Numerical solution of the Duffing equation with three types of boundary conditions using shifted Legendre polynomials

Abstract In this study, a reproducing kernel method based on shifted Legendre polynomials is proposed for obtaining numerical solutions to the nonlinear Duffing equation involving both integral and non-integral forcing terms subject to three types of boundary conditions. Unlike the classical reproducing kernel method, the proposed method employs Legendre polynomials to construct the reproducing kernel function. Furthermore, instead of homogenising the boundary conditions, the method incorporates additional basis functions to handle the three types of boundary conditions considered: Bitsadze–Samarskii boundary conditions, separated boundary conditions, and integral boundary conditions. The construction of the applied method is established by theorems, and the iteration process of the approximate solution is presented in detail. To show the accuracy and effectiveness of the presented method, the comparison of the numerical results obtained for six different problems is illustrated through tables and graphs.