{"id":"https://openalex.org/W4415813985","doi":"https://doi.org/10.3390/sym17111845","title":"Variational Analysis and Integration of the (2 + 1) Fourth-Order Time-Dependent Biharmonic Equation via Energy and Momentum Conservation","display_name":"Variational Analysis and Integration of the (2 + 1) Fourth-Order Time-Dependent Biharmonic Equation via Energy and Momentum Conservation","publication_year":2025,"publication_date":"2025-11-03","ids":{"openalex":"https://openalex.org/W4415813985","doi":"https://doi.org/10.3390/sym17111845"},"language":"en","primary_location":{"id":"doi:10.3390/sym17111845","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111845","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1845/pdf?version=1762155479","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/2073-8994/17/11/1845/pdf?version=1762155479","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049368083","display_name":"Yasir Masood","orcid":"https://orcid.org/0000-0003-1079-8012"},"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":true,"raw_author_name":"Yasir Masood","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/A5032975641","display_name":"A. H. Kara","orcid":"https://orcid.org/0000-0002-0231-0198"},"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":"A. H. Kara","raw_affiliation_strings":["School of Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa"],"raw_orcid":"https://orcid.org/0000-0002-0231-0198","affiliations":[{"raw_affiliation_string":"School of Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa","institution_ids":["https://openalex.org/I192619145"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5038369224","display_name":"F. D. Zaman","orcid":"https://orcid.org/0000-0002-6498-0664"},"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":"F. D. Zaman","raw_affiliation_strings":["Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-6498-0664","affiliations":[{"raw_affiliation_string":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan","institution_ids":["https://openalex.org/I43879129"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101823828","display_name":"Ali Raza","orcid":"https://orcid.org/0000-0002-7593-9923"},"institutions":[{"id":"https://openalex.org/I26092322","display_name":"Stellenbosch University","ror":"https://ror.org/05bk57929","country_code":"ZA","type":"education","lineage":["https://openalex.org/I26092322"]},{"id":"https://openalex.org/I4210106521","display_name":"National Institute for Theoretical Physics","ror":"https://ror.org/01jgv6w72","country_code":"ZA","type":"facility","lineage":["https://openalex.org/I4210106521"]}],"countries":["ZA"],"is_corresponding":false,"raw_author_name":"Ali Raza","raw_affiliation_strings":["Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7602, South Africa","National Institute for Theoretical and Computational Sciences (NITheCS), Stellenbosch 7600, South Africa"],"raw_orcid":"https://orcid.org/0000-0002-7593-9923","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7602, South Africa","institution_ids":["https://openalex.org/I26092322"]},{"raw_affiliation_string":"National Institute for Theoretical and Computational Sciences (NITheCS), Stellenbosch 7600, South Africa","institution_ids":["https://openalex.org/I4210106521"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5049368083"],"corresponding_institution_ids":["https://openalex.org/I43879129"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.19592079,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"17","issue":"11","first_page":"1845","last_page":"1845"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9387999773025513,"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9387999773025513,"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/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.03669999912381172,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T11416","display_name":"Numerical methods for differential equations","score":0.006899999920278788,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/noethers-theorem","display_name":"Noether's theorem","score":0.9083999991416931},{"id":"https://openalex.org/keywords/biharmonic-equation","display_name":"Biharmonic equation","score":0.7484999895095825},{"id":"https://openalex.org/keywords/conservation-law","display_name":"Conservation law","score":0.6834999918937683},{"id":"https://openalex.org/keywords/conserved-quantity","display_name":"Conserved quantity","score":0.6033999919891357},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5205000042915344},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.49630001187324524},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.4657000005245209},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.41830000281333923},{"id":"https://openalex.org/keywords/homogeneous-space","display_name":"Homogeneous space","score":0.4000999927520752}],"concepts":[{"id":"https://openalex.org/C80892491","wikidata":"https://www.wikidata.org/wiki/Q578555","display_name":"Noether's theorem","level":3,"score":0.9083999991416931},{"id":"https://openalex.org/C145784156","wikidata":"https://www.wikidata.org/wiki/Q859808","display_name":"Biharmonic equation","level":3,"score":0.7484999895095825},{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.6834999918937683},{"id":"https://openalex.org/C71914180","wikidata":"https://www.wikidata.org/wiki/Q1071013","display_name":"Conserved quantity","level":2,"score":0.6033999919891357},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.5408999919891357},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5288000106811523},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5205000042915344},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.49630001187324524},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4657000005245209},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45500001311302185},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.43959999084472656},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.41830000281333923},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.4000999927520752},{"id":"https://openalex.org/C520301825","wikidata":"https://www.wikidata.org/wiki/Q380170","display_name":"Energy conservation","level":2,"score":0.38989999890327454},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.382099986076355},{"id":"https://openalex.org/C52957995","wikidata":"https://www.wikidata.org/wiki/Q11382","display_name":"Conservation of energy","level":2,"score":0.37700000405311584},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.36550000309944153},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3628000020980835},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.35580000281333923},{"id":"https://openalex.org/C9259290","wikidata":"https://www.wikidata.org/wiki/Q745215","display_name":"Variational principle","level":2,"score":0.3474999964237213},{"id":"https://openalex.org/C60718061","wikidata":"https://www.wikidata.org/wiki/Q1414747","display_name":"Momentum (technical analysis)","level":2,"score":0.34150001406669617},{"id":"https://openalex.org/C206237540","wikidata":"https://www.wikidata.org/wiki/Q5163158","display_name":"Conserved current","level":4,"score":0.33390000462532043},{"id":"https://openalex.org/C163105828","wikidata":"https://www.wikidata.org/wiki/Q2734005","display_name":"Integrating factor","level":5,"score":0.30880001187324524},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.30640000104904175},{"id":"https://openalex.org/C97985569","wikidata":"https://www.wikidata.org/wiki/Q216861","display_name":"Calculus of variations","level":2,"score":0.2825999855995178},{"id":"https://openalex.org/C115071613","wikidata":"https://www.wikidata.org/wiki/Q1047080","display_name":"Differential form","level":2,"score":0.28060001134872437},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.27559998631477356},{"id":"https://openalex.org/C14037181","wikidata":"https://www.wikidata.org/wiki/Q215007","display_name":"Equations of motion","level":2,"score":0.26109999418258667},{"id":"https://openalex.org/C64057670","wikidata":"https://www.wikidata.org/wiki/Q4381442","display_name":"First-order partial differential equation","level":3,"score":0.2590999901294708},{"id":"https://openalex.org/C11302649","wikidata":"https://www.wikidata.org/wiki/Q6927157","display_name":"Moving frame","level":3,"score":0.25690001249313354},{"id":"https://openalex.org/C191640071","wikidata":"https://www.wikidata.org/wiki/Q5377056","display_name":"Energy functional","level":2,"score":0.2540999948978424},{"id":"https://openalex.org/C28062496","wikidata":"https://www.wikidata.org/wiki/Q1145653","display_name":"Hamilton's principle","level":3,"score":0.25060001015663147}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17111845","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111845","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1845/pdf?version=1762155479","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/sym17111845","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111845","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1845/pdf?version=1762155479","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415813985.pdf","grobid_xml":"https://content.openalex.org/works/W4415813985.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W80544570","https://openalex.org/W90800293","https://openalex.org/W2012126777","https://openalex.org/W2032355643","https://openalex.org/W2062561585","https://openalex.org/W2063907161","https://openalex.org/W2080707851","https://openalex.org/W2083744164","https://openalex.org/W2095181409","https://openalex.org/W2152619933","https://openalex.org/W2493562759","https://openalex.org/W3034915735","https://openalex.org/W3182321990","https://openalex.org/W4292170110","https://openalex.org/W4292394877","https://openalex.org/W4394969159"],"related_works":[],"abstract_inverted_index":{"We":[0],"consider":[1],"the":[2,17,30,35],"fourth-order":[3],"PDE":[4,18],"uxxxx+2uxxyy+uyyyy\u2212utt=h(u).":[5],"The":[6],"Lie":[7],"and":[8,14,47],"Noether":[9],"symmetry":[10],"generators":[11],"are":[12],"constructed,":[13],"we":[15,23],"reduce":[16],"to":[19,28],"simpler":[20],"ODEs.":[21],"Furthermore,":[22],"use":[24],"some":[25],"well-known":[26],"methods":[27],"compute":[29],"conserved":[31],"vectors":[32],"associated":[33],"with":[34],"PDE.":[36],"An":[37],"analysis":[38],"of":[39],"reduced":[40],"ordinary":[41],"differential":[42],"equations":[43],"(ODEs),":[44],"invariant":[45],"solutions,":[46],"their":[48],"physical":[49],"interpretations":[50],"is":[51],"presented.":[52]},"counts_by_year":[],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-11-03T00:00:00"}