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.

We introduce iterative methods for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points for an infinite family of nonexpansive mappings and a family of strictly pseudocontractive mappings, and the set of solutions of the Variational inequalities for a family of alpha-inverse-strongly monotone mappings in a Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our proposed schemes. The strong convergence results are obtained via the CQ method. 

In this paper, the Variational iteration and the homotopy perturbation methods are adopted to obtain the solution of the one-dimensional wave equation. The solutions procedure reveal that the applied methods are very efficient. In addition, the result obtained is the same as the D' Alembert's formula for the wave equation.

In this article, we implement Variational iteration method (VIM) and Adomian decomposition method (ADM) for finding exact solutions of Poisson equation with Dirichlet or Neumann boundary conditions. To illustrate the simplicity and reliability of the methods, three examples are provided. The results obtained by VIM are compared with those of ADM, and then the ability of each method is also discussed. 

The aim of this work is to investigate the Variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discreted. Optimal error estimates in L2 are established for the state and the control variable. As a result, it can be proved that the discrete solutions possess the convergence property of order. Finally, a numerical example is presented which confirms the theoretical results.

In this paper, a Klein-Gordon equation is solved by using the Adomian’s decomposition method, Variational iteration method and modified form of these methods. The approxi-mate solution of this equation is calculated in the form of series in which its components are computed by applying a recursive relation. The existence and uniqueness of the so-lution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods.

In this paper we prove that if X is a Banach space, then for every lower semi-continuous bounded below function f; there exists a ('1; '2)-convex function g; with arbitrarily small norm, such that f + g attains its strong minimum on X: This result extends some of the well-known varitional principles as that of Ekeland [On the Variational principle, J. Math. Anal. Appl. 47 (1974) 323{ 353], that of Borwein-Preiss [A smooth Variational principle with applications to subdi erentiability and to di erentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987) 517{527] and that of Deville-Godefroy-Zizler [Un principe variationel utilisant des fonctions bosses, C. R. Acad. Sci. (Paris). Ser. I 312 (1991) 281{286] and [A smooth Variational principle with applications to Hamilton-Jacobi equations in in nite dimensions, J. Funct. Anal. 111 (1993) 197{212].