Browsing by Author "Albalawi, Sami M"

Now showing 1 - 2 of 2

Restricted
A novel motivation for the unstable nonlinear Schrödinger equation through random inputs
(Saudi Digital Library, 2024) Albalawi, Sami M; M.A. Shohaly and M.E. Fares
We investigate the stochastic unstable nonlinear Schrödinger equation through bi-random sources. Specifically, we solve this equation via Itô sense, with the parameter following Laplace and Gumbel distributions. We provide vital stochastic solutions in applied sciences. We employ He’s semi-inverse technique in order to provide these solutions in a unified way. Actually, this is the first time that the same model has been taken into account in these circumstances. In order to investigate the real relevance of the stochastic unstable nonlinear Schrödinger equation, we provide the simulations for some of the collected solutions using the appropriate parameter settings provided by the MATLAB software. Finally, our renewed drive might expand to incorporate further emerging natural science models.
3 0
Restricted
Investigation of the deterministic and stochastic waves for some nonlinear partial differential equations with their applications
(Saudi Digital Library, 2025) Albalawi, Sami M; M.E, Fares; M.A, Shohaly
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.
4 0