http://localhost:8081/jspui/handle/123456789/19062 Sun, 05 Apr 2026 22:56:19 GMT 2026-04-05T22:56:19Z http://localhost:8081/jspui/handle/123456789/19902 Title: STUDY ON SHOCK WAVES PROPAGATION IN GASEOUS MEDIA AND SYMMETRY ANALYSIS OF COUPLED EQUAL WIDTH WAVE EQUATION Authors: Chauhan, Swati Abstract: Gasdynamics is a sub eld of hydrodynamics, which is the study of uids, liquids, and gases. It emerged at the tail end of the nineteenth century as a consequence of e orts to comprehend the fundamentals of high-speed compressible ow theories. The equations of gasdynamics have traditionally been used to study shock wave propagation. The phenomenon of shock waves is mathematically and physically interesting, and study into them has been extremely important over the past few decades. One of the primary reasons for this is that it can be applied in a wide variety of domains. In brief, a shock wave is a non-linear wave that travels faster than the speed of sound in the medium. Shock waves occur frequently in everyday life, although they go unnoticed the majority of the time. Physically, the emergence of a shock wave in a uid ow is always characterized by instant changes in the ow velocity, pressure and temperature. This change is not reversible; inside the shock wave, dissipation of energy occurs and the entropy increases. Shock waves have gained much importance due to instantaneous changes in velocity and pressure, and are being investigated for potential applications in a variety of elds. In the eld of aerodynamics, when designing the optimal geometry of objects moving faster than the speed of sound, such as supersonic aircraft or spacecraft re-entering the atmospheres of planets, having an understanding of shock waves is extremely important. A shock wave can promote the transition of graphite into diamond, which is a classic example of this phenomenon. In the eld of astrophysics, two common scenarios that involve shock waves are the explosions of supernovae and the collisions of clouds. Numerous applications of shock waves may be found in the realm of medical science. One such application is the removal of kidney stones from the human body using a technique known as \Extracorporeal Shock Wave Lithotripsy (ESWL)", which involves the use of relatively modest shock waves. To begin, we will conduct a quick review of shock waves. In this type vii viii of wave, the shock that is associated with hyperbolic systems of conservation laws can be identi ed by its front. This front is a moving surface that divides the space into two subspaces in which a continuous solution exists but there is a jump in the eld variables across the shock front. The existence of shock waves was predicted by considering certain waves traveling in a uid governed by the Euler equations. Thu, 01 Sep 2022 00:00:00 GMT http://localhost:8081/jspui/handle/123456789/19902 2022-09-01T00:00:00Z http://localhost:8081/jspui/handle/123456789/19667 Title: HYBRID DECISION-MAKING MODELS FOR MULTI-CRITERIA OPTIMIZATION PROBLEMS Authors: Singh, Meenu Abstract: Most of the real-world problems are dependent on a set of criteria rather than a single criterion, which are usually tangible and their corresponding information is merely impossible to get in quantitative form. Hence, this increases difficulties for a method to handle such complexities and provide a feasible solution. If the Decision Makers (DMs) are involved too, it can worsen the decision output. Thus, the development of efficient Hybrid Decision-making (HDM) models is a need of the hour that can provide the optimal solutions to the DMs. The main focus of this thesis is to develop efficient Hybrid Multi-Criteria Decision Making (HMCDM) models for handling fuzzy environment, sustainability issues, rankability issues, multi-dimension data and incomplete data. The HMCDM models are developed by integrating the different Multi-Criteria Decision Making (MCDM) methods, followed by the sensitivity analysis and correlation analysis for validation purposes in different chapters. The application of the proposed integrated model is demonstrated on a set of three problems taken from different Indian industries, including the pulp and papermaking industry, packaging industry, and paper mills. Four types of HMCDM models are considered (I) integration of MCDM methods with MCDM methods, (II) integration of MCDM with Fuzzy Set Theory, (III) integration of MCDM with MODM method, and (IV) integration of MCDM with a sports ranking method (D-matrix). The entire work done for this research is organized into six chapters of the thesis. Chapter 1 defines multicriteria decision-making in the context of decision analysis in general and draws the distinction between single and multicriteria decision-making. It provides a classification of MCDM methods into those based on multi-attribute utility theory, outranking methods and hierarchical approaches. It further provides elementary decisions and distinguishes between multi-objective decision making and multi-attribute decision making and illustrates their historical development along with a brief literature review of some selected weighting methods. An analysis of published literature provides some insights into recent trends. Chapter two provides a brief summary of the main MADM and MODM methods used in this thesis. This includes mathematical formulations and brief descriptions of the steps required for application of a method.Chapters 3 to 5 contain the main contributions of the thesis. Each of them develops one or two HMCDM methods and applies them to a particular industrial application. Third chapter deal with the selection problem while considering the fuzziness in the DMs’ opinions and information collected. In this chapter, two different HMCDM models are proposed for both the selection problems of the pulp and papermaking industry and the packaging industry in India. The fourth chapter draws the attention towards another ranking problem in MCDM. Here, MCDM method is integrated with a minimization model to overcome the drawback of MCDM methods of providing conflicting rankings. An example of Pulp and paper industry in context to India is considered to demonstrate the proposed framework. Here, the performance of twenty-two Indian paper mills is measured by seven MCDM methods and finally, an optimal aggregated rank is acquired. The fifth chapter covers the issue of incomplete information in the MCDM problem by discussing the various imputation in the past and then proposes a method Modified D-Matrix (MoDM) method. This method can handle biased and bias free evaluations in both result-separating and result-merging contexts. The proposed method is applied to the supplier selection data, and the results were found to be in accordance with experts. Finally, the major contributions of this thesis along with concluding remarks in theoretical and application facets are presented in Chapter six. The results obtained in all the chapters indicate that the performance of MCDM can be improved significantly by integrating it with either classical MCDM or artificial intelligence techniques like Fuzzy set theory, Genetic Algorithm. Thu, 01 Sep 2022 00:00:00 GMT http://localhost:8081/jspui/handle/123456789/19667 2022-09-01T00:00:00Z http://localhost:8081/jspui/handle/123456789/19507 Title: STUDY OF SHOCK WAVES IN GASEOUS MEDIUM AND A SHALLOW WATER WAVE MODEL Authors: Singh, Deepika Abstract: Gas dynamics is a particular branch of uid dynamics, which is the science of uids, liquids, and gases. It developed around the end of the nineteenth century as a result of attempts to comprehend the principles of high-speed compressible ow theories and their applications. The equations of gas dynamics have traditionally been used to study shock wave propagation. A shock wave can be brie y described as a non-linear wave traveling faster than the sound speed in the medium. Shock waves are a common occurrence in everyday life, yet they go unnoticed most of the time. The bangs of an applauding crowd or an explosion, the sound of thunderstorms, earthquakes and volcanic eruptions are some examples of situations in which shock waves are noticeable. Physically, the emergence of a shock wave in a uid ow is always characterized by instant changes in the ow velocity, pressure and temperature. This change is not reversible; inside the shock wave, dissipation of energy occurs and the entropy increases. Shock waves have gained much importance due to instantaneous changes in velocity and pressure, and are being investigated for potential applications in a variety of elds. In the eld of medical science, shock waves are used in traumatology and orthopedics to treat various insertional tendinopathies (enthesopathies) and delayed unions and nonunions of fracture. Shock waves are bene cial in material synthesis and industrial elds too. Physically, the shock wave is very thin and the thickness of a shock is about 0.2 m (10􀀀5 in), or roughly four times the mean free path of the gas molecules for air at ambient conditions. The entire width of a shock wave therefore only contains a small number of particles in the longitudinal direction and thus the shock appears as nearly a singularity in the continuum model, yet the existence of shock waves was predicted by considering certain waves traveling in a uid governed by the Euler equations. The Euler equations are a non-linear hyperbolic system of partial di erential equations (PDEs) that are derived from the compressible Navier-Stokes equations by omitting the i e ects of viscosity and heat conduction. Mathematically, the fascinating aspect of such systems is that they admit shock waves in their solution; discontinuities in the solution that can form even from the smooth initial data. Since the superposition of the solutions is not possible for the system of non-linear PDEs, we still do not have a general approach for nding the solution of an initial or boundary value problem associated with a system of nonlinear PDEs. It is necessary to look for approximate analytical and numerical techniques whose objective is precisely to nd the solutions of the system of non-linear PDEs. The present thesis deals with the mathematical study of shock wave propagation in the gaseous medium by considering the e ects of dust particles and magnetic eld, and a shallow waterwave model of non-linear PDEs. Shallow water waves correspond to the ow beneath a horizontal pressure surface in a uid or the ow at the free surface of a body of shallow water under the pull of gravity. In atmospheric science, oceanography, and many other elds, the shallow water-wave models are of particular importance. The present thesis is organized into six chapters as brie y described below: Chapter 1: This is an introductory chapter. We present our objectives and the motivation behind them supported by a literature survey. Chapter 2: This chapter concerns the study of converging shock waves in a dusty gas of uniform density. The dusty gas is assumed to be a mixture of an ideal gas and a large number of dust particles. The dust particles are of the micrometric size and uniformly distributed in the mixture. The dusty gas is lled into a cylindrical/spherical piston, which then begins to compress at a faster pace than the medium's acoustic speed, generating a cylindrically/spherically symmetric shock wave within the piston. The position of the shock wave is unknown and has to be determined. In this chapter, the perturbation series method is used to solve the implosion problem in a dusty gas, providing a global solution and yielding accurately the results of Guderley's local similarity solution, which holds only in the neighborhood of the axis/center of implosion. The similarity exponents are determined together with the corresponding amplitudes in the vicinity of the shock collapse by extending the ow variables and shock position in the Taylor series in time t. A comparison is done between the computed values of the similarity exponents and the numerical results obtained by other methods. The shock position and ow variables are analyzed graphically in the region extending from the piston to the axis/center of collapse for di erent values of the adiabatic exponent ( ), the wavefront curvature ( ) and for various dusty gas parameters, namely the mass concentration of the dust particles (Kp), the ratio of the density of ii the dust particles to the initial density of the gas (G0), and the relative speci c heat ( ). Chapter 3: With the impact of an azimuthal magnetic eld, the strong converging cylindrically symmetric shock waves collapsing at the axis are investigated for a non-ideal gas. It is demonstrated in this chapter that the perturbation series approach, when applied to the shock implosion problem in non-ideal magnetogasdynamics, yields a global solution, in contrast to Guderley's asymptotic solution, which holds only in the immediate neighborhood of the axis of implosion. The similarity exponents and the corresponding amplitudes are found near the shock collapse. The re nement of the leading similarity exponents near the axis of implosion is also made. Figures depicting the distributions of the ow variables and shock trajectory for various values of the adiabatic exponent ( ), non-ideal parameter (b) and shock Cowling number (C0) have been presented. Chapter 4: This chapter demonstrates the study of the propagation of converging cylindrical shock waves in a non-ideal gas (van der Waals type) with the e ect of magnetic eld and isothermal ow conditions via the Lie group theoretic method. The ambient gas ahead of the shock is considered to be homogeneous. The Lie group of transformations is used to determine the entire class of self-similar solutions to the problem involving strong converging shock waves. The surface invariance conditions are used to determine the in nitesimal generators of the Lie group of transformations. Based on the arbitrary constants that occur in the expressions for the generators, two di erent cases of possible solutions with power-law shock path and exponential shock path are obtained. A particular case of the power-law shock path is worked out in detail. The similarity exponents are obtained numerically for di erent values of the non-ideal parameter (b) and shock Cowling number (C0), and then compared to the similarity exponents obtained using the \Guderley's approach". All the ow variables are graphically analyzed behind the shock for di erent values of the non-ideal parameter and shock Cowling number. Chapter 5: In this chapter, the propagation of cylindrical shock waves produced on account of a strong explosion in a non-ideal gas under the e ect of an azimuthal magnetic eld is studied. The problem can be expressed mathematically by a non-linear hyperbolic system of PDEs with jump conditions at the shock as the boundary conditions. The approximate analytic solutions to the considered problem are obtained using the power series method (Sakurai's approach) by extending the power series of the ow variables in terms of (a0=V )2, where a0 and V are the velocities of sound in the undisturbed medium and shock front, respectively. This chapter discusses the rst-order and second-order approximate soiii lutions and provides closed-form solutions for the rst-order approximation. The behavior of the ow variables is depicted via gures behind the shock front for the rst-order approximation along with the variation in the values of the non-ideal parameter (b) and shock Cowling number (C0). To verify the obtained results, numerical calculations are performed in the absence of the magnetic eld which showed that the existing results in the literature are recovered very well. Also, it is observed that these results are in good agreement with the results obtained by the Runge-Kutta method of fourth-order (RK4 method). Chapter 6: This chapter provides an analysis of a (2+1)-dimensional modi ed dispersive water-wave (MDWW) system of partial di erential equations, which describes the non-linear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. The Lie group theoretic approach is employed in this chapter to nd the similarity reductions and analytic solutions of the (2+1)-dimensional MDWW system. The in nitesimal generators for the considered system are obtained under the invariance property of the Lie group of transformations. The one-dimensional optimal system of subalgebras is established. Finally, based on the optimal system, the similarity reductions and invariant solutions of the (2+1)-dimensional MDWW system are obtained. Moreover, the dynamical behaviors of the obtained solutions such as multi-soliton, doubly soliton, single soliton, solitary waves and stationary waves are graphically shown using 2D, 3D and corresponding contour plots. Fri, 01 Apr 2022 00:00:00 GMT http://localhost:8081/jspui/handle/123456789/19507 2022-04-01T00:00:00Z http://localhost:8081/jspui/handle/123456789/19404 Title: STUDY OF CERTAIN QUASI-LINEAR HYPERBOLIC SYSTEMS OF PDEs Authors: Singh, Mayank 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. Wed, 01 Feb 2023 00:00:00 GMT http://localhost:8081/jspui/handle/123456789/19404 2023-02-01T00:00:00Z