{"id":"https://openalex.org/W4229372237","doi":"https://doi.org/10.3390/sym14050907","title":"Approximate Solution of Nonlinear Time-Fractional Klein-Gordon Equations Using Yang Transform","display_name":"Approximate Solution of Nonlinear Time-Fractional Klein-Gordon Equations Using Yang Transform","publication_year":2022,"publication_date":"2022-04-29","ids":{"openalex":"https://openalex.org/W4229372237","doi":"https://doi.org/10.3390/sym14050907"},"language":"en","primary_location":{"id":"doi:10.3390/sym14050907","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050907","pdf_url":"https://www.mdpi.com/2073-8994/14/5/907/pdf?version=1651212059","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/2073-8994/14/5/907/pdf?version=1651212059","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043251411","display_name":"Jinxing Liu","orcid":"https://orcid.org/0000-0003-1071-2231"},"institutions":[{"id":"https://openalex.org/I4210131342","display_name":"Yibin University","ror":"https://ror.org/03w8m2977","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210131342"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jinxing Liu","raw_affiliation_strings":["Faculty of Science, Yibin University, Yibin 644000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Science, Yibin University, Yibin 644000, China","institution_ids":["https://openalex.org/I4210131342"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069957230","display_name":"Muhammad Nadeem","orcid":"https://orcid.org/0000-0002-9349-4729"},"institutions":[{"id":"https://openalex.org/I4210131342","display_name":"Yibin University","ror":"https://ror.org/03w8m2977","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210131342"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Muhammad Nadeem","raw_affiliation_strings":["Faculty of Science, Yibin University, Yibin 644000, China"],"raw_orcid":"https://orcid.org/0000-0002-9349-4729","affiliations":[{"raw_affiliation_string":"Faculty of Science, Yibin University, Yibin 644000, China","institution_ids":["https://openalex.org/I4210131342"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5049564865","display_name":"Mustafa Habib","orcid":"https://orcid.org/0000-0002-2797-1865"},"institutions":[{"id":"https://openalex.org/I142732210","display_name":"University of Engineering and Technology Lahore","ror":"https://ror.org/0051w2v06","country_code":"PK","type":"education","lineage":["https://openalex.org/I142732210"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Mustafa Habib","raw_affiliation_strings":["Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan","institution_ids":["https://openalex.org/I142732210"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084821839","display_name":"Ali Akg\u00fcl","orcid":"https://orcid.org/0000-0001-9832-1424"},"institutions":[{"id":"https://openalex.org/I149717392","display_name":"Siirt \u00dcniversitesi","ror":"https://ror.org/05ptwtz25","country_code":"TR","type":"education","lineage":["https://openalex.org/I149717392"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Ali Akg\u00fcl","raw_affiliation_strings":["Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey","institution_ids":["https://openalex.org/I149717392"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5069957230"],"corresponding_institution_ids":["https://openalex.org/I4210131342"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":3.0721,"has_fulltext":true,"cited_by_count":26,"citation_normalized_percentile":{"value":0.92443636,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"14","issue":"5","first_page":"907","last_page":"907"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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":0.9998999834060669,"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9801999926567078,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.649601399898529},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6271877884864807},{"id":"https://openalex.org/keywords/klein\u2013gordon-equation","display_name":"Klein\u2013Gordon equation","score":0.5663623809814453},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.4921936094760895},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.4768686890602112},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.43257245421409607},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.413493812084198},{"id":"https://openalex.org/keywords/homotopy-analysis-method","display_name":"Homotopy analysis method","score":0.41191181540489197},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.3911631405353546},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37755075097084045},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.23532989621162415},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22150307893753052},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.16162794828414917}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.649601399898529},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6271877884864807},{"id":"https://openalex.org/C101682238","wikidata":"https://www.wikidata.org/wiki/Q868967","display_name":"Klein\u2013Gordon equation","level":3,"score":0.5663623809814453},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.4921936094760895},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.4768686890602112},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.43257245421409607},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.413493812084198},{"id":"https://openalex.org/C173636693","wikidata":"https://www.wikidata.org/wiki/Q17030668","display_name":"Homotopy analysis method","level":3,"score":0.41191181540489197},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.3911631405353546},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37755075097084045},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.23532989621162415},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22150307893753052},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.16162794828414917},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14050907","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050907","pdf_url":"https://www.mdpi.com/2073-8994/14/5/907/pdf?version=1651212059","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:19f8ac2b118d4f56a40c62e2349ca19b","is_oa":true,"landing_page_url":"https://doaj.org/article/19f8ac2b118d4f56a40c62e2349ca19b","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 5, p 907 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/5/907/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14050907","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/sym14050907","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050907","pdf_url":"https://www.mdpi.com/2073-8994/14/5/907/pdf?version=1651212059","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":[{"id":"https://openalex.org/G7585428671","display_name":null,"funder_award_id":"2019QD07","funder_id":"https://openalex.org/F4320324815","funder_display_name":"Yibin University"}],"funders":[{"id":"https://openalex.org/F4320324815","display_name":"Yibin University","ror":"https://ror.org/03w8m2977"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4229372237.pdf","grobid_xml":"https://content.openalex.org/works/W4229372237.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W1850511188","https://openalex.org/W1968819460","https://openalex.org/W1987078599","https://openalex.org/W2002006488","https://openalex.org/W2018836488","https://openalex.org/W2023229972","https://openalex.org/W2025001386","https://openalex.org/W2059062496","https://openalex.org/W2064354079","https://openalex.org/W2069948307","https://openalex.org/W2090303355","https://openalex.org/W2097777580","https://openalex.org/W2271858403","https://openalex.org/W2294662144","https://openalex.org/W2511749680","https://openalex.org/W2521146637","https://openalex.org/W2572888587","https://openalex.org/W2584401307","https://openalex.org/W2609540538","https://openalex.org/W2783497650","https://openalex.org/W2901796792","https://openalex.org/W2904557328","https://openalex.org/W3015001775","https://openalex.org/W3029729324","https://openalex.org/W3088435881","https://openalex.org/W3088661392","https://openalex.org/W3096936012","https://openalex.org/W3110904444","https://openalex.org/W3156837567","https://openalex.org/W3189224876","https://openalex.org/W3199032845","https://openalex.org/W3199183383","https://openalex.org/W4207040300","https://openalex.org/W4210816620","https://openalex.org/W4293104531","https://openalex.org/W4304607001","https://openalex.org/W6731931253","https://openalex.org/W6800862419","https://openalex.org/W6807603593"],"related_works":["https://openalex.org/W2391752819","https://openalex.org/W2015757695","https://openalex.org/W1976870670","https://openalex.org/W2337755963","https://openalex.org/W1990902692","https://openalex.org/W2113046061","https://openalex.org/W2362256999","https://openalex.org/W2470744056","https://openalex.org/W2007245702","https://openalex.org/W2048873866"],"abstract_inverted_index":{"The":[0,116],"algebras":[1],"of":[2,49,109,118,150],"the":[3,7,29,44,65,76,81,87,100,107,119,148],"symmetry":[4],"operators":[5],"for":[6,12,42,123],"Klein\u2013Gordon":[8,54],"equation":[9],"are":[10],"important":[11],"a":[13,24,47,93,110,139],"charged":[14],"test":[15],"particle,":[16],"moving":[17],"in":[18,23,57,92,106],"an":[19,37],"external":[20],"electromagnetic":[21],"field":[22],"space":[25],"time":[26],"manifold":[27],"on":[28],"isotropic":[30],"hydrosulphate.":[31],"In":[32],"this":[33],"paper,":[34],"we":[35],"develop":[36],"analytical":[38],"and":[39,51,60,80,104,141],"numerical":[40,133],"approach":[41],"providing":[43],"solution":[45,101,122],"to":[46,145],"class":[48],"linear":[50],"nonlinear":[52],"fractional":[53,88],"equations":[55],"arising":[56],"classical":[58],"relativistic":[59],"quantum":[61],"mechanics.":[62],"We":[63],"study":[64],"Yang":[66,77],"homotopy":[67,82],"perturbation":[68,83],"transform":[69,78],"method":[70,84],"(YHPTM),":[71],"which":[72],"is":[73,90,138],"associated":[74],"with":[75,112,147],"(YT)":[79],"(HPM),":[85],"where":[86],"derivative":[89],"taken":[91],"Caputo\u2013Fabrizio":[94],"(CF)":[95],"sense.":[96],"This":[97],"technique":[98],"provides":[99],"very":[102],"accurately":[103],"efficiently":[105],"form":[108],"series":[111,121],"easily":[113],"computable":[114],"coefficients.":[115],"behavior":[117],"approximate":[120],"different":[124],"fractional-order":[125],"\u2118":[126],"values":[127],"has":[128],"been":[129],"shown":[130],"graphically.":[131],"Our":[132],"investigations":[134],"indicate":[135],"that":[136],"YHPTM":[137],"simple":[140],"powerful":[142],"mathematical":[143],"tool":[144],"deal":[146],"complexity":[149],"such":[151],"problems.":[152]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":9},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":6}],"updated_date":"2026-08-10T07:06:39.071514","created_date":"2025-10-10T00:00:00"}
