Numerical Solutions for Nonlinear Ordinary and Fractional Duffing Equations Using Combined Fibonacci–Lucas Polynomials

Abstract

Two nonlinear Duffing equations are numerically treated in this article. The nonlinear fractional-order Duffing equations and the second-order nonlinear Duffing equations are handled. Based on the collocation technique, we provide two numerical algorithms. To achieve this goal, a new family of basis functions is built by combining the sets of Fibonacci and Lucas polynomials. Several new formulae for these polynomials are developed. The operational matrices of integer and fractional derivatives of these polynomials, as well as some new theoretical results of these polynomials, are presented and used in conjunction with the collocation method to convert nonlinear Duffing equations into algebraic systems of equations by forcing the equation to hold at certain collocation points. To numerically handle the resultant nonlinear systems, one can use symbolic algebra solvers or Newton’s approach. Some particular inequalities are proved to investigate the convergence analysis. Some numerical examples show that our suggested strategy is effective and accurate. The numerical results demonstrate that the suggested collocation approach yields accurate solutions by utilizing Fibonacci–Lucas polynomials as basis functions.

Keywords

• Fibonacci–Lucas polynomials
• spectral methods
• nonlinear Duffing equation
• convergence analysis