A Survey of Direct Methods for Solving Variational Problems

Abstract

This study presents a comparative survey of direct methods for solving Variational Problems. This<br />problems can be used to solve various differential equations in physics and chemistry like Rate<br />Equation for a chemical reaction. There are procedures that any type of a differential equation is<br />convertible to a variational problem. Therefore finding the solution of a differential equation is<br />equivalent to solving its related variational problem. The objective of this paper is to describe the<br />major direct methods that have been developed over the years for solving these types of problems. In<br />this paper we focus on using orthogonal polynomials and triangular functions as basis functions. Each<br />method needs computing operational matrices and some other parameters which are presented as<br />well. Several numerical examples are then included to demonstrate the accuracy and applicability of<br />the reviewed methods. Computational concerns are then discussed to provide a guideline to the<br />preferred and the most accurate method.