ANALYTICAL AND NUMERICAL SOLUTIONS OF SELECTED NON-LINEAR PDEs

Abstract

This thesis addresses the detailed analytical and numerical investigation of the solutions of selected non-linear partial differential equations (PDEs). Chapter-1 is of introductory nature that provides an overview of the subject matter, developed over the years. Chapter-2 is based on to find the analytical solutions of the Riemann problem for a quasi-linear hyperbolic system of equations, which describes the unsteady and onedimensional flow of an ideal polytropic dusty gas in the presence of the magnetic field. The problem is solved without any restriction on the initial data. The explicit expressions of the elementary wave solutions (i.e., shock waves, simple waves and contact discontinuities) are derived by using the Rankine-Hugoniot (R-H) conditions and Lax conditions. In the flow-field, the velocity and density distributions for the compressive and rarefaction waves are discussed and shown graphically. It is also shown that how the presence of small solid particles and the magnetic field affect the velocity and density across the elementary waves. Chapter-3 is focused on the self-similar solutions of a one-dimensional unsteady and cylindrically symmetric flow, driven by a moving piston in a dusty gas. Solid particles of the gas are continuously distributed. The medium has been taken to be uniform. It is supposed that the equilibrium condition for the flow holds and the piston continuously supplies the variable energy input. The solutions are determined for both types of flows, i.e., for the isothermal flow and adiabatic flow. The effects of different values of the parameter of non-idealness, mass concentration, magnetic field, and the density ratio of the solid particles with the initial density of the mixture are shown in detail. A comparison has been shown between the adiabatic and isothermal flows.