{"id":"https://openalex.org/W2913692467","doi":"https://doi.org/10.3390/sym11020205","title":"An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations","display_name":"An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations","publication_year":2019,"publication_date":"2019-02-12","ids":{"openalex":"https://openalex.org/W2913692467","doi":"https://doi.org/10.3390/sym11020205","mag":"2913692467"},"language":"en","primary_location":{"id":"doi:10.3390/sym11020205","doi":"https://doi.org/10.3390/sym11020205","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym11020205","pdf_url":"https://www.mdpi.com/2073-8994/11/2/205/pdf?version=1550461334","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,"listed_in":["anvur-scientific","cwts-core","norway-1","poland-70","russia-white-list-1","vabb-shw"],"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"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/11/2/205/pdf?version=1550461334","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004534298","display_name":"Mohammad Alaroud","orcid":"https://orcid.org/0000-0001-9099-5619","observed_orcids":["https://orcid.org/0000-0001-9099-5619"]},"institutions":[{"id":"https://openalex.org/I885383172","display_name":"National University of Malaysia","ror":"https://ror.org/00bw8d226","country_code":"MY","type":"education","lineage":["https://openalex.org/I885383172"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Mohammad Alaroud","raw_affiliation_strings":["Center for Modelling and Data Science, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor DE, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Modelling and Data Science, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor DE, Malaysia","institution_ids":["https://openalex.org/I885383172"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029673983","display_name":"Mohammed Al\u2010Smadi","orcid":"https://orcid.org/0000-0003-0226-7254","observed_orcids":["https://orcid.org/0000-0003-0226-7254","https://orcid.org/0000-0002-5910-3843"]},"institutions":[{"id":"https://openalex.org/I33926330","display_name":"Al-Balqa Applied University","ror":"https://ror.org/00qedmt22","country_code":"JO","type":"education","lineage":["https://openalex.org/I33926330"]}],"countries":["JO"],"is_corresponding":false,"raw_author_name":"Mohammed Al-Smadi","raw_affiliation_strings":["Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan","institution_ids":["https://openalex.org/I33926330"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059989391","display_name":"Rokiah Rozita Ahmad","orcid":"https://orcid.org/0000-0002-1781-3716","observed_orcids":["https://orcid.org/0000-0002-1781-3716"]},"institutions":[{"id":"https://openalex.org/I885383172","display_name":"National University of Malaysia","ror":"https://ror.org/00bw8d226","country_code":"MY","type":"education","lineage":["https://openalex.org/I885383172"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Rokiah Rozita Ahmad","raw_affiliation_strings":["Center for Modelling and Data Science, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor DE, Malaysia"],"raw_orcid":"https://orcid.org/0000-0002-1781-3716","affiliations":[{"raw_affiliation_string":"Center for Modelling and Data Science, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor DE, Malaysia","institution_ids":["https://openalex.org/I885383172"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019903939","display_name":"Ummul Khair Salma Din","orcid":"https://orcid.org/0000-0003-4407-1241","observed_orcids":["https://orcid.org/0000-0003-4407-1241"]},"institutions":[{"id":"https://openalex.org/I885383172","display_name":"National University of Malaysia","ror":"https://ror.org/00bw8d226","country_code":"MY","type":"education","lineage":["https://openalex.org/I885383172"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Ummul Khair Salma Din","raw_affiliation_strings":["Center for Modelling and Data Science, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor DE, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Modelling and Data Science, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor DE, Malaysia","institution_ids":["https://openalex.org/I885383172"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5059989391"],"corresponding_institution_ids":["https://openalex.org/I885383172"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1418},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1418},"fwci":5.9419,"has_fulltext":true,"cited_by_count":56,"citation_normalized_percentile":{"value":0.97397213,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"11","issue":"2","first_page":"205","last_page":"205"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":1.0,"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":1.0,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":9.999999747378752e-05,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.0,"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/volterra-integro-differential-equations","display_name":"Volterra integro-differential equations","score":0.8989999890327454},{"id":"https://openalex.org/keywords/symmetric-triangular-fuzzy-numbers","display_name":"symmetric triangular fuzzy numbers","score":0.828000009059906},{"id":"https://openalex.org/keywords/fuzzy-fractional-differential-equations","display_name":"fuzzy fractional differential equations","score":0.7329999804496765}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7888503074645996},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.6227421164512634},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5059897303581238},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.46146658062934875},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4561145007610321},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.41892462968826294},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.41180577874183655},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3523384630680084},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.18589508533477783},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13442474603652954},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym11020205","doi":"https://doi.org/10.3390/sym11020205","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym11020205","pdf_url":"https://www.mdpi.com/2073-8994/11/2/205/pdf?version=1550461334","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,"listed_in":["anvur-scientific","cwts-core","norway-1","poland-70","russia-white-list-1","vabb-shw"],"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"}],"best_oa_location":{"id":"doi:10.3390/sym11020205","doi":"https://doi.org/10.3390/sym11020205","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym11020205","pdf_url":"https://www.mdpi.com/2073-8994/11/2/205/pdf?version=1550461334","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,"listed_in":["anvur-scientific","cwts-core","norway-1","poland-70","russia-white-list-1","vabb-shw"],"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":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3016819552","display_name":null,"funder_award_id":"GP-K007788 and GP-K006926","funder_id":"https://openalex.org/F4320322699","funder_display_name":"Universiti Kebangsaan Malaysia"}],"funders":[{"id":"https://openalex.org/F4320322699","display_name":"Universiti Kebangsaan Malaysia","ror":"https://ror.org/00bw8d226"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2913692467.pdf","grobid_xml":"https://content.openalex.org/works/W2913692467.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1940323032","https://openalex.org/W1965955764","https://openalex.org/W1990950138","https://openalex.org/W1992012098","https://openalex.org/W1994178762","https://openalex.org/W1999080212","https://openalex.org/W2004610033","https://openalex.org/W2009342598","https://openalex.org/W2040987249","https://openalex.org/W2042459455","https://openalex.org/W2052105007","https://openalex.org/W2070545443","https://openalex.org/W2071773958","https://openalex.org/W2075014533","https://openalex.org/W2076628063","https://openalex.org/W2107983068","https://openalex.org/W2132943071","https://openalex.org/W2146747427","https://openalex.org/W2320122857","https://openalex.org/W2321307647","https://openalex.org/W2380392230","https://openalex.org/W2470505627","https://openalex.org/W2527845113","https://openalex.org/W2550822554","https://openalex.org/W2609299810","https://openalex.org/W2757171398","https://openalex.org/W2772923827","https://openalex.org/W2799105508","https://openalex.org/W2809989124","https://openalex.org/W2819231394","https://openalex.org/W2897394303","https://openalex.org/W2907073664","https://openalex.org/W2962902678","https://openalex.org/W7054860566"],"related_works":["https://openalex.org/W4200433985","https://openalex.org/W4224083797","https://openalex.org/W2809989124","https://openalex.org/W2945475136","https://openalex.org/W2973807632","https://openalex.org/W4244663968","https://openalex.org/W2811250917","https://openalex.org/W7167216733","https://openalex.org/W4412058838","https://openalex.org/W2956152145"],"abstract_inverted_index":{"The":[0,70,86,114,126,190],"modeling":[1],"of":[2,44,50,83,110,131,156,176,185,208],"fuzzy":[3,45,64,115,123,158,166,171,209],"fractional":[4,28,46,98,116,210],"integro-differential":[5,48],"equations":[6,49],"is":[7,198],"a":[8,20,42,174,183],"very":[9],"significant":[10],"matter":[11],"in":[12,92],"engineering":[13],"and":[14,36,79,108,129,142,151,160,179,188,202],"applied":[15],"sciences.":[16],"This":[17],"paper":[18],"presents":[19],"novel":[21],"treatment":[22],"algorithm":[23,72,88],"based":[24],"on":[25,80,104,173],"utilizing":[26],"the":[27,38,75,81,105,121,132,146,149,157,162,169,195],"residual":[29,76],"power":[30,99],"series":[31,100],"(FRPS)":[32],"method":[33],"to":[34,60,182,204],"study":[35],"interpret":[37],"approximated":[39,90],"solutions":[40,91,172],"for":[41],"class":[43],"Volterra":[47],"order":[51],"0":[52],"<":[53],"\u03b2":[54],"\u2264":[55],"1":[56],"which":[57],"are":[58,118,135],"subject":[59],"appropriate":[61],"symmetric":[62,164,170],"triangular":[63,165],"conditions":[65],"under":[66],"strongly":[67],"generalized":[68,84],"differentiability.":[69],"proposed":[71,133],"relies":[73],"upon":[74],"error":[77],"concept":[78],"formula":[82],"Taylor.":[85],"FRPS":[87,196],"provides":[89],"parametric":[93],"form":[94],"with":[95],"rapidly":[96],"convergent":[97],"without":[101],"linearization,":[102],"limitation":[103],"problem\u2019s":[106],"nature,":[107],"sort":[109],"classification":[111],"or":[112],"perturbation.":[113],"derivatives":[117],"described":[119],"via":[120],"Caputo":[122],"H":[124],"-differentiable.":[125],"ability,":[127],"effectiveness,":[128],"simplicity":[130],"technique":[134],"demonstrated":[136],"by":[137],"testing":[138],"two":[139],"applications.":[140],"Graphical":[141],"numerical":[143],"results":[144,192],"reveal":[145,193],"symmetry":[147],"between":[148],"lower":[150],"upper":[152],"r":[153],"-cut":[154],"representations":[155],"solution":[159],"satisfy":[161],"convex":[163],"number.":[167],"Notably,":[168],"focus":[175],"their":[177],"core":[178],"support":[180],"refer":[181],"sense":[184],"proportion,":[186],"harmony,":[187],"balance.":[189],"obtained":[191],"that":[194],"scheme":[197],"simple,":[199],"straightforward,":[200],"accurate":[201],"convenient":[203],"solve":[205],"different":[206],"forms":[207],"differential":[211],"equations.":[212]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":9},{"year":2022,"cited_by_count":12},{"year":2021,"cited_by_count":10},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":7}],"updated_date":"2026-11-13T13:42:25.865203","created_date":"2019-02-21T00:00:00"}