STUDY OF CERTAIN QUASI-LINEAR HYPERBOLIC SYSTEMS OF PDEs

Abstract

This thesismainlyfocusesonthestudyofthequasi-linearhyperbolicsystemsofpartial di erentialequations.Thethesisconsistsofsixchapters,whicharebrie ydescribedas follows: The chapter 1 is introductoryandprovidesanoverviewoftheworkpresentedinthis thesis. Chapter 2 deals withasystemofquasi-linearpartialdi erentialequations(PDEs), whichdescribestheone-dimensionalmotionofaninviscid,self-gravitatingandspherically symmetric vanderWaalsgascloud.Byusingthemethodbasedonthekinematicsofshock waves,theevolutionequationforsphericalshockwavesininterstellarvanderWaalsgas clouds isderived.Byapplyingthetruncationapproximationprocedure,anin nitesystem of transportequations,whichgovernstheshockpropagation,isderivedtostudythekine- matics ofshockwavesfortheone-dimensionalmotion.The rst,secondandthird-order transportequations,whichdescribetheshockstrengthandtheinduceddiscontinuitybe- hind it,areusedtoanalyzethedecayandgrowthbehavioroftheshockwavesinanon-ideal gas. Theresultsfortheexponentareobtainedfrom rst,secondandthird-orderapproxi- mations, andcomparedwiththeresultsobtainedbyWhitham'scharacteristicrule(CCW approximation).Also,thee ectsoftheparametersofnon-idealnessandcooling-heating function ontheevolutionarybehaviorofshocksarediscussedandshowngraphically. Chapter 3 seeks toinvestigateaquasi-linearhyperbolicsystemofpartialdi erential equations (PDEs)whichdescribestheunsteadyone-dimensionalmotionofashockwave of arbitrarystrengthpropagatingthroughanon-idealradiatinggas.Wehavederivedan in nite hierarchyofthetransportequationwhichisbasedonthekinematicsoftheone- dimensional motionoftheshockfront.Byusingthetruncationapproximationmethod,an in nite hierarchyoftransportequations,whichgovernstheshockstrengthandtheinduced discontinuitiesbehindit,isderivedtostudythekinematicsoftheshockfront.The rst three transportequations(i.e., rst,secondandthird-order)areusedtostudythegrowth and decaybehaviorofshocksinvanderWaalsradiatinggas.Thedecaylawsforweak shockwavesinnon-radiatinggasareentirelyrecoveredinthesecond-ordertruncation approximation.Theresultsobtainedbythe rstthreeapproximationsforshockwavesof arbitrary strengtharecomparedwiththeresultspredictedbythecharacteristicrule.Also, the e ectsofnon-idealparametersandradiationontheevolutionarybehaviorofshock wavesarediscussedanddepictedpictorially. In chapter 4, byusingtheLiegroupoftransformations,acompleterangeofself-similar solutions isobtainedforaquasi-linearhyperbolicsystemofpartialdi erentialequations (PDEs) whichdescribesaproblemofimplodingcylindricalshockwavesinavanderWaals dustygasthroughtheaxialmagnetic eld.Thenecessaryconditionsfortheexistenceof similaritysolutionsforstrongshocksarediscussed.Thecollapseofimploding(converging) cylindrical shocksisalsostudiedandtheimpactsofnon-idealparameters,therelative speci cheat,theratioofthedensityofsolidparticlestotheinitialdensityofthegas,the mass concentrationofdustparticlesandshockCowlingnumberonshockevolutionarealso discussed indetailanddepictedgraphically. In chapter 5, byusingtheperturbationseriesapproach,aglobalsolutiontotheimplo- sion problemisobtainedforaquasi-linearhyperbolicsystemofpartialdi erentialequations (PDEs) describingaproblemofstrongconvergingcylindricalshockwavescollapsingatthe axis ofsymmetryinavanderWaalsdustygasunderthee ectoftheaxialmagnetic eld. This globalsolutionprovidestheresultsofGuderley'slocalself-similarsolutionwhichis validonlyintheneighborhoodoftheaxisofimplosion.Thesimilarityexponentsand correspondingamplitudesareobtainedintheneighborhoodoftheshockcollapse.Inaddi- tion, thevaluesofleadingsimilarityexponentsarecomparedwiththeresultsobtainedby Whitham's method.Theshocktrajectoryandthepro lesof owvariables(i.e.,density, velocity,pressureandmagneticpressure)havebeendrawnfordi erentvaluesoftherela- tivespeci cheat,themassconcentrationofdustparticles,theratioofthedensityofsolid particles totheinitialdensityofthegas,vanderWaalsexcludedgasvolumeandshock Cowlingnumber. In chapter 6, weobtainthesolutiontothegeneralizedRiemannproblem(GRP)forthe one-dimensional Euler'sequationforChaplygingasequationswithCoulomb-likefriction term byusingthedi erentialconstraintmethod.Thecompatibilityconditionbetweendif- ferentialconstraintsandthebasicgoverningmodelisderivedhere.Fornon-constantand smoothinitialdata,thesolutiontothegeneralizedRiemannproblemispresentedwitha complete characterizationofthesolutions.