Numerical Solutions To Parabolic Differential Equations And Application.

ABSTRACT

Partial Differential Equations (PDEs) of parabolic type arise naturally in the modeling of many phenomenon of in various fields of physics, engineering and economics. The main aim of this research is to study finite difference methods with numerical solutions of this class of equations. Both one and two dimensions have discussed in this research. I have investigated stability and convergence analysis of different schemes and obtained convergence error estimates. It is important to note that convergence results hold also on Finite- element methods but in this study we didn‟t worked on in. The discretization in time using Explicit, Implicit, Crank-Nicolson, Richardson and 𝜃 − method are addressed and stability, consistency and convergence estimates analyzed well. The mathematical modeling and Numerical method of non-reactive solute in porous media is also in the scope of this study. Among the applications like time estimates and solutions of dynamic model of reactor have discussed in this research. This fluid dynamics plays a major rule in hydrology, medical science and petroleum industry. Also few problems have discussed in this work to show how these methods like Crank-Nicolson, Explicit and implicit method works in parabolic equations (heat equations). According to Farlow 1982 , Most physical phenomenon, whether in domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow can be described in general by partial differential equations (PDEs); In fact, most of mathematical physics are PDEs. It‟s true that the simplification can be made that reduce the equations in equation to ordinary differential equations, but, nevertheless, the complete description of these systems resides in the general area of PDEs. In this research we are aiming to show what parabolic partial differential equations are and how they are solved by Numerical solutions.