SOLUTIONS OF SOME NON-LINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN SHOWK WAVES

Abstract

Partial di erential equations have become powerful tools to model and describe the fundamental theories in engineering science and physics. In the context of mathematical modeling, most of the physical phenomena in our surroundings are non-linear, hence are modeled by non-linear partial di erential equations. Shock wave is the most remarkable non-linear wave propagation phenomenon which is mathematically and physically interesting. It has been a very important research topic in the last few decades. This may be just because of its many applications in various elds. In aerodynamics, the knowledge of shock waves is essential for designing the optimal geometry of objects moving faster than the sound velocity, e.g., supersonic aircraft or spacecraft re-entering the atmosphere of planets. Researchers have developed many systems to understand the behavior of natural phenomena in our surroundings. For non-linear partial di erential equations, a solution is not always possible. There is no single methodology for dealing with all types of non-linear partial di erential equations. Special techniques must be used to obtain the analytical and numerical solutions to these non-linear partial di erential equations. This thesis deals with the study of shock waves in magnetogasdynamics and involves analytical and numerical solutions of some higher-order non-linear partial di erential equations. The whole thesis has been divided into six chapters which are brie y described as follows: Chapter 1 of the thesis is introductory and provides an overview of the work presented in this thesis. Chapter 2 is focused on the self-similar solutions of the cylindrical shock waves for one-dimensional adiabatic unsteady ow in a rotating non-ideal gas, which has a varying azimuthal uid velocity together with a varying axial uid velocity, with radiation heat ux using the method of invariance of Lie group. The problem involving strong converging shock waves is solved by identifying the whole class of self-similar solutions using the Lie group of transformations. The invariant surface 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. All the ow variables are graphically analyzed behind the shock for di erent values of the adiabatic index of the gas, non-ideal parameter, ambient azimuthal velocity exponent and Alfven-Mach number. Chapter 3 In this chapter, the power series technique (Sakurai's approach) is used to study the propagation of cylindrical shock waves in rotating axisymmetric dusty gas under isothermal ow conditions. The approximate analytic solutions to the considered problem are found with the help of the Sakurai's approach by expanding the power series of the ow variables in terms of (C=V )2; where C and V are the velocities of sound in the undisturbed medium and shock front, respectively. This method is used to derive the zeroth and the rst-order approximations. The solutions for the zeroth-order approximation are constructed in analytical form. Figures behind the shock front for the zeroth-order approximation show the e ects of variation in the values of shock Cowling number (c0); adiabatic exponent ( ); mass fraction of the solid particles in the mixture (kp); and rotational parameter (L) on the ow variables. The results obtained in this chapter match well with the existing results in the literature. Chapter 4 is based on the use of the one-parameter Lie group method of invariance to nd the optimal system and some exact explicit solutions of a (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Under the invariance property of Lie groups, the in nitesimal generators, commutator table, adjoint table, invariant functions and one-dimensional optimal system of subalgebras for the (3+1)-dimensional non-linear wave equation are obtained. Based on the optimal system, symmetry reductions are presented and similarity variables are obtained. Similarity variables reduce the higher order equations into the less order PDEs, later we reduce these PDEs into ordinary di erential equations (ODEs) and obtain the exact solutions of the (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. The generalized group invariant solutions are obtained with the help of the reduced equations. To show the physical a rmation of the results, the obtained solutions are traced graphically. Thus, the solitary wave, single soliton, double soliton, multi soliton and parabolic pro les of the obtained results are found. Chapter 5 provides an analysis of a (2+1)-dimensional Vakhnenko equation. The Lie group theoretic approach is employed in this chapter to nd the similarity reductions and analytic solutions of the considered equation. The in nitesimal generators for the considered problem 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 exacts solutions of the (2+1)- dimensional Vakhnenko equation are obtained. Moreover, the dynamical behaviors of the obtained solutions are graphically shown. Chapter 6 deals with a problem of cylindrical strong shock waves collapsing at the axis of symmetry in non-ideal gas in the presence of an axial magnetic eld. The perturbation series technique used in this chapter provides a global solution to the shock implosion problem in non-ideal magnetogasdynamics in contrast to Guderley's local solution that holds in the neighborhood of implosion. We analyzed the ow parameters by expanding them in powers of time and found the similarity exponents as well as the corresponding amplitudes. All the ow variables and the shock path have been shown graphically in the region from the piston to the axis of collapse. The impacts of the variation of adiabatic coe cient, shock cowling number, and non-ideal parameter are shown on the ow variables and shock path.Solutions, Invariant Solutions, Solitons, Optimal System, (2+1)-dimensional Vakhnenko equation, (3+1)-dimensional non-linear wave equation in liquid with gas bubbles.