Law of transfer physical substances described using differential equations, where the unknown functions are subject to the given boundary conditions. These conditions depend on the position in which the object under study. Such equations can be regarded as the operator equations acting in specific functional areas. Information obtained from this formulation can be used to build an efficient numerical methods for solving problems with great practical importance. In this paper the relationship between the operator equations are considered in a Banach space with a quadratic functional and its application for a modification postreniyavariational method. Considered in the work of the physical problem: transverse bending beams with constant stiffness, which lies on an elastic foundation. A mathematical model of this process is described by a differential equation of the fourth order with the given boundary conditions, regarded as as an operator equation with a positive operator on the left side. A solution of this equation is studied by using a special form of a quadratic functional. It is proved that the problem of finding solutions of this equation with the given boundary conditions, equivalent to the problem of finding the minimum of the functional. Further, the variational method is based on minimization of a quadratic functional, numerical calculation which gives solutions of the original physical problem. The practical significance of the work is that the numerical calculation of finding the minimum of the functional is easily accomplished rapidly convergent numerical algorithm based minimization of a quadratic functiona
