Paper Title
Application Of Variational Methods And Galerkin Method In Solving Engineering Problems Represented By Ordinary Differential Equations

Nowadays the accuracy of problem solving is very important. In olden days the Variational methods were used to solve all engineering problems like structural, heat transfer and fluid mechanics problems. With the emergence of Finite Element Method (FEM) those methods are become less important, although FEM is also an approximate method of numerical technique. The concept of variational methods is inducted to solve majority of engineering problems, which gives more accurate results than any other type of approximate methods. The engineering problems like uniform bar, beams, heat transfer and fluid flow problems are used in our daily life and they play an important role in the development of our society. To achieve drastic development in the society, it is a must to focus on adopting approximation methods that improve the accuracy of engineering solution. Of all the methods, Galerkin method is emerging as an alternative and more accurate method than those of Ritz, Rayleigh – Ritz methods. Any physical problem in nature can be transformed into an equivalent mathematical model by idealization process and describing its behavior by a suitable governing equation with associated boundary conditions. Against this backdrop, the present work focuses on application of different variational methods in solving ordinary differential equations. The reason behind choosing second order differential equation (ODE) is that most of the structural and heat transfer problems can well be represented by an ODE. As an illustration the work herein reported highlights the utility of above cited methods with a simple bar problem. Furthermore the numerical part of this work is carried out on a MATLAB platform. Keywords— Variational methods, Second order differential equation, elastic bar, Ritz method, Rayleigh – Ritz method, Galerkin method and MATLAB