{"id":"https://openalex.org/W4412945550","doi":"https://doi.org/10.3390/sym17081218","title":"Comparative Analysis of the Gardner Equation in Plasma Physics Using Analytical and Neural Network Methods","display_name":"Comparative Analysis of the Gardner Equation in Plasma Physics Using Analytical and Neural Network Methods","publication_year":2025,"publication_date":"2025-08-01","ids":{"openalex":"https://openalex.org/W4412945550","doi":"https://doi.org/10.3390/sym17081218"},"language":"en","primary_location":{"id":"doi:10.3390/sym17081218","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17081218","pdf_url":"https://www.mdpi.com/2073-8994/17/8/1218/pdf?version=1754050786","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"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/17/8/1218/pdf?version=1754050786","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041836818","display_name":"Zain Majeed","orcid":null},"institutions":[{"id":"https://openalex.org/I43879129","display_name":"Government College University, Lahore","ror":"https://ror.org/040gec961","country_code":"PK","type":"education","lineage":["https://openalex.org/I43879129"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Zain Majeed","raw_affiliation_strings":["Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan","institution_ids":["https://openalex.org/I43879129"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5049325996","display_name":"Adil Jhangeer","orcid":"https://orcid.org/0000-0001-6747-425X"},"institutions":[{"id":"https://openalex.org/I113330528","display_name":"Khazar University","ror":"https://ror.org/014te7048","country_code":"AZ","type":"education","lineage":["https://openalex.org/I113330528"]},{"id":"https://openalex.org/I142208455","display_name":"VSB - Technical University of Ostrava","ror":"https://ror.org/05x8mcb75","country_code":"CZ","type":"education","lineage":["https://openalex.org/I142208455"]}],"countries":["AZ","CZ"],"is_corresponding":false,"raw_author_name":"Adil Jhangeer","raw_affiliation_strings":["Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan","IT4Innovations, VSB\u2014Technical University of Ostrava, Poruba, 708 00 Ostrava, Czech Republic"],"raw_orcid":"https://orcid.org/0000-0001-6747-425X","affiliations":[{"raw_affiliation_string":"Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan","institution_ids":["https://openalex.org/I113330528"]},{"raw_affiliation_string":"IT4Innovations, VSB\u2014Technical University of Ostrava, Poruba, 708 00 Ostrava, Czech Republic","institution_ids":["https://openalex.org/I142208455"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072280941","display_name":"F. M. Mahomed","orcid":"https://orcid.org/0000-0002-6995-5820"},"institutions":[{"id":"https://openalex.org/I192619145","display_name":"University of the Witwatersrand","ror":"https://ror.org/03rp50x72","country_code":"ZA","type":"education","lineage":["https://openalex.org/I192619145"]},{"id":"https://openalex.org/I43879129","display_name":"Government College University, Lahore","ror":"https://ror.org/040gec961","country_code":"PK","type":"education","lineage":["https://openalex.org/I43879129"]}],"countries":["PK","ZA"],"is_corresponding":false,"raw_author_name":"F. M. Mahomed","raw_affiliation_strings":["Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan","School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa"],"raw_orcid":"https://orcid.org/0000-0002-6995-5820","affiliations":[{"raw_affiliation_string":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan","institution_ids":["https://openalex.org/I43879129"]},{"raw_affiliation_string":"School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa","institution_ids":["https://openalex.org/I192619145"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013954454","display_name":"Hassan Almusawa","orcid":"https://orcid.org/0000-0001-5024-866X"},"institutions":[{"id":"https://openalex.org/I91450128","display_name":"Jazan University","ror":"https://ror.org/02bjnq803","country_code":"SA","type":"education","lineage":["https://openalex.org/I91450128"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Hassan Almusawa","raw_affiliation_strings":["Department of Mathematics, College of Sciences, Jazan University, Jazan 45142, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-5024-866X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Sciences, Jazan University, Jazan 45142, Saudi Arabia","institution_ids":["https://openalex.org/I91450128"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5038369224","display_name":"F. D. Zaman","orcid":"https://orcid.org/0000-0002-6498-0664"},"institutions":[{"id":"https://openalex.org/I192619145","display_name":"University of the Witwatersrand","ror":"https://ror.org/03rp50x72","country_code":"ZA","type":"education","lineage":["https://openalex.org/I192619145"]}],"countries":["ZA"],"is_corresponding":false,"raw_author_name":"F. D. Zaman","raw_affiliation_strings":["School of Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa","institution_ids":["https://openalex.org/I192619145"]}]}],"institutions":[],"countries_distinct_count":5,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5013954454"],"corresponding_institution_ids":["https://openalex.org/I91450128"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.3609,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.89174246,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":96},"biblio":{"volume":"17","issue":"8","first_page":"1218","last_page":"1218"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9983999729156494,"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"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9983999729156494,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9941999912261963,"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":0.9926000237464905,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6486952304840088},{"id":"https://openalex.org/keywords/plasma","display_name":"Plasma","score":0.4817085862159729},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.4351489245891571},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.42151230573654175},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.34574851393699646},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.34547048807144165},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3368472456932068},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.262067973613739},{"id":"https://openalex.org/keywords/nuclear-physics","display_name":"Nuclear physics","score":0.16903585195541382}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6486952304840088},{"id":"https://openalex.org/C82706917","wikidata":"https://www.wikidata.org/wiki/Q10251","display_name":"Plasma","level":2,"score":0.4817085862159729},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4351489245891571},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.42151230573654175},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34574851393699646},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34547048807144165},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3368472456932068},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.262067973613739},{"id":"https://openalex.org/C185544564","wikidata":"https://www.wikidata.org/wiki/Q81197","display_name":"Nuclear physics","level":1,"score":0.16903585195541382}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17081218","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17081218","pdf_url":"https://www.mdpi.com/2073-8994/17/8/1218/pdf?version=1754050786","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"}],"best_oa_location":{"id":"doi:10.3390/sym17081218","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17081218","pdf_url":"https://www.mdpi.com/2073-8994/17/8/1218/pdf?version=1754050786","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/G1408456483","display_name":null,"funder_award_id":"CZ.10.03.01/00/22_003/0000048","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G3771185154","display_name":null,"funder_award_id":"CZ.10.03","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G4273056302","display_name":null,"funder_award_id":".10.03.01/00/22_003/0000048","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4412945550.pdf","grobid_xml":"https://content.openalex.org/works/W4412945550.grobid-xml"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W1849716","https://openalex.org/W1506397612","https://openalex.org/W1979447154","https://openalex.org/W2085668771","https://openalex.org/W2166166690","https://openalex.org/W2318127549","https://openalex.org/W2501549795","https://openalex.org/W2506786835","https://openalex.org/W2541170635","https://openalex.org/W2593501769","https://openalex.org/W2786762283","https://openalex.org/W2793473605","https://openalex.org/W2810687640","https://openalex.org/W2899283552","https://openalex.org/W2912554575","https://openalex.org/W2961856433","https://openalex.org/W2976642150","https://openalex.org/W3036631204","https://openalex.org/W3107930199","https://openalex.org/W3120495973","https://openalex.org/W3195602193","https://openalex.org/W4206726492","https://openalex.org/W4214902984","https://openalex.org/W4221107806","https://openalex.org/W4224214581","https://openalex.org/W4282941394","https://openalex.org/W4292170110","https://openalex.org/W4292415314","https://openalex.org/W4317426778","https://openalex.org/W4320481962","https://openalex.org/W4360850923","https://openalex.org/W4378652639","https://openalex.org/W4380762716","https://openalex.org/W4385215773","https://openalex.org/W4392117812","https://openalex.org/W4392718076","https://openalex.org/W4394977080","https://openalex.org/W4399119243","https://openalex.org/W4399301029","https://openalex.org/W4403734597","https://openalex.org/W4408251007","https://openalex.org/W4409349828","https://openalex.org/W6810441208"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2962838298","https://openalex.org/W2600246793"],"abstract_inverted_index":{"In":[0],"the":[1,8,49,53,68,74,104,107,119,123,131,135,146,163,180,184,188,194,200,210,227,232],"present":[2],"paper,":[3],"a":[4,19,33,41,45,88,151,160,254],"mathematical":[5],"analysis":[6,36],"of":[7,23,43,52,83,106,118,159,166,187,199,213,234,253],"Gardner":[9],"equation":[10,63,75],"with":[11,103,142,250],"varying":[12],"coefficients":[13,55,168,252],"has":[14,93,169,175],"been":[15,94,127,139,171,177],"performed":[16],"to":[17,29,73,79,101,193,206,225,245],"give":[18,80],"more":[20],"realistic":[21],"model":[22],"physical":[24],"phenomena,":[25],"especially":[26],"in":[27,162,215,231],"regards":[28],"plasma":[30,216],"physics.":[31,217],"First,":[32],"Lie":[34],"symmetry":[35,46],"was":[37,56],"carried":[38],"out,":[39],"as":[40,96],"result":[42],"which":[44,149,229],"classification":[47],"following":[48],"different":[50,167],"representations":[51],"variable":[54,251],"systematically":[57],"derived.":[58],"The":[59,114,157,197],"reduced":[60,108],"ordinary":[61],"differential":[62,248],"obtained":[64],"is":[65,150,224],"solved":[66],"using":[67,111],"power-series":[69],"method":[70,100],"and":[71,116,130,153,173,236,239],"solutions":[72,120,132],"are":[76],"represented":[77],"graphically":[78],"an":[81,97],"idea":[82],"their":[84],"dynamical":[85],"behavior.":[86],"Moreover,":[87],"fully":[89],"connected":[90],"neural":[91,124,136,243],"network":[92],"included":[95],"efficient":[98,222],"computation":[99],"deal":[102],"complexity":[105],"equation,":[109],"by":[110,122,134,145],"traveling-wave":[112],"transformation.":[113],"validity":[115],"correctness":[117],"provided":[121,133],"networks":[125,137],"have":[126,138],"rigorously":[128],"tested":[129],"thoroughly":[140],"compared":[141,192],"those":[143],"generated":[144],"Runge\u2013Kutta":[147],"method,":[148],"conventional":[152],"well-recognized":[154],"numerical":[155],"method.":[156],"impact":[158],"variation":[161],"loss":[164,189],"function":[165],"also":[170,176,219],"discussed,":[172],"it":[174,223],"found":[178],"that":[179],"dispersive":[181],"coefficient":[182],"affects":[183],"convergence":[185],"rate":[186],"contribution":[190],"considerably":[191],"other":[195],"coefficients.":[196],"results":[198],"current":[201],"work":[202],"can":[203],"be":[204],"used":[205],"improve":[207],"knowledge":[208],"on":[209,242],"nonlinear":[211,247],"dynamics":[212],"waves":[214],"They":[218],"show":[220],"how":[221],"combine":[226],"approaches,":[228],"consists":[230],"use":[233],"analytical":[235],"semi-analytical":[237],"methods":[238,240],"based":[241],"networks,":[244],"solve":[246],"equations":[249],"complex":[255],"nature.":[256]},"counts_by_year":[{"year":2025,"cited_by_count":2}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}
