Investigation of the deterministic and stochastic waves for some nonlinear partial differential equations with their applications

Abstract

This thesisfocusesonthestudyofnonlinearstochasticmodels,particularlythosearis- ing inmathematicalphysics.Stochasticmodelinghasbecomeincreasinglyessentialin understanding real-worldphenomena,whereuncertaintyplaysacrucialrole.Unlike deterministic models,stochasticmodelspreservealltypesofuncertaintiesandprovide more realisticsimulations.Theworkpresentedinthisthesisinvestigatestheimpactof stochasticeffectsonnonlinearevolutionequations,withaspecificfocusonthe unstable nonlinear Schr¨odingerequation(UNLSE) and othernonlinearwavemodels. Variousmathematicaltechniquesareemployedtoderiveanalyticalsolutionsforthese stochasticmodels.The RB sub-ODEmethod and He’s semi-inversetechnique are appliedtoobtainexactsolutionsfornonlinearwaveequationsundertheinfluenceof randomness. Thestochasticnatureoftheseequationsisexploredusingdifferenttypes of randomvariables,including Laplace andGumbeldistributions. Additionally, simulationsareprovidedtovisualizethebehavioroftheobtainedsolutionsunderdifferent parameter settings Chapter 1:Introduction This chapterintroducesfundamentalconceptsrelatedtorandomvariables,stochastic processes,andBrownianmotion,alongwithkeystatisticaldistributionsusedinthe thesis. Ithighlightsthesignificantadvancementsinappliedmathematicsoverthelast fiftyyears,particularlyinenergy-relatedapplications,whichhavedriventhedevelop- mentofsophisticatedcomputingtechniques.Thechapteremphasizestheimportanceof nonlinear partialdifferentialequations(NPDEs)inmodelingvariousnaturalphenomena across multiplescientificdisciplines,includingsolidstatephysics,quantummechanics, and chemicalphysics.Italsodiscussestheroleoffirst,second,andthird-orderNPDEs in modelingnonlinearwaves,diffusionprocesses,anddispersivewavemotion.Addition- ally,thechapterintroducessolitarywavesandsolitons,explainingtheirsignificancein understanding complexphysicalsystems.Thediscussionsetsthefoundationforfurther exploration ofstochasticnonlinearpartialdifferentialequations(SNPDEs),aimingto modelreal-worldsystemswithgreateraccuracy.. Chapter 2:MathematicalMethods This chapterintroducesthefundamentalconceptsofstochasticmodelinganditssignif- icance innonlinearsystems.Itdiscussesthenecessityofusingstochasticratherthan deterministic approachestostudynonlinearmodels,astheyaccountforuncertainties more effectively.Thechapteralsoprovidesanoverviewof Brownianmotion, whichis a keystochasticprocess,anditsapplicationsinphysics,chemistry,andengineering.Ad-ditionally,itintroducesthestochasticunstablenonlinearSchr¨odingerequation(UNLSE) and outlinesthemainobjectivesofthisthesis. Chapter 3:AnalyticalSolutionsforNonlinearWaveEquations This chapterfocusesonanalyticalmethodsforsolvingnonlinearwaveequations.The RB sub-ODEtechnique is appliedtoobtainexactsolutionsforthe cubic Boussinesq equation and the modifiedequal-width(MEW)equation. Thesemodelsdescribe long wavesinshallowwaterandwavepropagationinnonlineardispersivemedia,respec- tively.Theobtainedsolutionsincludesoliton,periodic,andrationalwaveforms,which are visualizedusingtwo-andthree-dimensionalgraphs. Chapter 4:StochasticNonlinearSchr¨odingerEquations This chapterexplorestheimpactofstochasticperturbationsonthenonlinearSchr¨odinger equation (NLSE).TheUNLSEisstudiedundertheinfluenceof additivenoise and uncertaintyinitsparameters.Thechapterpresentsvariousnumericalandanalytical methodsusedinrecentresearchonstochasticNLSEs.Inaddition,thesignificanceofthe Laplace andGumbelrandomvariablesinmodelinguncertaintyisdiscussed. Chapter 5:StochasticSolutionsforUNLSE This chapterapplies He’s semi-inversetechnique to solvethestochasticUNLSE. Examines theinfluenceofrandomnessonsolitarywavepropagation,consideringboth Laplace andGumbelrandomvariables. Themeanoftheserandomsolutionsis calculated andnumericalsimulationsareprovidedtoillustratethestochasticbehaviorof the system.Thefindingshighlighttheadvantagesoftheproposedapproachinreducing computational complexitywhileobtainingaccuratesolutions.