On the Collocation Method in Constructing a Solution to the Bending Equation for a Long Rectangular Nanoplate

Within the framework of the theory of microstructural deformation, a new approach is proposed for constructing a solution to the bending equation of a long rectangular nanoplate that is under the influence of a transverse load. The proposed approach is based on the collocation method using a system of orthogonal Chebyshev polynomials of the first kind. The bending function is represented as a partial sum of a series of these polynomials. The roots of Chebyshev polynomials of the first kind are chosen as the collocation points. By sequentially multiplying the left and right sides of the resulting matrix equation by the inverse matrix to the matrix with the values of Chebyshev polynomials at the collocation points and by the generalized inverse matrix to the degenerate matrix of differentiation of these polynomials, the equation of the bending surface, taking into account boundary conditions, is reduced to a system of linear algebraic equations with respect to unknown coefficients in the representation of the solution. In this case, the elements of each of these matrices are presented explicitly. An estimate of the error of the constructed solution based on an infinite norm is obtained. The results of the conducted computational experiments are presented, which demonstrate the effectiveness of the proposed approach.