{"id":"https://openalex.org/W4224023123","doi":"https://doi.org/10.3390/axioms11040178","title":"Weighted Generalized Fractional Integration by Parts and the Euler\u2013Lagrange Equation","display_name":"Weighted Generalized Fractional Integration by Parts and the Euler\u2013Lagrange Equation","publication_year":2022,"publication_date":"2022-04-15","ids":{"openalex":"https://openalex.org/W4224023123","doi":"https://doi.org/10.3390/axioms11040178"},"language":"en","primary_location":{"id":"doi:10.3390/axioms11040178","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11040178","pdf_url":"https://www.mdpi.com/2075-1680/11/4/178/pdf?version=1650002877","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/11/4/178/pdf?version=1650002877","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003920258","display_name":"Houssine Zine","orcid":"https://orcid.org/0000-0002-8412-7783"},"institutions":[{"id":"https://openalex.org/I60858718","display_name":"University of Aveiro","ror":"https://ror.org/00nt41z93","country_code":"PT","type":"education","lineage":["https://openalex.org/I60858718"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Houssine Zine","raw_affiliation_strings":["Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"],"affiliations":[{"raw_affiliation_string":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal","institution_ids":["https://openalex.org/I60858718"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051424822","display_name":"El Mehdi Lotfi","orcid":"https://orcid.org/0000-0002-3604-3841"},"institutions":[{"id":"https://openalex.org/I99297268","display_name":"University of Hassan II Casablanca","ror":"https://ror.org/001q4kn48","country_code":"MA","type":"education","lineage":["https://openalex.org/I99297268"]}],"countries":["MA"],"is_corresponding":false,"raw_author_name":"El Mehdi Lotfi","raw_affiliation_strings":["Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M\u2019sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco","Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco"],"affiliations":[{"raw_affiliation_string":"Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M\u2019sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco","institution_ids":["https://openalex.org/I99297268"]},{"raw_affiliation_string":"Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco","institution_ids":["https://openalex.org/I99297268"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057131265","display_name":"Delfim F. M. Torres","orcid":"https://orcid.org/0000-0001-8641-2505"},"institutions":[{"id":"https://openalex.org/I60858718","display_name":"University of Aveiro","ror":"https://ror.org/00nt41z93","country_code":"PT","type":"education","lineage":["https://openalex.org/I60858718"]}],"countries":["PT"],"is_corresponding":true,"raw_author_name":"Delfim F. M. Torres","raw_affiliation_strings":["Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"],"affiliations":[{"raw_affiliation_string":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal","institution_ids":["https://openalex.org/I60858718"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090815657","display_name":"Noura Yousfi","orcid":"https://orcid.org/0000-0001-8435-5981"},"institutions":[{"id":"https://openalex.org/I99297268","display_name":"University of Hassan II Casablanca","ror":"https://ror.org/001q4kn48","country_code":"MA","type":"education","lineage":["https://openalex.org/I99297268"]}],"countries":["MA"],"is_corresponding":false,"raw_author_name":"Noura Yousfi","raw_affiliation_strings":["Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M\u2019sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco","Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco"],"affiliations":[{"raw_affiliation_string":"Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M\u2019sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco","institution_ids":["https://openalex.org/I99297268"]},{"raw_affiliation_string":"Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco","institution_ids":["https://openalex.org/I99297268"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5057131265"],"corresponding_institution_ids":["https://openalex.org/I60858718"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.4109,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.52909844,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"11","issue":"4","first_page":"178","last_page":"178"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9940000176429749,"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":0.9786999821662903,"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/applied-mathematics","display_name":"Applied mathematics","score":0.5426881313323975},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.542419970035553},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.5187819004058838},{"id":"https://openalex.org/keywords/euler\u2013lagrange-equation","display_name":"Euler\u2013Lagrange equation","score":0.4155493378639221},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3919851779937744},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3358473777770996},{"id":"https://openalex.org/keywords/lagrangian","display_name":"Lagrangian","score":0.19685545563697815},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.0478232204914093}],"concepts":[{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5426881313323975},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.542419970035553},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.5187819004058838},{"id":"https://openalex.org/C2776392105","wikidata":"https://www.wikidata.org/wiki/Q875744","display_name":"Euler\u2013Lagrange equation","level":3,"score":0.4155493378639221},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3919851779937744},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3358473777770996},{"id":"https://openalex.org/C53469067","wikidata":"https://www.wikidata.org/wiki/Q505735","display_name":"Lagrangian","level":2,"score":0.19685545563697815},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0478232204914093},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/axioms11040178","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11040178","pdf_url":"https://www.mdpi.com/2075-1680/11/4/178/pdf?version=1650002877","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2204.07659","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2204.07659","pdf_url":"https://arxiv.org/pdf/2204.07659","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:b4e3e9d740d244ffbea83a9027caa5fc","is_oa":true,"landing_page_url":"https://doaj.org/article/b4e3e9d740d244ffbea83a9027caa5fc","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":"Axioms, Vol 11, Iss 4, p 178 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/11/4/178/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms11040178","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":"Axioms; Volume 11; Issue 4; Pages: 178","raw_type":"Text"},{"id":"pmh:oai:ria.ua.pt:10773/34273","is_oa":true,"landing_page_url":"http://hdl.handle.net/10773/34273","pdf_url":null,"source":{"id":"https://openalex.org/S4306402433","display_name":"Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT)","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"journal article"}],"best_oa_location":{"id":"doi:10.3390/axioms11040178","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11040178","pdf_url":"https://www.mdpi.com/2075-1680/11/4/178/pdf?version=1650002877","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1709823736","display_name":null,"funder_award_id":"UIDB/04106/202","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G2590986966","display_name":null,"funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G2867470761","display_name":null,"funder_award_id":"UIDB/04106/2020 (CIDMA)","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3069120617","display_name":null,"funder_award_id":"04106","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3272457656","display_name":null,"funder_award_id":"UIDB/","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3461949579","display_name":null,"funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320329951","funder_display_name":"Center for Research and Development in Mathematics and Applications"},{"id":"https://openalex.org/G6772520733","display_name":null,"funder_award_id":"IDB/04106/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G878132358","display_name":null,"funder_award_id":"CIDMA","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"}],"funders":[{"id":"https://openalex.org/F4320329951","display_name":"Center for Research and Development in Mathematics and Applications","ror":null},{"id":"https://openalex.org/F4320334779","display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","ror":"https://ror.org/00snfqn58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4224023123.pdf","grobid_xml":"https://content.openalex.org/works/W4224023123.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W260145074","https://openalex.org/W1584611818","https://openalex.org/W1900194605","https://openalex.org/W1966397236","https://openalex.org/W2068345729","https://openalex.org/W2949305105","https://openalex.org/W2964046643","https://openalex.org/W2969445909","https://openalex.org/W3027645126","https://openalex.org/W3046280651","https://openalex.org/W3105963035","https://openalex.org/W3109634920","https://openalex.org/W3126196578","https://openalex.org/W3164248124","https://openalex.org/W4205649352","https://openalex.org/W4210417410","https://openalex.org/W4210903706","https://openalex.org/W4226189346","https://openalex.org/W4249833219","https://openalex.org/W4388925118","https://openalex.org/W6759743601","https://openalex.org/W6807867690"],"related_works":["https://openalex.org/W1973641256","https://openalex.org/W2089274067","https://openalex.org/W56994397","https://openalex.org/W1121942432","https://openalex.org/W2418069309","https://openalex.org/W2376158287","https://openalex.org/W1526165314","https://openalex.org/W2359859943","https://openalex.org/W2357986411","https://openalex.org/W2811200908"],"abstract_inverted_index":{"Integration":[0],"by":[1,27,87],"parts":[2,88],"plays":[3],"a":[4,32],"crucial":[5],"role":[6],"in":[7,18,38,62,108],"mathematical":[8],"analysis,":[9],"e.g.,":[10],"during":[11],"the":[12,19,39,47,71,74,84,91,109,114,122],"proof":[13],"of":[14,21,116],"necessary":[15],"optimality":[16],"conditions":[17],"calculus":[20],"variations":[22],"and":[23,73],"optimal":[24],"control.":[25],"Motivated":[26],"this":[28],"fact,":[29],"we":[30,95],"construct":[31],"new,":[33],"right-weighted":[34],"generalized":[35,51],"fractional":[36,52,76],"derivative":[37,53],"Riemann\u2013Liouville":[40],"sense":[41],"with":[42,54,113],"its":[43],"associated":[44],"integral":[45],"for":[46,102],"recently":[48],"introduced":[49],"weighted":[50,99],"Mittag\u2013Leffler":[55],"kernel.":[56],"We":[57,111],"rewrite":[58],"these":[59],"operators":[60],"equivalently":[61],"effective":[63],"series,":[64],"proving":[65],"some":[66],"interesting":[67],"properties":[68],"relating":[69],"to":[70,82,121],"left":[72],"right":[75],"operators.":[77],"These":[78],"results":[79],"permit":[80],"us":[81],"obtain":[83,96],"corresponding":[85],"integration":[86],"formula.":[89],"With":[90],"new":[92],"general":[93],"formula,":[94],"an":[97,117],"appropriate":[98],"Euler\u2013Lagrange":[100],"equation":[101],"dynamic":[103],"optimization,":[104],"extending":[105],"those":[106],"existing":[107],"literature.":[110],"end":[112],"application":[115],"optimization":[118],"variational":[119],"problem":[120],"quantum":[123],"mechanics":[124],"framework.":[125]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
