In this findings, a numerical meshless solution algorithm for 1D oscillatory Fredholm integral equation (OFIE) is put forward. The proposed algorithm is based on Levin's quadrature theory (LQT) incorporating multi-quadric radial basis function (MQ-RBF). The procedure involves local approach of MQ-RBF differentiation matrix. The proposed method is specially designed to handle the case when the kernel function (KF) involves stationary point(s) (SP(s)). In addition to that, the model without SP(s) is also considered. The main advantage of the meshless procedure is that it can be easily extended to multi-dimensional geometry. These models have several physical applications in the area of engineering and sciences. The existence of the SP(s) in such models has numerous applications in the field of scattering and acoustics etc. (see [1, 2, 4, 6–8]). The proposed meshless method is accurate and cost-effective and provides a trustworthy platform to solve OFIE(s)