Computational method for solving boundary value problems of mechanics deformable body using non-orthogonal functions

¹ Institute of Applied Mechanics of Russian Academy of Sciences, 7, Leningradsky prospekt, Moscow, 125040, Russia
² Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NASU, 3-b, Naukova st., Lviv, 79060, Ukraine

^* e-mail: vbak@yandex.ru
^** e-mail: victor.rev.12@gmail.com

Abstract

In this paper a complete system of non-orthogonal functions was built on the basis of orthogonal sines and cosines. It is shown that the known orthogonal systems of functions are a degenerate case of non-orthogonal systems of functions. It has been proven that the continuous function can be approximated non-orthogonal functions in such a way that one selected nonorthogonal function will not included in this amount. The boundary value problem of the elasticity theory has been considered for an inhomogeneous plate. A new method for solving the boundary value problem is developed for the fourth-order equation with variable coefficients. The proposed method is based on separating of the stress state of the plate, use of complete systems of nonorthogonal functions and a generalized quadratic form. Criteria have been established under which the constructed approximate solution coincides with the exact solution. This method has been adapted to solving a boundary value problem for high-order differential equations. The high accuracy of the method has been confirmed by numerical calculations.