{"id":"https://openalex.org/W3112377812","doi":null,"title":"Numerical solution of fractional partial differential equations by spectral methods","display_name":"Numerical solution of fractional partial differential equations by spectral methods","publication_year":2019,"publication_date":"2019-09-01","ids":{"openalex":"https://openalex.org/W3112377812","mag":"3112377812"},"language":"en","primary_location":{"id":"pmh:oai:eprints.uthm.edu.my:44","doi":null,"is_oa":true,"landing_page_url":"http://eprints.uthm.edu.my/44/1/24p%20AFSHAN%20KANWAL.pdf","pdf_url":"http://eprints.uthm.edu.my/44/1/24p%20AFSHAN%20KANWAL.pdf","source":{"id":"https://openalex.org/S4306400770","display_name":"UTHM Institutional Repository (Universiti Tun Hussein Onn Malaysia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I930072361","host_organization_name":"Tun Hussein Onn University of Malaysia","host_organization_lineage":["https://openalex.org/I930072361"],"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":"Thesis"},"volume":null,"type":"dissertation","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://eprints.uthm.edu.my/44/1/24p%20AFSHAN%20KANWAL.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023251238","display_name":"Afshan Kanwal","orcid":"https://orcid.org/0000-0001-7062-7624","observed_orcids":["https://orcid.org/0000-0001-7062-7624"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Kanwal, Afshan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5023251238"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9961000084877014,"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":0.9961000084877014,"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/T11416","display_name":"Numerical Solvers for Ordinary Differential Equations","score":0.0010000000474974513,"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":0.0006000000284984708,"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/fractional-partial-differential-equations","display_name":"fractional partial differential equations","score":1.0},{"id":"https://openalex.org/keywords/genocchi-polynomials","display_name":"Genocchi polynomials","score":0.9399999976158142},{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.9459999799728394},{"id":"https://openalex.org/keywords/ritz-galerkin-method","display_name":"Ritz-Galerkin method","score":0.953000009059906},{"id":"https://openalex.org/keywords/operational-matrix-of-derivatives","display_name":"operational matrix of derivatives","score":0.7630000114440918},{"id":"https://openalex.org/keywords/collocation-method","display_name":"collocation method","score":0.8629999756813049},{"id":"https://openalex.org/keywords/tau-method","display_name":"tau method","score":0.8059999942779541},{"id":"https://openalex.org/keywords/variable-order-fractional-derivative","display_name":"variable-order fractional derivative","score":0.8899999856948853},{"id":"https://openalex.org/keywords/numerical-solution-of-fractional-equations","display_name":"numerical solution of fractional equations","score":0.5099999904632568}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8154147267341614},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.7725732326507568},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.6942160129547119},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.6788891553878784},{"id":"https://openalex.org/C38961997","wikidata":"https://www.wikidata.org/wiki/Q7575186","display_name":"Spectral element method","level":4,"score":0.6739168167114258},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5612512230873108},{"id":"https://openalex.org/C48490523","wikidata":"https://www.wikidata.org/wiki/Q371631","display_name":"Jacobi polynomials","level":3,"score":0.5421638488769531},{"id":"https://openalex.org/C86287605","wikidata":"https://www.wikidata.org/wiki/Q4454970","display_name":"Legendre's equation","level":3,"score":0.5154176950454712},{"id":"https://openalex.org/C123769847","wikidata":"https://www.wikidata.org/wiki/Q6517888","display_name":"Legendre wavelet","level":5,"score":0.5134802460670471},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.48894158005714417},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.4655478894710541},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.4575374126434326},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.4555952548980713},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.4389054477214813},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.4314550459384918},{"id":"https://openalex.org/C159674985","wikidata":"https://www.wikidata.org/wiki/Q6823718","display_name":"Method of mean weighted residuals","level":4,"score":0.4220591187477112},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4189125895500183},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.40079793334007263},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.3946838974952698},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.37935999035835266},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.31659847497940063},{"id":"https://openalex.org/C144468803","wikidata":"https://www.wikidata.org/wiki/Q16965966","display_name":"Mixed finite element method","level":3,"score":0.1672551929950714},{"id":"https://openalex.org/C196216189","wikidata":"https://www.wikidata.org/wiki/Q2867","display_name":"Wavelet transform","level":3,"score":0.1644679605960846},{"id":"https://openalex.org/C46286280","wikidata":"https://www.wikidata.org/wiki/Q2414958","display_name":"Discrete wavelet transform","level":4,"score":0.12085554003715515},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10413491725921631},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.07552337646484375},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"pmh:oai:eprints.uthm.edu.my:44","doi":null,"is_oa":true,"landing_page_url":"http://eprints.uthm.edu.my/44/1/24p%20AFSHAN%20KANWAL.pdf","pdf_url":"http://eprints.uthm.edu.my/44/1/24p%20AFSHAN%20KANWAL.pdf","source":{"id":"https://openalex.org/S4306400770","display_name":"UTHM Institutional Repository (Universiti Tun Hussein Onn Malaysia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I930072361","host_organization_name":"Tun Hussein Onn University of Malaysia","host_organization_lineage":["https://openalex.org/I930072361"],"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":"Thesis"}],"best_oa_location":{"id":"pmh:oai:eprints.uthm.edu.my:44","doi":null,"is_oa":true,"landing_page_url":"http://eprints.uthm.edu.my/44/1/24p%20AFSHAN%20KANWAL.pdf","pdf_url":"http://eprints.uthm.edu.my/44/1/24p%20AFSHAN%20KANWAL.pdf","source":{"id":"https://openalex.org/S4306400770","display_name":"UTHM Institutional Repository (Universiti Tun Hussein Onn Malaysia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I930072361","host_organization_name":"Tun Hussein Onn University of Malaysia","host_organization_lineage":["https://openalex.org/I930072361"],"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":"Thesis"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320310112","display_name":"Universiti Tun Hussein Onn Malaysia","ror":"https://ror.org/01c5wha71"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3112377812.pdf","grobid_xml":"https://content.openalex.org/works/W3112377812.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W2809985998"],"related_works":["https://openalex.org/W2984818439","https://openalex.org/W2941664585","https://openalex.org/W1997724051","https://openalex.org/W2570973091","https://openalex.org/W2942696942","https://openalex.org/W4226174873","https://openalex.org/W2982531365","https://openalex.org/W4213189554","https://openalex.org/W2608397475","https://openalex.org/W3081582585"],"abstract_inverted_index":{"Fractional":[0],"partial":[1],"differential":[2],"equations":[3],"(FPDEs)":[4],"have":[5,26,101,154,185,191,234],"become":[6],"essential":[7],"tool":[8],"for":[9,29,77,239],"the":[10,20,30,59,78,116,136,158,188,193,214,217,225,231],"modeling":[11],"of":[12,32,42,80,139,206,216],"physical":[13],"models":[14],"by":[15,223],"using":[16,115],"spectral":[17,24,43],"methods.":[18,53],"In":[19],"last":[21],"few":[22],"decades,":[23],"methods":[25,44,50,63,70,160,190,200,219,233],"been":[27,102,155,221],"developed":[28],"solution":[31,79],"time":[33],"and":[34,51,61,74,82,94,108,130,149,178],"space":[35],"dimensional":[36,84],"FPDEs.":[37,85,111,242],"There":[38],"are":[39],"different":[40],"types":[41],"such":[45,161],"as":[46,145,162,196],"collocation":[47,60],"methods,":[48],"Tau":[49,62],"Galerkin":[52,99],"This":[54],"research":[55],"work":[56],"focuses":[57],"on":[58],"to":[64,104,123,134,147,198,202],"propose":[65],"an":[66],"efficient":[67],"operational":[68,137,208],"matrix":[69,138],"via":[71,210],"Genocchi":[72,90,95,117,211],"polynomials":[73,76,96,118],"Legendre":[75,150,166],"two":[81],"three":[83],"Moreover,":[86],"in":[87],"this":[88],"study,":[89],"wavelet-like":[91],"basis":[92],"method":[93,100,177],"based":[97],"Ritz-":[98],"derived":[103,207],"deal":[105],"with":[106,127,141,157],"FPDEs":[107],"variable-":[109],"order":[110],"The":[112,152,182],"reason":[113],"behind":[114],"is":[119],"that,":[120],"it":[121],"helps":[122],"generate":[124],"functional":[125],"expansions":[126],"less":[128,142],"degree":[129],"small":[131],"coefficients":[132],"values":[133,238],"derive":[135],"derivative":[140],"computational":[143,204],"complexity":[144,205],"compared":[146,156,197],"Chebyshev":[148,163],"Polynomials.":[151],"results":[153,184,195],"existing":[159,199],"wavelets":[164,167],"method,":[165,168,171,174],"Adomian":[169],"decomposition":[170],"Variational":[172],"iteration":[173],"Finite":[175,179],"difference":[176],"element":[180],"method.":[181],"numerical":[183],"revealed":[186],"that":[187,230],"proposed":[189,218,232],"provided":[192,235],"better":[194,236],"due":[201],"minimum":[203],"matrices":[209],"polynomials.":[212],"Additionally,":[213],"significance":[215],"has":[220],"verified":[222],"finding":[224],"error":[226],"bound,":[227],"which":[228],"shows":[229],"approximation":[237],"under":[240],"consideration":[241]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2025-10-10T00:00:00"}