex.org/W2792191619","https://openalex.org/W3046886605","https://openalex.org/W3008631921","https://openalex.org/W4313364931","https://openalex.org/W4302307873","https://openalex.org/W2898762024"],"abstract_inverted_index":{"This":[0],"work":[1],"proposes":[2],"a":[3],"valuable":[4],"and":[5],"successful":[6],"strategy":[7],"for":[8],"approximating":[9],"the":[10,13,31,70,85],"solutions":[11],"to":[12,29,95],"Bagley\u2013Torvik":[14,36,73,78],"system,":[15],"which":[16,80],"plays":[17],"an":[18,45],"essential":[19],"role":[20],"in":[21,41],"fractional":[22,72],"calculus.":[23],"The":[24,35,59],"Caputo":[25],"sense":[26],"is":[27,38],"used":[28],"derive":[30],"basic":[32],"conformable":[33],"fractional.":[34],"problem":[37],"numerically":[39],"solved":[40,83,94],"this":[42,51],"study":[43],"using":[44,84],"effective":[46],"symmetric":[47],"projection":[48],"method.":[49,87],"From":[50],"symmetry,":[52],"there":[53],"are":[54,93],"some":[55],"interesting":[56],"original":[57],"results.":[58],"proposed":[60],"approach":[61],"has":[62],"two":[63,76,89],"key":[64],"benefits.":[65],"We":[66],"began":[67],"by":[68],"converting":[69],"connected":[71],"equations":[74],"into":[75],"fractional-order":[77],"equations,":[79],"we":[81],"then":[82],"current":[86],"Second,":[88],"linear":[90],"equation":[91],"systems":[92],"obtain":[96],"approximate":[97],"solutions.":[98]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}