The operational matrices methods for solving the Poisson equation with nonlocal boundary conditions

The objective of this paper is to implement some operational matrices methods for solving the two-dimensional Poisson equation with nonlocal boundary conditions using orthogonal polynomials with their operational matrices. Polynomials, namely of the form Standard, Chebyshev, Legendre, Bernoulli, Genocchi and Boubaker are used to find the approximate solution for the problem under Dirichlet and two types of nonlocal boundary conditions. The partial differential equation is converted to a set of linear algebraic equations that can utilize Mathematica®12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Mathematica}}^{\circledR }12$$\end{document} for the linear algebra task. The mean square error Lrms\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}_{rms}$$\end{document} and error norm L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}_{\infty }$$\end{document} are then computed to compare the errors of proposed approaches. A different set of examples is solved to prove the effectiveness and accuracy of the methods. Three examples with classical Dirichlet boundary conditions and two types of nonlocal boundary conditions, namely two-point boundary conditions and integral boundary condition, are treated by the suggested polynomials.