Mohammad Reza Doostdar - Department of Mathematics, Zarandieh Branch, Islamic Azad University, Zarandieh, Iran
Manochehr Kazemi - Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran
Alireza Vahidi - Department of Mathematics, Yadegar -e-Imam Khomeini (RAH) Shahre Rey Branch, Islamic Azad University, Tehran, Iran

This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Block-pulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results.

Integral boundary conditions, Boundary value problem, Duffing equation, Hybrid functions, Legendre polynomials