A Survey of Direct Methods for Solving Variational Problems

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