{"id":"https://openalex.org/W4283329840","doi":"https://doi.org/10.3390/sym14061279","title":"Stability of Euler Methods for Fuzzy Differential Equation","display_name":"Stability of Euler Methods for Fuzzy Differential Equation","publication_year":2022,"publication_date":"2022-06-20","ids":{"openalex":"https://openalex.org/W4283329840","doi":"https://doi.org/10.3390/sym14061279"},"language":"en","primary_location":{"id":"doi:10.3390/sym14061279","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061279","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1279/pdf?version=1655794624","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/6/1279/pdf?version=1655794624","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088538223","display_name":"Cuilian You","orcid":"https://orcid.org/0000-0002-9855-7370"},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Cuilian You","raw_affiliation_strings":["College of Mathematics and Information Science, Hebei University, Baoding 071002, China","Hebei Key Lab of Machine Learning and Computational Intelligence, Hebei University, Baoding 071002, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Hebei University, Baoding 071002, China","institution_ids":["https://openalex.org/I43337087"]},{"raw_affiliation_string":"Hebei Key Lab of Machine Learning and Computational Intelligence, Hebei University, Baoding 071002, China","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032605369","display_name":"Yan Cheng","orcid":"https://orcid.org/0000-0002-3208-2536"},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yan Cheng","raw_affiliation_strings":["College of Mathematics and Information Science, Hebei University, Baoding 071002, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Hebei University, Baoding 071002, China","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5107459843","display_name":"Hongyan Ma","orcid":null},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Hongyan Ma","raw_affiliation_strings":["College of Mathematics and Information Science, Hebei University, Baoding 071002, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Hebei University, Baoding 071002, China","institution_ids":["https://openalex.org/I43337087"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5088538223","https://openalex.org/A5107459843"],"corresponding_institution_ids":["https://openalex.org/I43337087"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.9312,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.74801193,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"14","issue":"6","first_page":"1279","last_page":"1279"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9933000206947327,"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"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9902999997138977,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7981197834014893},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.7245427966117859},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.6178096532821655},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5539209842681885},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5296076536178589},{"id":"https://openalex.org/keywords/euler-method","display_name":"Euler method","score":0.5244708061218262},{"id":"https://openalex.org/keywords/backward-euler-method","display_name":"Backward Euler method","score":0.4941089451313019},{"id":"https://openalex.org/keywords/semi-implicit-euler-method","display_name":"Semi-implicit Euler method","score":0.49015292525291443},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4155518710613251},{"id":"https://openalex.org/keywords/euler-equations","display_name":"Euler equations","score":0.3944553732872009},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13702377676963806},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.04389962553977966}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7981197834014893},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.7245427966117859},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.6178096532821655},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5539209842681885},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5296076536178589},{"id":"https://openalex.org/C75380026","wikidata":"https://www.wikidata.org/wiki/Q868454","display_name":"Euler method","level":3,"score":0.5244708061218262},{"id":"https://openalex.org/C768646","wikidata":"https://www.wikidata.org/wiki/Q2736820","display_name":"Backward Euler method","level":3,"score":0.4941089451313019},{"id":"https://openalex.org/C93082080","wikidata":"https://www.wikidata.org/wiki/Q5475314","display_name":"Semi-implicit Euler method","level":4,"score":0.49015292525291443},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4155518710613251},{"id":"https://openalex.org/C38409319","wikidata":"https://www.wikidata.org/wiki/Q375175","display_name":"Euler equations","level":2,"score":0.3944553732872009},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13702377676963806},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.04389962553977966},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14061279","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061279","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1279/pdf?version=1655794624","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:f5b943237bac4c03a0e53be6ed69ff08","is_oa":true,"landing_page_url":"https://doaj.org/article/f5b943237bac4c03a0e53be6ed69ff08","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":"Symmetry, Vol 14, Iss 6, p 1279 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/6/1279/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14061279","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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14061279","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061279","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1279/pdf?version=1655794624","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4283329840.pdf","grobid_xml":"https://content.openalex.org/works/W4283329840.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1980681767","https://openalex.org/W1993344872","https://openalex.org/W2017725028","https://openalex.org/W2044637309","https://openalex.org/W2075572284","https://openalex.org/W2081026153","https://openalex.org/W2127740852","https://openalex.org/W2128530362","https://openalex.org/W2151755084","https://openalex.org/W2187611310","https://openalex.org/W2213086817","https://openalex.org/W2268289160","https://openalex.org/W2772730311","https://openalex.org/W2791071105","https://openalex.org/W2794981702","https://openalex.org/W2806406393","https://openalex.org/W2811198387","https://openalex.org/W2909949298","https://openalex.org/W2947197557","https://openalex.org/W2984237296","https://openalex.org/W3023199142","https://openalex.org/W6678868654","https://openalex.org/W6686915614"],"related_works":["https://openalex.org/W2089955609","https://openalex.org/W2466040413","https://openalex.org/W4382053489","https://openalex.org/W2189596786","https://openalex.org/W2009711572","https://openalex.org/W2039299714","https://openalex.org/W2365810470","https://openalex.org/W2134248673","https://openalex.org/W2378504881","https://openalex.org/W1674877062"],"abstract_inverted_index":{"The":[0,18],"Liu":[1,30],"process":[2,6],"is":[3,10,87,128,137],"a":[4,11,24,78],"fuzzy":[5,25,35,74,95,100,113,125,134],"whose":[7],"membership":[8],"function":[9,13],"symmetric":[12],"on":[14,141],"an":[15,93,111,123],"expected":[16],"value.":[17],"object":[19],"of":[20,144],"this":[21],"paper":[22],"was":[23],"differential":[26,75],"equation":[27,76],"driven":[28],"by":[29],"process.":[31],"Since":[32],"the":[33,50,142],"existing":[34],"Euler":[36,40,43,47,96,101,114,126,135],"solving":[37],"methods":[38],"(explicit":[39],"scheme,":[41,44],"semi-implicit":[42,99,133],"and":[45,68,98,106,131,146],"implicit":[46,124],"scheme)":[48],"have":[49],"same":[51],"convergence,":[52],"to":[53,89,117],"compare":[54],"them,":[55],"we":[56,81],"presented":[57],"four":[58],"stabilities,":[59],"i.e.,":[60],"asymptotical":[61,104],"stability,":[62,65,67,109],"mean":[63,84,107],"square":[64,85,108],"exponential":[66,90],"A":[69,119,129,138],"stability.":[70,91],"By":[71],"choosing":[72],"special":[73],"as":[77],"test":[79],"equation,":[80],"deduced":[82],"that":[83,122],"stability":[86,105,120],"equivalent":[88],"Furthermore,":[92],"explicit":[94,112],"scheme":[97,102,115,127,136],"showed":[103],"while":[110],"failed":[116],"meet":[118],"but":[121],"stable,":[130],"whether":[132],"stable":[139],"depends":[140],"values":[143],"\u03b1":[145],"\u03bb.":[147]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}