A Survey of Direct Methods for Solving Variational Problems

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