{"id":"https://openalex.org/W4309215061","doi":"https://doi.org/10.3390/axioms11110642","title":"Solving Time-Fractional Partial Differential Equation Using Chebyshev Cardinal Functions","display_name":"Solving Time-Fractional Partial Differential Equation Using Chebyshev Cardinal Functions","publication_year":2022,"publication_date":"2022-11-14","ids":{"openalex":"https://openalex.org/W4309215061","doi":"https://doi.org/10.3390/axioms11110642"},"language":"en","primary_location":{"id":"doi:10.3390/axioms11110642","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11110642","pdf_url":"https://www.mdpi.com/2075-1680/11/11/642/pdf?version=1668414665","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":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/11/11/642/pdf?version=1668414665","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051367541","display_name":"Haifa Bin Jebreen","orcid":"https://orcid.org/0000-0001-9394-7305"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Haifa Bin Jebreen","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-9394-7305","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5022405104","display_name":"Carlo Cattani","orcid":"https://orcid.org/0000-0002-7504-0424"},"institutions":[{"id":"https://openalex.org/I138938424","display_name":"Universit\u00e0 degli Studi della Tuscia","ror":"https://ror.org/03svwq685","country_code":"IT","type":"education","lineage":["https://openalex.org/I138938424"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Carlo Cattani","raw_affiliation_strings":["Engineering School (DEIM), University of Tuscia, Largo dell\u2019Universit a, 01100 Viterbo, Italy","Engineering School (DEIM), University of Tuscia, Largo dell'Universit a, 01100 Viterbo, Italy"],"raw_orcid":"https://orcid.org/0000-0002-7504-0424","affiliations":[{"raw_affiliation_string":"Engineering School (DEIM), University of Tuscia, Largo dell\u2019Universit a, 01100 Viterbo, Italy","institution_ids":["https://openalex.org/I138938424"]},{"raw_affiliation_string":"Engineering School (DEIM), University of Tuscia, Largo dell'Universit a, 01100 Viterbo, Italy","institution_ids":["https://openalex.org/I138938424"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5051367541"],"corresponding_institution_ids":["https://openalex.org/I28022161"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":0.7677,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.66278393,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":97},"biblio":{"volume":"11","issue":"11","first_page":"642","last_page":"642"},"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9973999857902527,"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":0.9821000099182129,"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":0.7575255632400513},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.6832890510559082},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.6446850895881653},{"id":"https://openalex.org/keywords/chebyshev-filter","display_name":"Chebyshev filter","score":0.5947632193565369},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.5800967812538147},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5749731063842773},{"id":"https://openalex.org/keywords/chebyshev-equation","display_name":"Chebyshev equation","score":0.5344921946525574},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5238311886787415},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.513601541519165},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.48166096210479736},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4610307812690735},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.4485085904598236},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.44416049122810364},{"id":"https://openalex.org/keywords/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.41342535614967346},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.21750906109809875},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.11651968955993652},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.06932663917541504},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06294447183609009}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7575255632400513},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.6832890510559082},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.6446850895881653},{"id":"https://openalex.org/C21424316","wikidata":"https://www.wikidata.org/wiki/Q718621","display_name":"Chebyshev filter","level":2,"score":0.5947632193565369},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.5800967812538147},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5749731063842773},{"id":"https://openalex.org/C142484921","wikidata":"https://www.wikidata.org/wiki/Q3890369","display_name":"Chebyshev equation","level":4,"score":0.5344921946525574},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5238311886787415},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.513601541519165},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.48166096210479736},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4610307812690735},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.4485085904598236},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.44416049122810364},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.41342535614967346},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.21750906109809875},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.11651968955993652},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.06932663917541504},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06294447183609009},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms11110642","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11110642","pdf_url":"https://www.mdpi.com/2075-1680/11/11/642/pdf?version=1668414665","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:doaj.org/article:72e071e6b8eb4df0a51d3c03a9c1acf0","is_oa":true,"landing_page_url":"https://doaj.org/article/72e071e6b8eb4df0a51d3c03a9c1acf0","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":"Axioms, Vol 11, Iss 11, p 642 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/11/11/642/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms11110642","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 11; Pages: 642","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms11110642","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11110642","pdf_url":"https://www.mdpi.com/2075-1680/11/11/642/pdf?version=1668414665","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/G3445937380","display_name":null,"funder_award_id":"RSP-2021/210","funder_id":"https://openalex.org/F4320321145","funder_display_name":"King Saud University"},{"id":"https://openalex.org/G7545718068","display_name":null,"funder_award_id":"RSP-2021","funder_id":"https://openalex.org/F4320321145","funder_display_name":"King Saud University"}],"funders":[{"id":"https://openalex.org/F4320321145","display_name":"King Saud University","ror":"https://ror.org/02f81g417"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4309215061.pdf","grobid_xml":"https://content.openalex.org/works/W4309215061.grobid-xml"},"referenced_works_count":33,"referenced_works":["https://openalex.org/W1879346481","https://openalex.org/W2000659785","https://openalex.org/W2022691550","https://openalex.org/W2023600694","https://openalex.org/W2035204586","https://openalex.org/W2038329281","https://openalex.org/W2040469380","https://openalex.org/W2041437724","https://openalex.org/W2056829543","https://openalex.org/W2061914086","https://openalex.org/W2065232540","https://openalex.org/W2146008531","https://openalex.org/W2605124421","https://openalex.org/W2793735827","https://openalex.org/W2804853878","https://openalex.org/W2894803565","https://openalex.org/W3014567336","https://openalex.org/W3025763651","https://openalex.org/W3126352790","https://openalex.org/W3185290179","https://openalex.org/W3192060108","https://openalex.org/W3197339611","https://openalex.org/W3215220336","https://openalex.org/W3215654771","https://openalex.org/W4206804158","https://openalex.org/W4210401923","https://openalex.org/W4223490408","https://openalex.org/W4236165936","https://openalex.org/W4247375400","https://openalex.org/W4283780553","https://openalex.org/W4285798191","https://openalex.org/W6681368628","https://openalex.org/W6840322709"],"related_works":["https://openalex.org/W2179563627","https://openalex.org/W2043532630","https://openalex.org/W2112919228","https://openalex.org/W2136992695","https://openalex.org/W2054912819","https://openalex.org/W2106962382","https://openalex.org/W2894195618","https://openalex.org/W2034868998","https://openalex.org/W4236068172","https://openalex.org/W3013278801"],"abstract_inverted_index":{"We":[0],"propose":[1],"a":[2,69],"numerical":[3,94],"scheme":[4],"based":[5,41],"on":[6,42],"the":[7,12,22,28,37,51,56,60,64,79,97,102],"Galerkin":[8,52],"method":[9,53,89],"for":[10,59,87],"solving":[11,76],"time-fractional":[13],"partial":[14],"differential":[15],"equations.":[16,74],"To":[17],"this":[18,77,88],"end,":[19],"after":[20],"introducing":[21],"Chebyshev":[23],"cardinal":[24],"functions":[25],"(CCFs),":[26],"using":[27,55],"relation":[29],"between":[30],"fractional":[31,39,62],"integral":[32],"and":[33,45,54,92,99],"derivative,":[34,63],"we":[35],"represent":[36],"Caputo":[38,61],"derivative":[40],"these":[43],"bases":[44],"obtain":[46],"an":[47],"operational":[48,57],"matrix.":[49],"Applying":[50],"matrix":[58],"desired":[65],"equation":[66],"reduces":[67],"to":[68],"system":[70],"of":[71,101],"linear":[72],"algebraic":[73],"By":[75],"system,":[78],"unknown":[80],"solution":[81],"is":[82,90],"obtained.":[83],"The":[84],"convergence":[85],"analysis":[86],"investigated,":[91],"some":[93],"simulations":[95],"show":[96],"accuracy":[98],"ability":[100],"technique.":[103]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}