{"id":"https://openalex.org/W2789698180","doi":"https://doi.org/10.3390/sym10020042","title":"Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives","display_name":"Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives","publication_year":2018,"publication_date":"2018-02-08","ids":{"openalex":"https://openalex.org/W2789698180","doi":"https://doi.org/10.3390/sym10020042","mag":"2789698180"},"language":"en","primary_location":{"id":"doi:10.3390/sym10020042","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10020042","pdf_url":"https://www.mdpi.com/2073-8994/10/2/42/pdf?version=1518428246","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/2073-8994/10/2/42/pdf?version=1518428246","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027603299","display_name":"Syahirbanun Isa","orcid":null},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]},{"id":"https://openalex.org/I930072361","display_name":"Tun Hussein Onn University of Malaysia","ror":"https://ror.org/01c5wha71","country_code":"MY","type":"education","lineage":["https://openalex.org/I930072361"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Syahirbanun Isa","raw_affiliation_strings":["Department of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn Malaysia (Pagoh Campus), Muar 84000, Johor, Malaysia","Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn Malaysia (Pagoh Campus), Muar 84000, Johor, Malaysia","institution_ids":["https://openalex.org/I930072361"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]},{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5085828782","display_name":"Zanariah Abdul Majid","orcid":"https://orcid.org/0000-0003-4263-2287"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Zanariah Abdul Majid","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":"https://orcid.org/0000-0003-4263-2287","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]},{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069012279","display_name":"Fudziah Ismail","orcid":"https://orcid.org/0000-0002-1548-8702"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Fudziah Ismail","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]},{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021855509","display_name":"Faranak Rabiei","orcid":"https://orcid.org/0000-0002-0648-0477"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Faranak Rabiei","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":"https://orcid.org/0000-0002-0648-0477","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]},{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5085828782"],"corresponding_institution_ids":["https://openalex.org/I130343225"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.7247,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.71703187,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"10","issue":"2","first_page":"42","last_page":"42"},"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":0.9919000267982483,"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/T12508","display_name":"Functional Equations Stability Results","score":0.9907000064849854,"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/mathematics","display_name":"Mathematics","score":0.7874680757522583},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.67662513256073},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.5802561044692993},{"id":"https://openalex.org/keywords/defuzzification","display_name":"Defuzzification","score":0.5144736766815186},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4935368299484253},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.45624983310699463},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.41491344571113586},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.39369913935661316},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.3372150659561157},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.3366659879684448},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1388569176197052},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.06495866179466248}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7874680757522583},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.67662513256073},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.5802561044692993},{"id":"https://openalex.org/C170260401","wikidata":"https://www.wikidata.org/wiki/Q5251735","display_name":"Defuzzification","level":5,"score":0.5144736766815186},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4935368299484253},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.45624983310699463},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.41491344571113586},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39369913935661316},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3372150659561157},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.3366659879684448},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1388569176197052},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.06495866179466248}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym10020042","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10020042","pdf_url":"https://www.mdpi.com/2073-8994/10/2/42/pdf?version=1518428246","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:0e65d53240174000b3287a26d0ee1498","is_oa":false,"landing_page_url":"https://doaj.org/article/0e65d53240174000b3287a26d0ee1498","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 10, Iss 2, p 42 (2018)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/10/2/42/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym10020042","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; Volume 10; Issue 2; Pages: 42","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym10020042","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10020042","pdf_url":"https://www.mdpi.com/2073-8994/10/2/42/pdf?version=1518428246","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/W2789698180.pdf","grobid_xml":"https://content.openalex.org/works/W2789698180.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W102596075","https://openalex.org/W1780876027","https://openalex.org/W1826050079","https://openalex.org/W1965955764","https://openalex.org/W1983161216","https://openalex.org/W1994178762","https://openalex.org/W2003184377","https://openalex.org/W2012580290","https://openalex.org/W2020508693","https://openalex.org/W2022135193","https://openalex.org/W2028592149","https://openalex.org/W2031628517","https://openalex.org/W2050444457","https://openalex.org/W2066084209","https://openalex.org/W2071305979","https://openalex.org/W2075014533","https://openalex.org/W2076628063","https://openalex.org/W2080430919","https://openalex.org/W2086144729","https://openalex.org/W2089021392","https://openalex.org/W2092092154","https://openalex.org/W2092781598","https://openalex.org/W2095478259","https://openalex.org/W2106751301","https://openalex.org/W2144091775","https://openalex.org/W2169253851","https://openalex.org/W2179883489","https://openalex.org/W2180113582","https://openalex.org/W2188107347","https://openalex.org/W2217271039","https://openalex.org/W2218874955","https://openalex.org/W2238088374","https://openalex.org/W2334383698","https://openalex.org/W2343019824","https://openalex.org/W2587856378","https://openalex.org/W6637879585","https://openalex.org/W6685646043","https://openalex.org/W6689884322","https://openalex.org/W6733584428"],"related_works":["https://openalex.org/W2773005380","https://openalex.org/W4388451300","https://openalex.org/W2060361571","https://openalex.org/W1879432897","https://openalex.org/W2256857057","https://openalex.org/W3167814444","https://openalex.org/W3176863924","https://openalex.org/W2023513928","https://openalex.org/W2182392910","https://openalex.org/W4361984403"],"abstract_inverted_index":{"In":[0,130],"this":[1,41],"paper,":[2],"the":[3,43,50,56,77,108,113,124,134,149,161,173,176,184,187],"solution":[4],"of":[5,18,79,104,116,154,160,175],"fuzzy":[6,51,63,67,82,90,95,119,135,196],"differential":[7,37,156,197],"equations":[8],"is":[9,25,59,69,92,121,140,144,164,180,190],"approximated":[10],"numerically":[11],"using":[12],"diagonally":[13],"implicit":[14],"multistep":[15,22],"block":[16,23],"method":[17,24,33,44,163,189],"order":[19,131],"four.":[20],"The":[21,65,87,158],"well":[26],"known":[27],"as":[28],"an":[29],"efficient":[30],"and":[31,101,123,192],"accurate":[32],"for":[34],"solving":[35,195],"ordinary":[36,155],"equations,":[38],"hence":[39],"in":[40,183,194],"paper":[42],"will":[45],"be":[46,85],"used":[47,146],"to":[48,111,132,147,171],"solve":[49],"initial":[52,57],"value":[53,58],"problems":[54,150],"where":[55],"a":[60,80,93,152],"symmetric":[61,88,117],"triangular":[62,66,81,89,94,118],"interval.":[64],"number":[68,83,120],"not":[70],"necessarily":[71],"symmetric,":[72],"however":[73],"by":[74],"imposing":[75],"symmetry":[76],"definition":[78],"can":[84],"simplified.":[86],"interval":[91,96],"that":[97,186],"has":[98],"same":[99],"left":[100],"right":[102],"width":[103],"membership":[105],"function":[106],"from":[107],"center.":[109],"Due":[110],"this,":[112],"parametric":[114],"form":[115],"simple":[122],"performing":[125],"arithmetic":[126],"operations":[127],"become":[128],"easier.":[129],"interpret":[133],"problems,":[136],"Seikkala\u2019s":[137],"derivative":[138],"approach":[139],"implemented.":[141],"Characterization":[142],"theorem":[143],"then":[145],"translate":[148],"into":[151],"system":[153],"equations.":[157,198],"convergence":[159],"introduced":[162],"also":[165],"proved.":[166],"Numerical":[167],"examples":[168],"are":[169],"given":[170],"investigate":[172],"performance":[174],"proposed":[177,188],"method.":[178],"It":[179],"clearly":[181],"shown":[182],"results":[185],"comparable":[191],"reliable":[193]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}
