The numerical solution of nonlinear delay Volterra integral equations using the thin plate spline collocation method with error analysis

Delay integral equations can be used to model a large variety of phenomena more realistically by intervening in the history of processes. Indeed, the past exerts its influences on the present and, hence, on the future of these models. This paper presents a numerical method for solving nonlinear Volterra integral equations of the second kind with delay arguments. The method uses the discrete collocation approach together with thin plate splines as a type of free-shape parameter radial basis functions. Therefore, the offered scheme establishes an effective and stable algorithm to estimate the solution, which can be easily implemented on a personal computer with normal specifications. We employ the composite Gauss–Legendre integration rule to estimate all integrals that appeared in the method. The error analysis of the presented method is provided. The convergence validity of the new technique is examined over several nonlinear delay integral equations, and obtained results confirm the theoretical error estimates.