Classical Solution of Some Nonlinear Partial Differential Equations

Abstract

Theoretical and applied research in the field of fluid flow through porous media has received increased attention during past three decades. This is due to the importance of this research area in various branches of engineering and science such as reservoir engineering, petroleum engineering, environmental engineering, civil engineering, ground water hydrology, soil science etc. Many mathematical models have been developed to explain fluid flow through porous media. When oil and water flowing simultaneously in porous medium, some physical phenomena occur. The mathematical formulation of these physical phenomena leads to the non-linear partial differential equations. It is a challenging task to solve nonlinear partial differential equation. Some standard transformation like similarity transformation is used to transform nonlinear partial differential equation into nonlinear ordinary differential equation but still it is difficult to get its exact solution. Many researchers are working on approximate solution of some nonlinear partial differential equation using different numerical techniques. In present work our attempt is to obtain classical exact solution of nonlinear partial differential equation of many real-world problems. The thesis discusses Functional separable method (FSM), Clarkson-Kruskal direct method (CKDM), Homotopy perturbation transform method (HPTM) and Variational iteration method (VIM). newlineWe have referred many research papers for the study of mathematical model of one-dimensional movement of ground water, fingering and imbibition phenomenon. With distinct perspectives, several researchers researched these phenomena. But, as yet, in vertical downward direction, no researcher had studied these phenomena in heterogeneous porous medium. In this work, we have analysed mathematical modelling of moisture content in unsaturated heterogeneous soil and its solution by using Functional separable method. We have obtained mathematical solution of fingering phenomenon and counter-current imbibition phenomenon