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Ph.D.
(Science)
CLASSICAL SOLUTION OF SOME NONLINEAR PARTIAL
DIFFERENTIAL EQUATIONS

Ph.D. Scholar : Shah Disha Arvindbhai
Research Supervisor : Prof. (Dr.) Amit K. Parikh

Regi. No.: 17146061002
Abstract :
Theoretical and applied research in the field of fluid flow through porous media has
received increased attention during the 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 non-linear partial differential equations. It is a
challenging task to solve the nonlinear partial differential equations. Some standard
transformation like similarity transformation is used to transform nonlinear partial
differential equation into the nonlinear ordinary differential equation but still, it is difficult
to get their exact solution. Many researchers are working on approximate solutions of
some nonlinear partial differential equations using different numerical techniques. In the
present work, our attempt is to obtain the classical exact solution of nonlinear partial
differential equations of many real-world problems. The thesis discusses the Functional
separable method (FSM), Clarkson-Kruskal direct method (CKDM),
Homotopyperturbation transform method (HPTM) and Variational iteration method (VIM).

We have referred many research papers for the study of the 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 inunsaturated heterogeneous soil and its solution by using
Functional separable method. We haveobtained mathematical solution of fingering
phenomenon and counter-current imbibition phenomenon in vertical downward direction
through heterogeneous porous medium using Variational iteration method.

Key words: Counter-current imbibition phenomenon, Fingering phenomenon,
Heterogeneous porous medium, Nonlinear partial differential equation, Functional separable
method, Clarkson-Kruskal direct method, Homotopy perturbation transform method,
Variational iteration method, Oil recovery process, Nonlinear source, Logarithmic source
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