Highly accurate method for solving forced Duffing equations with integral boundary conditions

Abstract The main goal of this paper is to come up with a new numerical algorithm for solving a second-order forced Duffing equation (FDUE) with integral boundary conditions (IBCs). This paper builds a modified shifted Legendre polynomials’ (SLPs) function basis that satisfies homogeneous IBCs, named IMSLP. We have also establish an operational matrix (OM) for the derivatives of IMSLP. The numerical solutions are spectral, obtained by applying the spectral collocation method (SCM). This approach converts the problem with its IBCs into a set of algebraic equations, allowing any suitable numerical solver to resolve them. In the end, we support the suggested theoretical analysis by giving four examples that show the developed method is correct, effective, and useful. We compare the acquired numerical findings with those derived from other methodologies. Tables and figures display the method’s highly accurate agreement between the exact and approximate solutions obtained.