{"id":"https://openalex.org/W4293771143","doi":"https://doi.org/10.3390/sym14091777","title":"An Efficient Technique to Solve Time-Fractional Kawahara and Modified Kawahara Equations","display_name":"An Efficient Technique to Solve Time-Fractional Kawahara and Modified Kawahara Equations","publication_year":2022,"publication_date":"2022-08-26","ids":{"openalex":"https://openalex.org/W4293771143","doi":"https://doi.org/10.3390/sym14091777"},"language":"en","primary_location":{"id":"doi:10.3390/sym14091777","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091777","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1777/pdf?version=1661507861","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/2073-8994/14/9/1777/pdf?version=1661507861","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062256159","display_name":"Pavani Koppala","orcid":null},"institutions":[{"id":"https://openalex.org/I876193797","display_name":"Vellore Institute of Technology University","ror":"https://ror.org/00qzypv28","country_code":"IN","type":"education","lineage":["https://openalex.org/I876193797"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Pavani Koppala","raw_affiliation_strings":["Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India","institution_ids":["https://openalex.org/I876193797"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023184359","display_name":"K. Raghavendar","orcid":null},"institutions":[{"id":"https://openalex.org/I876193797","display_name":"Vellore Institute of Technology University","ror":"https://ror.org/00qzypv28","country_code":"IN","type":"education","lineage":["https://openalex.org/I876193797"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Raghavendar Kondooru","raw_affiliation_strings":["Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India"],"raw_orcid":"https://orcid.org/0000-0001-9582-6662","affiliations":[{"raw_affiliation_string":"Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India","institution_ids":["https://openalex.org/I876193797"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5023184359"],"corresponding_institution_ids":["https://openalex.org/I876193797"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.7593,"has_fulltext":true,"cited_by_count":13,"citation_normalized_percentile":{"value":0.83649066,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"14","issue":"9","first_page":"1777","last_page":"1777"},"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":0.9940000176429749,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9747999906539917,"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":0.7410136461257935},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5952088236808777},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.5705058574676514},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.5332911014556885},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5211379528045654},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5202825665473938},{"id":"https://openalex.org/keywords/exact-solutions-in-general-relativity","display_name":"Exact solutions in general relativity","score":0.4986100196838379},{"id":"https://openalex.org/keywords/traveling-wave","display_name":"Traveling wave","score":0.43773242831230164},{"id":"https://openalex.org/keywords/homotopy-analysis-method","display_name":"Homotopy analysis method","score":0.4186769723892212},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4089621603488922},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.2663196325302124},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.24828511476516724},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14000222086906433},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.0921926498413086}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7410136461257935},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5952088236808777},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.5705058574676514},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.5332911014556885},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5211379528045654},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5202825665473938},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.4986100196838379},{"id":"https://openalex.org/C2988173416","wikidata":"https://www.wikidata.org/wiki/Q37172","display_name":"Traveling wave","level":2,"score":0.43773242831230164},{"id":"https://openalex.org/C173636693","wikidata":"https://www.wikidata.org/wiki/Q17030668","display_name":"Homotopy analysis method","level":3,"score":0.4186769723892212},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4089621603488922},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.2663196325302124},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.24828511476516724},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14000222086906433},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0921926498413086},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14091777","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091777","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1777/pdf?version=1661507861","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:8baf1d1b83344135ad42dcb30f7df125","is_oa":true,"landing_page_url":"https://doaj.org/article/8baf1d1b83344135ad42dcb30f7df125","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 9, p 1777 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/9/1777/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14091777","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/sym14091777","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091777","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1777/pdf?version=1661507861","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":[{"display_name":"Clean water and sanitation","score":0.8100000023841858,"id":"https://metadata.un.org/sdg/6"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320322838","display_name":"VIT University","ror":"https://ror.org/00qzypv28"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4293771143.pdf","grobid_xml":"https://content.openalex.org/works/W4293771143.grobid-xml"},"referenced_works_count":61,"referenced_works":["https://openalex.org/W31877248","https://openalex.org/W188390741","https://openalex.org/W433794350","https://openalex.org/W1567760004","https://openalex.org/W1964997874","https://openalex.org/W1982911875","https://openalex.org/W1984853762","https://openalex.org/W2000053425","https://openalex.org/W2030484509","https://openalex.org/W2037013349","https://openalex.org/W2041585148","https://openalex.org/W2045949287","https://openalex.org/W2060841776","https://openalex.org/W2078643611","https://openalex.org/W2087605340","https://openalex.org/W2093812902","https://openalex.org/W2112690937","https://openalex.org/W2146200458","https://openalex.org/W2153261118","https://openalex.org/W2155611072","https://openalex.org/W2165917005","https://openalex.org/W2187932833","https://openalex.org/W2301173684","https://openalex.org/W2549914376","https://openalex.org/W2620923119","https://openalex.org/W2887445041","https://openalex.org/W2912488487","https://openalex.org/W2912498332","https://openalex.org/W2922135896","https://openalex.org/W2953563272","https://openalex.org/W2991362353","https://openalex.org/W2996379788","https://openalex.org/W3023112710","https://openalex.org/W3037097151","https://openalex.org/W3182024391","https://openalex.org/W3185181261","https://openalex.org/W3198364556","https://openalex.org/W3201979259","https://openalex.org/W3207093090","https://openalex.org/W4220816079","https://openalex.org/W4225982945","https://openalex.org/W4226039848","https://openalex.org/W4226504738","https://openalex.org/W4229045171","https://openalex.org/W4230762202","https://openalex.org/W4233256340","https://openalex.org/W4233616901","https://openalex.org/W4280490815","https://openalex.org/W4280546404","https://openalex.org/W4282837105","https://openalex.org/W4283795877","https://openalex.org/W4285801267","https://openalex.org/W6601305217","https://openalex.org/W6607757599","https://openalex.org/W6660865925","https://openalex.org/W6665508466","https://openalex.org/W6677098996","https://openalex.org/W6686907548","https://openalex.org/W6739055203","https://openalex.org/W6758704570","https://openalex.org/W6795843812"],"related_works":["https://openalex.org/W2470744056","https://openalex.org/W2088164634","https://openalex.org/W2313539504","https://openalex.org/W1584105255","https://openalex.org/W201376893","https://openalex.org/W2016523436","https://openalex.org/W2052573303","https://openalex.org/W2055669245","https://openalex.org/W2066436349","https://openalex.org/W2322006330"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"we":[3],"analysed":[4],"the":[5,9,19,27,34,42,50,64,75,86,101,105,109,115,122,136,149,153,157],"approximate":[6],"solutions":[7,146],"of":[8,21,29,79,100,108,121,129,152,159],"time-fractional":[10],"Kawahara":[11,15,102,154],"equation":[12,103],"and":[13,26,60,77,104,140],"modified":[14],"equation,":[16],"which":[17],"describe":[18],"propagation":[20],"signals":[22],"in":[23,33,49,57,63,89],"transmission":[24],"lines":[25],"formation":[28],"nonlinear":[30],"water":[31,87],"waves":[32,88],"long":[35,90],"wavelength":[36,91],"region.":[37],"An":[38],"efficient":[39],"technique,":[40],"namely":[41],"natural":[43],"transform":[44],"decomposition":[45],"method,":[46],"is":[47,96],"used":[48],"present":[51],"study.":[52],"Fractional":[53],"derivatives":[54,80],"are":[55,125],"considered":[56],"Caputo,":[58],"Caputo\u2013Fabrizio,":[59],"Atangana\u2013Baleanu":[61],"operative":[62],"Caputo":[65],"manner.":[66],"We":[67],"have":[68],"presented":[69],"numerical":[70,132],"results":[71],"graphically":[72],"to":[73,84,127,148],"demonstrate":[74],"applicability":[76],"efficiency":[78],"with":[81],"fractional":[82],"order":[83],"depict":[85],"regions.":[92],"The":[93,118],"symmetry":[94],"pattern":[95],"a":[97],"fundamental":[98],"feature":[99],"symmetrical":[106],"aspect":[107],"solution":[110,151],"can":[111],"be":[112],"seen":[113],"from":[114],"graphical":[116],"representations.":[117],"obtained":[119],"outcomes":[120],"proposed":[123,161],"method":[124,139],"compared":[126],"those":[128],"other":[130],"well-known":[131],"techniques,":[133],"such":[134],"as":[135],"homotopy":[137],"analysis":[138],"residual":[141],"power":[142],"series":[143],"method.":[144,162],"Numerical":[145],"converge":[147],"exact":[150],"equations,":[155],"demonstrating":[156],"significance":[158],"our":[160]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}