Numerical solutions of ordinary differential equations using element free galerkin method

Abstract

Differential equations play an important role in almost all areas of science and engineering, for example physics, economics, biology, fluid dynamics, fluid mechanics, aerodynamics, etc. But the analytic solution can not be determined for most of the differential equations governing realistic model problems using analytical techniques. Therefore, we need to depend upon the numerical methods to find the approximate solution. Finite element techniques have been widely used for solving the differential equations. But from the last two decades, element free Galerkin methods have drawn attention of the research community to solve real model problems. The element free Galerkin techniques have many benefits over finite element techniques. In the present study, elements free Galerkin methods have been applied to solve the fluid dynamics problems.

The whole work is divided into three chapters.

Chapter 1 introduces the basic concepts of differential equations and the solution methodology. Finite element technique have been elaborated with the help of an example. Also the concept behind the element free Galerkin methods and the need for the method have been discussed. In Chapter 2 a review of research paper entitled "Element free Galerkin solution of radiative hydromagnetic micropolar flow saturated Darcy medium with heat transfer over a stretching sheet with Joule heating" has been carried out. The paper has been discussed thoroughly. Apart from this, the concept of shape functions, weight functions and moving least square strategy has been presented.

Chapter 3 reviews the research paper entitled "Numerical simulation of unsteady MHD flow and heat transfer of second grade fluid with viscous dissipation and Joule heating using element free approach". The paper has been discussed thoroughly. The element free Galerkin technique based on moving least squares has been explained for solving the model problem. Numerical results have been discussed in the last. Numerical results shows that the results obtained using the element free Galerkin method are in good agreements with those listed in the literature.

In the last, references have been presented.