Solutions of two (3+1)-dimensional Gardner-type equations using Lie symmetry analysis

Abstract

ThisΓresearchΓprojectΓseeksΓtoΓinvestigateΓtheΓtwoΓ(3+1)-dimensionalΓGardner-typeΓequationsΓthatΓfallΓunderΓtheΓnonlinearΓpartialΓdifferentialΓequationsΓ(NPDEs),ΓwhichΓappearΓinΓsoundΓwaveΓnonlinearΓphysicalΓphenomena.ΓTheΓstudyΓusesΓLieΓsymmetryΓtoΓobtainΓtheΓclassΓofΓexactΓsolutionsΓofΓNPDEΓmodelsΓunderΓinvestigation.ΓEx-ΓactΓsolutionsΓareΓobtainedΓfromΓtransformedΓsolvableΓnonlinearΓordinaryΓdifferentialΓequationsΓ(NODEs)ΓrepresentingΓtheΓmodelsΓobtainedΓfromΓLieΓpointΓsymmetriesΓwithoutΓchangingΓtheΓdynamicsΓofΓtheΓmodels.ΓTheΓobtainedΓsolutionsΓareΓrepre-ΓsentedΓgraphicallyΓandΓanalysedΓtoΓstudyΓmomentumΓ(velocity)ΓandΓenergyΓvariationΓinΓorderΓtoΓexploreΓtheΓsoundΓwaveΓdynamicsΓmodelledΓbyΓtheΓGardner-typeΓequationsΓstudiedΓinΓthisΓproject.ΓInΓadditionΓtoΓexactΓsolutions,ΓconservationΓlawsΓthatΓrepre-ΓsentΓconservedΓquantitiesΓofΓtheΓmodelsΓunderΓinvestigationΓareΓobtainedΓbyΓinvokingΓIbragimovsΓtheoremΓandΓmultipliersΓapproach.