A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equation

Abstract

In this paper, a new method based on hybridization of Lucas and Fibonacci polynomials is developed for approximate solutions of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equations. Firstly time discretization is made by using finite difference approaches. After that unknown function and its derivatives are expanded to Lucas series. Based on these series expansion, differentiation matrices are derived by utilizing Fibonacci polynomials. By doing so, the solution of the mentioned equations is reduced to the solution of an algebraic system of equations. By solving this system of equations the Lucas series coefficients are obtained. Then substituting these coefficients into Lucas series expansion approximate solutions can be constructed successively. The main goal of this paper is to indicate that Lucas polynomial based method is appropriate for 1D and 2D nonlinear problems. Efficiency and performance of the proposed method are judged on six test problems which consists of the 1D and 2D version of mentioned equation by calculating L2 and L∞ error norms. Feasibility of the method is verified by obtained accurate results.