Operational matrix approach based on two-dimensional Boubaker polynomials for solving nonlinear two-dimensional integral equations

Abstract

Two-dimensional First Boubaker polynomials (2D-FBPs) have been formulated and developed as the set of basis for the expansion of bivariate functions. These polynomials are an extension of one-dimensional (1D) FBPs that have previously been used to solve 1D nonlinear integral equations. The dual operational matrix, four operational matrices for integration and operational matrix of product for 2D-FBPs are presented in this study newly. The explicit formula for the operational matrix of product is based on the operational matrix of product by 2D Taylor polynomials (TPs), which has been determined too. Moreover, 2D-FBPs and their operational matrices are used to approximate solution of four types of 2D nonlinear integral equations, that are 2D nonlinear Fredholm, Volterra, Volterra–Fredholm and the system of Volterra integral equations. Using Newton–Cotes collocation nodes, our approach leads to the solution of the set of nonlinear algebraic equations and can be solved uniquely by an iteration method. Since the proposed approach does not require any integration, so all the computations can be easily performed. An explicit and simpler upper bound is obtained for the error vector function of the operational matrix of integration P2Db. In addition, an explicit and simpler upper bound is presented for each bivariate function that expand by 2D-FBPs, as well as the convergence analysis has been proved. Six test problems are given to verify the validity and applicability of the presented methods. The numerical results are compared with exciting methods in literature.