# Solution of Pde Essay

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Chapter-1 Introduction The exact solution to a boundary value problem can rarely be found even if the governing differential equation is a linear one. The finite element is essentially a powerful numerical method for obtaining approximate solutions to boundary value problems in which approximate trial solutions are used. If , in one dimension, or in two dimensions denotes the unknown exact solution to a boundary value problem then we shall denote approximate trial solution by or . The label finite element method first appeared in 1960, when it was used by Clough  in a paper on plane elasticity problems, the ideas of finite element analysis date back much further. The finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. Although originally developed to study stresses in complex airframe structures, it has since been extended and applied to the broad field of continuum mechanics. Because of its diversity and flexibility as an analysis tool, it is receiving much attention in engineering schools and in industry. One of the major feature of the Finite Element Method is the subdivision of the domain of the problem into smaller regions or elements and the construction of simple trial solutions within each element. However, long prior to the development of finite element, various numerical techniques has been developed for solving boundary value problems with out subdivision. One of the methods based on Weighted residual techniques. Summary This thesis is organized by four chapters. In Chapter 1, the introduction and summary are included. In chapter 2, first we discuss the Galerkin weighted residual method to approximate the numerical solutions of BVP. On the basis of the concept of the Galerkin Method, modified Galerkin method is developed for which the trial solutions