{"id":"https://openalex.org/W2937264697","doi":"https://doi.org/10.1080/00207160.2019.1608968","title":"Galerkin\u2013Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation","display_name":"Galerkin\u2013Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation","publication_year":2019,"publication_date":"2019-04-19","ids":{"openalex":"https://openalex.org/W2937264697","doi":"https://doi.org/10.1080/00207160.2019.1608968","mag":"2937264697"},"language":"en","primary_location":{"id":"doi:10.1080/00207160.2019.1608968","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1608968","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","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"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/A5008145354","display_name":"Mingfa Fei","orcid":"https://orcid.org/0000-0001-7397-9342"},"institutions":[{"id":"https://openalex.org/I47720641","display_name":"Huazhong University of Science and Technology","ror":"https://ror.org/00p991c53","country_code":"CN","type":"education","lineage":["https://openalex.org/I47720641"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Mingfa Fei","raw_affiliation_strings":["School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, People's Republic of China","institution_ids":["https://openalex.org/I47720641"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100651262","display_name":"Chengming Huang","orcid":"https://orcid.org/0000-0001-8556-1111"},"institutions":[{"id":"https://openalex.org/I47720641","display_name":"Huazhong University of Science and Technology","ror":"https://ror.org/00p991c53","country_code":"CN","type":"education","lineage":["https://openalex.org/I47720641"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Chengming Huang","raw_affiliation_strings":["Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, People's Republic of China","School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, People's Republic of China","institution_ids":["https://openalex.org/I47720641"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, People's Republic of China","institution_ids":["https://openalex.org/I47720641"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5100651262"],"corresponding_institution_ids":["https://openalex.org/I47720641"],"apc_list":null,"apc_paid":null,"fwci":1.5351,"has_fulltext":false,"cited_by_count":30,"citation_normalized_percentile":{"value":0.81365921,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"97","issue":"6","first_page":"1183","last_page":"1196"},"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":0.996399998664856,"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":0.9907000064849854,"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":0.828062891960144},{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.8184372186660767},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6531299352645874},{"id":"https://openalex.org/keywords/spectral-method","display_name":"Spectral method","score":0.6140497922897339},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.6071332693099976},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.562994122505188},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.548916220664978},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5112042427062988},{"id":"https://openalex.org/keywords/legendre-wavelet","display_name":"Legendre wavelet","score":0.4733025133609772},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.43718421459198},{"id":"https://openalex.org/keywords/order-of-accuracy","display_name":"Order of accuracy","score":0.42988696694374084},{"id":"https://openalex.org/keywords/legendre-function","display_name":"Legendre function","score":0.4274097979068756},{"id":"https://openalex.org/keywords/diffusion-equation","display_name":"Diffusion equation","score":0.4248884916305542},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42129185795783997},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4204520285129547},{"id":"https://openalex.org/keywords/legendres-equation","display_name":"Legendre's equation","score":0.416535884141922},{"id":"https://openalex.org/keywords/method-of-characteristics","display_name":"Method of characteristics","score":0.1560148000717163},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.1528448462486267}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.828062891960144},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.8184372186660767},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6531299352645874},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.6140497922897339},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.6071332693099976},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.562994122505188},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.548916220664978},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5112042427062988},{"id":"https://openalex.org/C123769847","wikidata":"https://www.wikidata.org/wiki/Q6517888","display_name":"Legendre wavelet","level":5,"score":0.4733025133609772},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.43718421459198},{"id":"https://openalex.org/C39177556","wikidata":"https://www.wikidata.org/wiki/Q7100567","display_name":"Order of accuracy","level":4,"score":0.42988696694374084},{"id":"https://openalex.org/C153262748","wikidata":"https://www.wikidata.org/wiki/Q17098449","display_name":"Legendre function","level":3,"score":0.4274097979068756},{"id":"https://openalex.org/C571446","wikidata":"https://www.wikidata.org/wiki/Q6510183","display_name":"Diffusion equation","level":3,"score":0.4248884916305542},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42129185795783997},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4204520285129547},{"id":"https://openalex.org/C86287605","wikidata":"https://www.wikidata.org/wiki/Q4454970","display_name":"Legendre's equation","level":3,"score":0.416535884141922},{"id":"https://openalex.org/C33163149","wikidata":"https://www.wikidata.org/wiki/Q1785085","display_name":"Method of characteristics","level":3,"score":0.1560148000717163},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.1528448462486267},{"id":"https://openalex.org/C2780378061","wikidata":"https://www.wikidata.org/wiki/Q25351891","display_name":"Service (business)","level":2,"score":0.0},{"id":"https://openalex.org/C196216189","wikidata":"https://www.wikidata.org/wiki/Q2867","display_name":"Wavelet transform","level":3,"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/C46286280","wikidata":"https://www.wikidata.org/wiki/Q2414958","display_name":"Discrete wavelet transform","level":4,"score":0.0},{"id":"https://openalex.org/C136264566","wikidata":"https://www.wikidata.org/wiki/Q159810","display_name":"Economy","level":1,"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160.2019.1608968","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1608968","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","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6203978640","display_name":null,"funder_award_id":"11771163","funder_id":"https://openalex.org/F4320332587","funder_display_name":"Natural Science Foundation of Ningbo"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320332587","display_name":"Natural Science Foundation of Ningbo","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":44,"referenced_works":["https://openalex.org/W374977727","https://openalex.org/W620183161","https://openalex.org/W794756600","https://openalex.org/W1584611818","https://openalex.org/W1870321888","https://openalex.org/W1975357829","https://openalex.org/W1991925980","https://openalex.org/W2001011967","https://openalex.org/W2008119119","https://openalex.org/W2009456713","https://openalex.org/W2014666630","https://openalex.org/W2026882859","https://openalex.org/W2026946703","https://openalex.org/W2030146312","https://openalex.org/W2041079303","https://openalex.org/W2058524173","https://openalex.org/W2065232540","https://openalex.org/W2082849992","https://openalex.org/W2095561603","https://openalex.org/W2110245394","https://openalex.org/W2290964926","https://openalex.org/W2304262673","https://openalex.org/W2310142350","https://openalex.org/W2480732224","https://openalex.org/W2510459129","https://openalex.org/W2567244142","https://openalex.org/W2577996109","https://openalex.org/W2582729665","https://openalex.org/W2753893094","https://openalex.org/W2782480706","https://openalex.org/W2787959293","https://openalex.org/W2790130100","https://openalex.org/W2790717678","https://openalex.org/W2790820636","https://openalex.org/W2793446492","https://openalex.org/W2795387349","https://openalex.org/W2799821673","https://openalex.org/W2807866283","https://openalex.org/W2883236899","https://openalex.org/W2889281424","https://openalex.org/W2893632445","https://openalex.org/W2963805080","https://openalex.org/W2964228534","https://openalex.org/W4253937035"],"related_works":["https://openalex.org/W1607269076","https://openalex.org/W2953361964","https://openalex.org/W2022538265","https://openalex.org/W3005861762","https://openalex.org/W2288349649","https://openalex.org/W3200178185","https://openalex.org/W2379187670","https://openalex.org/W4296546724","https://openalex.org/W2591407508","https://openalex.org/W2025296287"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,31],"consider":[4],"the":[5,11,22,28,33,44,51,57,64,105],"Galerkin\u2013Legendre":[6],"spectral":[7,59,93],"method":[8,60],"for":[9,63],"solving":[10],"two-dimensional":[12],"distributed-order":[13,29],"time":[14,39,91],"fractional":[15,40,54],"fourth-order":[16,83],"partial":[17],"differential":[18],"equation.":[19,42],"By":[20],"utilizing":[21],"composite":[23],"Simpson":[24],"formula":[25,46],"to":[26,49,71,103],"discretize":[27],"integral,":[30],"transform":[32],"considered":[34],"equation":[35],"into":[36],"a":[37],"multi-term":[38,52],"sub-diffusion":[41],"Then":[43],"L2-1\u03c3":[45],"is":[47,61,69],"used":[48],"approximate":[50],"Caputo":[53],"derivatives":[55],"and":[56,75,80,92],"Legendre":[58],"employed":[62],"spatial":[65],"discretization.":[66],"The":[67],"scheme":[68],"proved":[70],"be":[72],"unconditionally":[73],"stable":[74],"convergent":[76],"in":[77,85,90,95],"both":[78],"L2-":[79],"L\u221e-norms":[81],"with":[82],"accuracy":[84,89,94],"distributed":[86],"order,":[87],"second-order":[88],"space.":[96],"Finally,":[97],"some":[98],"numerical":[99],"tests":[100],"are":[101],"performed":[102],"verify":[104],"theoretical":[106],"results.":[107]},"counts_by_year":[{"year":2025,"cited_by_count":10},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":1}],"updated_date":"2026-08-15T07:11:24.734988","created_date":"2025-10-10T00:00:00"}
