{"id":"https://openalex.org/W4398218340","doi":"https://doi.org/10.1007/s11786-024-00580-3","title":"Solving Partial Differential Problems with Tau Toolbox","display_name":"Solving Partial Differential Problems with Tau Toolbox","publication_year":2024,"publication_date":"2024-05-22","ids":{"openalex":"https://openalex.org/W4398218340","doi":"https://doi.org/10.1007/s11786-024-00580-3"},"language":"en","primary_location":{"id":"doi:10.1007/s11786-024-00580-3","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s11786-024-00580-3","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11786-024-00580-3.pdf","source":{"id":"https://openalex.org/S136038430","display_name":"Mathematics in Computer Science","issn_l":"1661-8270","issn":["1661-8270","1661-8289"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics in Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s11786-024-00580-3.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102674548","display_name":"Nilson J. Lima","orcid":null},"institutions":[{"id":"https://openalex.org/I182534213","display_name":"Universidade do Porto","ror":"https://ror.org/043pwc612","country_code":"PT","type":"education","lineage":["https://openalex.org/I182534213"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Nilson J. Lima","raw_affiliation_strings":["CMUP, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade do Porto, Rua do Campo Alegre s/n, 4169-007, Porto, Portugal"],"affiliations":[{"raw_affiliation_string":"CMUP, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade do Porto, Rua do Campo Alegre s/n, 4169-007, Porto, Portugal","institution_ids":["https://openalex.org/I182534213"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014495885","display_name":"Jos\u00e9 M. A. Matos","orcid":"https://orcid.org/0000-0003-3512-5930"},"institutions":[{"id":"https://openalex.org/I83863532","display_name":"Polytechnic Institute of Porto","ror":"https://ror.org/04988re48","country_code":"PT","type":"education","lineage":["https://openalex.org/I83863532"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Jos\u00e9 M. A. Matos","raw_affiliation_strings":["CMUP, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Instituto Superior de Engenharia, Instituto Polit\u00e9cnico do Porto, Rua do Campo Alegre s/n, 4169-007, Porto, Portugal"],"affiliations":[{"raw_affiliation_string":"CMUP, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Instituto Superior de Engenharia, Instituto Polit\u00e9cnico do Porto, Rua do Campo Alegre s/n, 4169-007, Porto, Portugal","institution_ids":["https://openalex.org/I83863532"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039423894","display_name":"Paulo B. Vasconcelos","orcid":"https://orcid.org/0000-0002-7132-880X"},"institutions":[{"id":"https://openalex.org/I182534213","display_name":"Universidade do Porto","ror":"https://ror.org/043pwc612","country_code":"PT","type":"education","lineage":["https://openalex.org/I182534213"]}],"countries":["PT"],"is_corresponding":true,"raw_author_name":"Paulo B. Vasconcelos","raw_affiliation_strings":["CMUP, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-464, Porto, Portugal"],"affiliations":[{"raw_affiliation_string":"CMUP, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-464, Porto, Portugal","institution_ids":["https://openalex.org/I182534213"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5039423894"],"corresponding_institution_ids":["https://openalex.org/I182534213"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.09082805,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"18","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5319486260414124},{"id":"https://openalex.org/keywords/toolbox","display_name":"Toolbox","score":0.521005392074585},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.5188125967979431},{"id":"https://openalex.org/keywords/numerical-stability","display_name":"Numerical stability","score":0.5172284841537476},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.508000910282135},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5020365715026855},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.4983069896697998},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49044081568717957},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4813690781593323},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.47450822591781616},{"id":"https://openalex.org/keywords/numerical-partial-differential-equations","display_name":"Numerical partial differential equations","score":0.44737234711647034},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.44181060791015625},{"id":"https://openalex.org/keywords/spectral-method","display_name":"Spectral method","score":0.43967312574386597},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.41119468212127686},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3977337181568146},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.33225351572036743},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18439403176307678},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1504892110824585}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5319486260414124},{"id":"https://openalex.org/C2777655017","wikidata":"https://www.wikidata.org/wiki/Q1501161","display_name":"Toolbox","level":2,"score":0.521005392074585},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.5188125967979431},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.5172284841537476},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.508000910282135},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5020365715026855},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.4983069896697998},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49044081568717957},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4813690781593323},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.47450822591781616},{"id":"https://openalex.org/C97826883","wikidata":"https://www.wikidata.org/wiki/Q7069659","display_name":"Numerical partial differential equations","level":3,"score":0.44737234711647034},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.44181060791015625},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.43967312574386597},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.41119468212127686},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3977337181568146},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33225351572036743},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18439403176307678},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1504892110824585},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s11786-024-00580-3","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s11786-024-00580-3","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11786-024-00580-3.pdf","source":{"id":"https://openalex.org/S136038430","display_name":"Mathematics in Computer Science","issn_l":"1661-8270","issn":["1661-8270","1661-8289"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics in Computer Science","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s11786-024-00580-3","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s11786-024-00580-3","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11786-024-00580-3.pdf","source":{"id":"https://openalex.org/S136038430","display_name":"Mathematics in Computer Science","issn_l":"1661-8270","issn":["1661-8270","1661-8289"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics in Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320324086","display_name":"Universidade do Porto","ror":"https://ror.org/043pwc612"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4398218340.pdf"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W1678272269","https://openalex.org/W1974809035","https://openalex.org/W1987449763","https://openalex.org/W2025485823","https://openalex.org/W2074193803","https://openalex.org/W2133643411","https://openalex.org/W2171692351","https://openalex.org/W2298381282","https://openalex.org/W2901197498"],"related_works":["https://openalex.org/W2102789413","https://openalex.org/W1965237235","https://openalex.org/W4230788905","https://openalex.org/W2052500721","https://openalex.org/W2940603871","https://openalex.org/W1532287630","https://openalex.org/W2028331352","https://openalex.org/W4299508867","https://openalex.org/W1555930509","https://openalex.org/W1985583842"],"abstract_inverted_index":{"Abstract":[0],"This":[1],"paper":[2],"briefly":[3],"examines":[4],"how":[5,104],"literature":[6],"addresses":[7],"the":[8,16,22,32,35,54,65,85,95,112],"numerical":[9,27,82,92],"solution":[10,28,41,86],"of":[11,24,34,51,56,87],"partial":[12,118],"differential":[13,119],"equations":[14],"by":[15],"spectral":[17,68],"Tau":[18],"method.":[19],"It":[20,48,78,90],"discusses":[21],"implementation":[23],"such":[25],"a":[26,81],"for":[29,84],"PDE\u2019s":[30],"presenting":[31],"construction":[33],"problem\u2019s":[36],"algebraic":[37],"representation":[38],"and":[39,53,60,75,98,106],"exploring":[40,57],"mechanisms":[42],"with":[43],"different":[44],"orthogonal":[45],"polynomial":[46],"bases.":[47],"highlights":[49],"contexts":[50],"opportunity":[52],"advantages":[55],"low-rank":[58],"approximations":[59],"well-conditioned":[61],"linear":[62],"systems,":[63],"despite":[64],"fact":[66],"that":[67],"methods":[69],"usually":[70],"give":[71],"rise":[72],"to":[73,110,113,117],"dense":[74],"ill-conditioned":[76],"matrices.":[77],"presents":[79],",":[80],"library":[83],"integro-differential":[88],"problems.":[89,120],"shows":[91,103],"experiments":[93],"illustrating":[94],"implementations\u2019":[96],"accuracy":[97],"computational":[99],"costs.":[100],"Finally,":[101],"it":[102,108],"simple":[105],"easy":[107],"is":[109],"use":[111],"obtain":[114],"approximate":[115],"solutions":[116]},"counts_by_year":[],"updated_date":"2025-12-22T23:10:17.713674","created_date":"2025-10-10T00:00:00"}