On the solution to a boundary value problem for an inhomogeneous elliptic equation by using Legendre and Chebyshev polynomials

This paper constructs a solution to a fourth-order inhomogeneous elliptic equation within the framework of the Kirchhoff-Love theory of thin isotropic plates using the Legendre and Chebyshev polynomials of the first kind. It is assumed that the integration domain is a rectangle. The types of boundary conditions that correspond to pinching along the contour of a rectangular plate, hinged support, and their combinations are used as boundary conditions. The function that approximates the solution of the equation under consideration is represented as a finite sum of a series of these polynomials for each independent variable. Using the collocation method in combination with matrix transformations and properties of Legendre and Chebyshev polynomials, the boundary value problem is reduced to solving a system of linear algebraic equations with respect to coefficients in the expansion of the desired function in these polynomials. In this case, the zeros of the Legendre and Chebyshev polynomials for each independent variable are used as collocation points. The results of calculations using the proposed method of bending a square thin isotropic plate under the considered boundary conditions under the influence of a distributed load of a certain type of intensity leading to an analytical solution of the corresponding boundary value problem are presented. According to the comparison, the constructed solutions coincide with the analytical solutions with a high degree of accuracy.