{"id":"https://openalex.org/W2343318377","doi":"https://doi.org/10.1137/16m1073121","title":"Petrov--Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations","display_name":"Petrov--Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2343318377","doi":"https://doi.org/10.1137/16m1073121","mag":"2343318377"},"language":"en","primary_location":{"id":"doi:10.1137/16m1073121","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1073121","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1604.08650","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026301782","display_name":"Ehsan Kharazmi","orcid":"https://orcid.org/0000-0002-3680-5500"},"institutions":[{"id":"https://openalex.org/I87216513","display_name":"Michigan State University","ror":"https://ror.org/05hs6h993","country_code":"US","type":"education","lineage":["https://openalex.org/I87216513"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ehsan Kharazmi","raw_affiliation_strings":["\u2002Michigan State University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\u2002Michigan State University","institution_ids":["https://openalex.org/I87216513"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051268633","display_name":"Mohsen Zayernouri","orcid":"https://orcid.org/0000-0002-0831-2610"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mohsen Zayernouri","raw_affiliation_strings":["Mechanical Engineering"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mechanical Engineering","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009658255","display_name":"George Em Karniadakis","orcid":"https://orcid.org/0000-0002-9713-7120"},"institutions":[{"id":"https://openalex.org/I175594653","display_name":"John Brown University","ror":"https://ror.org/02ct41q97","country_code":"US","type":"education","lineage":["https://openalex.org/I175594653"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"George Em Karniadakis","raw_affiliation_strings":["Brown University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Brown University","institution_ids":["https://openalex.org/I175594653"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.00340441,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"39","issue":"3","first_page":"A1003","last_page":"A1037"},"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9995999932289124,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9986000061035156,"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":0.7623893022537231},{"id":"https://openalex.org/keywords/collocation","display_name":"Collocation (remote sensing)","score":0.687185525894165},{"id":"https://openalex.org/keywords/petrov\u2013galerkin-method","display_name":"Petrov\u2013Galerkin method","score":0.5681238174438477},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.5462770462036133},{"id":"https://openalex.org/keywords/eigenfunction","display_name":"Eigenfunction","score":0.5449537038803101},{"id":"https://openalex.org/keywords/collocation-method","display_name":"Collocation method","score":0.5372839570045471},{"id":"https://openalex.org/keywords/orthogonal-collocation","display_name":"Orthogonal collocation","score":0.5227615833282471},{"id":"https://openalex.org/keywords/spectral-method","display_name":"Spectral method","score":0.5172029733657837},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5051196217536926},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.49577173590660095},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4812062084674835},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.46000033617019653},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.43284517526626587},{"id":"https://openalex.org/keywords/differential-operator","display_name":"Differential operator","score":0.4307466149330139},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3923286497592926},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.37256139516830444},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.36775487661361694},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.16851899027824402},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12370234727859497},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.0832468569278717}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7623893022537231},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.687185525894165},{"id":"https://openalex.org/C2778540877","wikidata":"https://www.wikidata.org/wiki/Q7179155","display_name":"Petrov\u2013Galerkin method","level":3,"score":0.5681238174438477},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.5462770462036133},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.5449537038803101},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.5372839570045471},{"id":"https://openalex.org/C114609681","wikidata":"https://www.wikidata.org/wiki/Q24839227","display_name":"Orthogonal collocation","level":5,"score":0.5227615833282471},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.5172029733657837},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5051196217536926},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.49577173590660095},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4812062084674835},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.46000033617019653},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.43284517526626587},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.4307466149330139},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3923286497592926},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.37256139516830444},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.36775487661361694},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.16851899027824402},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12370234727859497},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.0832468569278717},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","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},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/16m1073121","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1073121","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1604.08650","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1604.08650","pdf_url":"https://arxiv.org/pdf/1604.08650","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":"doi:10.48550/arxiv.1604.08650","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1604.08650","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"},{"id":"mag:2343318377","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1604.08650","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1604.08650","pdf_url":"https://arxiv.org/pdf/1604.08650","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"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8944192383","display_name":null,"funder_award_id":"W911NF-15-1-0562","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"}],"funders":[{"id":"https://openalex.org/F4320338281","display_name":"Army Research Office","ror":"https://ror.org/05epdh915"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2343318377.pdf","grobid_xml":"https://content.openalex.org/works/W2343318377.grobid-xml"},"referenced_works_count":59,"referenced_works":["https://openalex.org/W104068007","https://openalex.org/W110340196","https://openalex.org/W620183161","https://openalex.org/W755756226","https://openalex.org/W1131718341","https://openalex.org/W1531510807","https://openalex.org/W1655132380","https://openalex.org/W1967569646","https://openalex.org/W1967595729","https://openalex.org/W1969312973","https://openalex.org/W1973110804","https://openalex.org/W1974267666","https://openalex.org/W1975900429","https://openalex.org/W1978349032","https://openalex.org/W1980479723","https://openalex.org/W1981384091","https://openalex.org/W1983159263","https://openalex.org/W1986915637","https://openalex.org/W1987597560","https://openalex.org/W1991982189","https://openalex.org/W1993445800","https://openalex.org/W2000988163","https://openalex.org/W2006907269","https://openalex.org/W2009456713","https://openalex.org/W2014254762","https://openalex.org/W2019469368","https://openalex.org/W2019659177","https://openalex.org/W2025020152","https://openalex.org/W2025672056","https://openalex.org/W2028420334","https://openalex.org/W2030146312","https://openalex.org/W2035830378","https://openalex.org/W2053256951","https://openalex.org/W2056829543","https://openalex.org/W2057127233","https://openalex.org/W2062650267","https://openalex.org/W2063589755","https://openalex.org/W2065254635","https://openalex.org/W2074077397","https://openalex.org/W2090239694","https://openalex.org/W2091740033","https://openalex.org/W2093225350","https://openalex.org/W2095472505","https://openalex.org/W2098803915","https://openalex.org/W2104055838","https://openalex.org/W2111271983","https://openalex.org/W2114010261","https://openalex.org/W2118288723","https://openalex.org/W2131079340","https://openalex.org/W2136196792","https://openalex.org/W2137120557","https://openalex.org/W2145803046","https://openalex.org/W2148320567","https://openalex.org/W2165076033","https://openalex.org/W2251960872","https://openalex.org/W2319089385","https://openalex.org/W2512232321","https://openalex.org/W2950576821","https://openalex.org/W4212834435"],"related_works":["https://openalex.org/W2248125967","https://openalex.org/W2040558735","https://openalex.org/W2914821196","https://openalex.org/W3002359135","https://openalex.org/W3196665045","https://openalex.org/W2142367591","https://openalex.org/W3119573630","https://openalex.org/W2741336247","https://openalex.org/W3164382911","https://openalex.org/W2785739022","https://openalex.org/W2792632122","https://openalex.org/W2810987144","https://openalex.org/W3133468970","https://openalex.org/W2805824372","https://openalex.org/W3093996726","https://openalex.org/W3161611021","https://openalex.org/W2970795967","https://openalex.org/W2402673101","https://openalex.org/W3014951111","https://openalex.org/W3021576674"],"abstract_inverted_index":{"Distributed":[0],"order":[1,54],"fractional":[2,55,65,88,157],"operators":[3],"offer":[4],"a":[5,21,29,43,48],"rigorous":[6],"tool":[7],"for":[8,52],"mathematical":[9],"modelling":[10],"of":[11,23,99,102,110,116,119,122,147,184,199],"multiphysics":[12],"phenomena,":[13],"where":[14,136],"the":[15,64,83,97,100,114,120,127,133,140,148,152,162,167,173,182,185,195],"differential":[16,56],"orders":[17],"are":[18,60,96],"distributed":[19,53,129,175],"over":[20],"range":[22],"values":[24],"rather":[25],"than":[26],"being":[27],"just":[28],"fixed":[30],"integer/fraction":[31],"as":[32,113],"in":[33,181],"standard/fractional":[34],"ODEs/PDEs.":[35],"We":[36,125,188],"develop":[37],"two":[38],"spectrally":[39],"accurate":[40],"schemes,":[41],"namely,":[42],"Petrov--Galerkin":[44,84],"spectral":[45,49],"method":[46,51],"and":[47,71,132,144,197],"collocation":[50,153,168],"equations.":[57],"These":[58],"schemes":[59],"developed":[61],"based":[62],"on":[63],"Sturm--Liouville":[66],"eigen-problems":[67],"(FSLPs)":[68],"[M.":[69],"Zayernouri":[70],"G.":[72],"E.":[73],"Karniadakis,":[74],"J.":[75],"Comput.":[76],"Phys.,":[77],"47":[78],"(2013),":[79],"pp.":[80],"2108--2131].":[81],"In":[82,151],"method,":[85],"we":[86,106,137,155,171],"employ":[87,107,156],"(nonpolynomial)":[89,158],"basis":[90],"functions,":[91],"called":[92],"Jacobi":[93],"polyfractonomials,":[94],"which":[95],"eigenfunctions":[98,118],"FSLP":[101,121],"first":[103],"kind,":[104],"while":[105],"another":[108],"space":[109,131],"test":[111],"functions":[112],"span":[115],"polyfractonomial":[117],"second":[123],"kind.":[124],"define":[126],"underlying":[128],"Sobolev":[130],"associated":[134],"norms,":[135],"carry":[138],"out":[139],"corresponding":[141,174],"discrete":[142],"stability":[143],"error":[145],"analyses":[146],"proposed":[149],"scheme.":[150],"scheme,":[154],"Lagrange":[159],"interpolants":[160],"satisfying":[161],"Kronecker":[163],"delta":[164],"property":[165],"at":[166],"points.":[169],"Subsequently,":[170],"obtain":[172],"differentiation":[176],"matrices":[177],"to":[178,193],"be":[179],"employed":[180],"discretization":[183],"strong":[186],"problem.":[187],"perform":[189],"systematic":[190],"numerical":[191],"tests":[192],"demonstrate":[194],"efficiency":[196],"conditioning":[198],"each":[200],"method.":[201]},"counts_by_year":[{"year":2016,"cited_by_count":1}],"updated_date":"2026-07-03T08:13:44.112507","created_date":"2025-10-10T00:00:00"}