{"id":"https://openalex.org/W2914021327","doi":"https://doi.org/10.1080/00207160.2019.1573992","title":"Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials","display_name":"Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials","publication_year":2019,"publication_date":"2019-01-25","ids":{"openalex":"https://openalex.org/W2914021327","doi":"https://doi.org/10.1080/00207160.2019.1573992","mag":"2914021327"},"language":"en","primary_location":{"id":"doi:10.1080/00207160.2019.1573992","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1573992","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"],"host_organization_lineage_names":["Taylor & Francis"],"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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076286645","display_name":"Liping Wang","orcid":"https://orcid.org/0000-0002-2913-311X"},"institutions":[{"id":"https://openalex.org/I39333907","display_name":"Yanshan University","ror":"https://ror.org/02txfnf15","country_code":"CN","type":"education","lineage":["https://openalex.org/I39333907"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Li-Ping Wang","raw_affiliation_strings":["School of Sciences, Yanshan University, Qinhuangdao, People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Sciences, Yanshan University, Qinhuangdao, People's Republic of China","institution_ids":["https://openalex.org/I39333907"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100333349","display_name":"Yiming Chen","orcid":"https://orcid.org/0000-0001-7040-8050"},"institutions":[{"id":"https://openalex.org/I28221208","display_name":"Institut National des Sciences Appliqu\u00e9es de Rennes","ror":"https://ror.org/04xaa4j22","country_code":"FR","type":"education","lineage":["https://openalex.org/I28221208"]},{"id":"https://openalex.org/I39333907","display_name":"Yanshan University","ror":"https://ror.org/02txfnf15","country_code":"CN","type":"education","lineage":["https://openalex.org/I39333907"]},{"id":"https://openalex.org/I4210096782","display_name":"Laboratoire Pluridisciplinaire de Recherche en Ing\u00e9nierie des Syst\u00e8mes, M\u00e9canique et Energ\u00e9tique","ror":"https://ror.org/00sbth994","country_code":"FR","type":"facility","lineage":["https://openalex.org/I12449238","https://openalex.org/I4210096782","https://openalex.org/I4210143826"]},{"id":"https://openalex.org/I4210129068","display_name":"Le Studium","ror":"https://ror.org/03qgg4v31","country_code":"FR","type":"facility","lineage":["https://openalex.org/I4210129068"]},{"id":"https://openalex.org/I80327900","display_name":"Chang Jung Christian University","ror":"https://ror.org/02s3d7j94","country_code":"TW","type":"education","lineage":["https://openalex.org/I80327900"]}],"countries":["CN","FR","TW"],"is_corresponding":false,"raw_author_name":"Yi-Ming Chen","raw_affiliation_strings":["LE STUDIUM RESEARCH PROFESSOR, Loire Valley Institute for Advanced Studies, Orl\u00e9ans, France","PRISME (INSA-Institut National des Sciences Appliqu\u00e9es) -88, Bourges, France","School of Sciences, Yanshan University, Qinhuangdao, People's Republic of China","Department of Information Management (Department of Information Management Chang Jung Christian University No.396, Sec. 1, Changrong Rd., Gueiren Dist., Tainan City 71101, Taiwan (R.O.C.) - Taiwan)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LE STUDIUM RESEARCH PROFESSOR, Loire Valley Institute for Advanced Studies, Orl\u00e9ans, France","institution_ids":["https://openalex.org/I4210129068"]},{"raw_affiliation_string":"PRISME (INSA-Institut National des Sciences Appliqu\u00e9es) -88, Bourges, France","institution_ids":["https://openalex.org/I28221208","https://openalex.org/I4210096782"]},{"raw_affiliation_string":"School of Sciences, Yanshan University, Qinhuangdao, People's Republic of China","institution_ids":["https://openalex.org/I39333907"]},{"raw_affiliation_string":"Department of Information Management (Department of Information Management Chang Jung Christian University No.396, Sec. 1, Changrong Rd., Gueiren Dist., Tainan City 71101, Taiwan (R.O.C.) - Taiwan)","institution_ids":["https://openalex.org/I80327900"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015407151","display_name":"Da\u2010Yan Liu","orcid":"https://orcid.org/0000-0003-2853-0129"},"institutions":[{"id":"https://openalex.org/I12449238","display_name":"Universit\u00e9 d'Orl\u00e9ans","ror":"https://ror.org/014zrew76","country_code":"FR","type":"education","lineage":["https://openalex.org/I12449238"]},{"id":"https://openalex.org/I4210129154","display_name":"Centre Val de Loire","ror":"https://ror.org/03ntmtb29","country_code":"FR","type":"facility","lineage":["https://openalex.org/I4210088668","https://openalex.org/I4210129154"]},{"id":"https://openalex.org/I4210143826","display_name":"Institut National des Sciences Appliqu\u00e9es Centre Val de Loire","ror":"https://ror.org/03y0qc033","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210143826"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Da-Yan Liu","raw_affiliation_strings":["INSA Centre Val de Loire, Universit\u00e9 d'Orl\u00e9ans, Bourges Cedex, France"],"raw_orcid":"https://orcid.org/0000-0003-2853-0129","affiliations":[{"raw_affiliation_string":"INSA Centre Val de Loire, Universit\u00e9 d'Orl\u00e9ans, Bourges Cedex, France","institution_ids":["https://openalex.org/I4210143826","https://openalex.org/I4210129154","https://openalex.org/I12449238"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5057148868","display_name":"Driss Boutat","orcid":"https://orcid.org/0000-0001-6026-5674"},"institutions":[{"id":"https://openalex.org/I12449238","display_name":"Universit\u00e9 d'Orl\u00e9ans","ror":"https://ror.org/014zrew76","country_code":"FR","type":"education","lineage":["https://openalex.org/I12449238"]},{"id":"https://openalex.org/I4210129154","display_name":"Centre Val de Loire","ror":"https://ror.org/03ntmtb29","country_code":"FR","type":"facility","lineage":["https://openalex.org/I4210088668","https://openalex.org/I4210129154"]},{"id":"https://openalex.org/I4210143826","display_name":"Institut National des Sciences Appliqu\u00e9es Centre Val de Loire","ror":"https://ror.org/03y0qc033","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210143826"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Driss Boutat","raw_affiliation_strings":["INSA Centre Val de Loire, Universit\u00e9 d'Orl\u00e9ans, Bourges Cedex, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INSA Centre Val de Loire, Universit\u00e9 d'Orl\u00e9ans, Bourges Cedex, France","institution_ids":["https://openalex.org/I4210143826","https://openalex.org/I4210129154","https://openalex.org/I12449238"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5015407151"],"corresponding_institution_ids":["https://openalex.org/I12449238","https://openalex.org/I4210129154","https://openalex.org/I4210143826"],"apc_list":null,"apc_paid":null,"fwci":1.7024,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.83266687,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"96","issue":"12","first_page":"2487","last_page":"2510"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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":0.9998999834060669,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9757999777793884,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9670000076293945,"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":0.8179035186767578},{"id":"https://openalex.org/keywords/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.7277746200561523},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.5845028162002563},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5521855354309082},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.5309106111526489},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.5018525123596191},{"id":"https://openalex.org/keywords/pantograph","display_name":"Pantograph","score":0.4972892105579376},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.4710150957107544},{"id":"https://openalex.org/keywords/chebyshev-equation","display_name":"Chebyshev equation","score":0.45255884528160095},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.43925023078918457},{"id":"https://openalex.org/keywords/chebyshev-filter","display_name":"Chebyshev filter","score":0.41372373700141907},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.41248464584350586},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3568304777145386},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.34447145462036133},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.30630674958229065}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8179035186767578},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.7277746200561523},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.5845028162002563},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5521855354309082},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.5309106111526489},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.5018525123596191},{"id":"https://openalex.org/C20756127","wikidata":"https://www.wikidata.org/wiki/Q722757","display_name":"Pantograph","level":2,"score":0.4972892105579376},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.4710150957107544},{"id":"https://openalex.org/C142484921","wikidata":"https://www.wikidata.org/wiki/Q3890369","display_name":"Chebyshev equation","level":4,"score":0.45255884528160095},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43925023078918457},{"id":"https://openalex.org/C21424316","wikidata":"https://www.wikidata.org/wiki/Q718621","display_name":"Chebyshev filter","level":2,"score":0.41372373700141907},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.41248464584350586},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3568304777145386},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.34447145462036133},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30630674958229065},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"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/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1080/00207160.2019.1573992","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1573992","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"],"host_organization_lineage_names":["Taylor & Francis"],"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:HAL:hal-02464216v1","is_oa":false,"landing_page_url":"https://hal.archives-ouvertes.fr/hal-02464216","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN: 0020-7160","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:HAL:hal-02884254v1","is_oa":false,"landing_page_url":"https://hal.science/hal-02884254","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"International Journal of Computer Mathematics, 2019, 96 (12), pp.2487-2510. ⟨10.1080/00207160.2019.1573992⟩","raw_type":"Journal articles"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4850625604","display_name":null,"funder_award_id":"A2017203100","funder_id":"https://openalex.org/F4320322163","funder_display_name":"Natural Science Foundation of Hebei Province"}],"funders":[{"id":"https://openalex.org/F4320322163","display_name":"Natural Science Foundation of Hebei Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W273630147","https://openalex.org/W1970005860","https://openalex.org/W1972995042","https://openalex.org/W1975383112","https://openalex.org/W1997651353","https://openalex.org/W2000924596","https://openalex.org/W2004446568","https://openalex.org/W2012854733","https://openalex.org/W2013958055","https://openalex.org/W2016755336","https://openalex.org/W2017595816","https://openalex.org/W2024562774","https://openalex.org/W2032647616","https://openalex.org/W2037423068","https://openalex.org/W2040329976","https://openalex.org/W2047173892","https://openalex.org/W2055701161","https://openalex.org/W2055861588","https://openalex.org/W2057056717","https://openalex.org/W2057152511","https://openalex.org/W2060225998","https://openalex.org/W2062060975","https://openalex.org/W2063133641","https://openalex.org/W2069379911","https://openalex.org/W2086600283","https://openalex.org/W2089348680","https://openalex.org/W2091877690","https://openalex.org/W2095151500","https://openalex.org/W2104205547","https://openalex.org/W2116807840","https://openalex.org/W2140377881","https://openalex.org/W2155980958","https://openalex.org/W2169378796","https://openalex.org/W2187242803","https://openalex.org/W2340668186","https://openalex.org/W2377515284","https://openalex.org/W2401294069","https://openalex.org/W2407973543","https://openalex.org/W2412352375","https://openalex.org/W2416676657","https://openalex.org/W2461570565","https://openalex.org/W2466433738","https://openalex.org/W2581158646","https://openalex.org/W2782480706","https://openalex.org/W2800087239","https://openalex.org/W2809529648","https://openalex.org/W2883236899","https://openalex.org/W2900435294","https://openalex.org/W4239838448"],"related_works":["https://openalex.org/W2106962382","https://openalex.org/W2062710710","https://openalex.org/W3091996841","https://openalex.org/W2093582315","https://openalex.org/W2970083787","https://openalex.org/W2518741117","https://openalex.org/W3120027569","https://openalex.org/W2759844628","https://openalex.org/W2294380385","https://openalex.org/W2761134672"],"abstract_inverted_index":{"In":[0,93],"this":[1],"paper,":[2],"an":[3],"efficient":[4],"numerical":[5,18,91,112],"technique":[6],"based":[7,68],"on":[8,36,69],"the":[9,47,50,54,62,72,97,102,109,125],"shifted":[10],"Chebyshev":[11],"polynomials":[12,29,41],"(SCPs)":[13],"is":[14,75],"established":[15],"to":[16,42],"obtain":[17,90],"solutions":[19],"of":[20,49,59,64,80,104,111,129],"generalized":[21,55],"fractional":[22],"pantograph":[23,56],"equations":[24],"with":[25,121],"variable":[26],"coefficients.":[27],"These":[28],"are":[30,132],"orthogonal":[31],"and":[32,61,100,119,127],"have":[33],"compact":[34],"support":[35],"0,L.":[37],"We":[38],"use":[39],"these":[40,70,85],"approximate":[43],"unknown":[44],"solutions.":[45,92,113],"Using":[46],"properties":[48],"SCPs,":[51],"we":[52,88,95],"derive":[53],"operational":[57],"matrix":[58],"SCPs":[60],"one":[63],"fractional-order":[65],"differentiation.":[66],"Then,":[67],"matrices":[71],"original":[73],"problem":[74],"transformed":[76],"into":[77],"a":[78],"system":[79],"algebraic":[81,86],"equations.":[82],"By":[83],"solving":[84],"equations,":[87],"can":[89],"addition,":[94],"investigate":[96],"error":[98,105],"analysis":[99],"introduce":[101],"process":[103],"correction":[106],"for":[107],"improving":[108],"precision":[110],"Lastly,":[114],"by":[115],"giving":[116],"some":[117],"examples":[118],"comparing":[120],"other":[122],"existing":[123],"methods,":[124],"validity":[126],"efficiency":[128],"our":[130],"method":[131],"demonstrated.":[133]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":3}],"updated_date":"2026-05-07T13:39:58.223016","created_date":"2025-10-10T00:00:00"}