Numerical solutions of a class of linear and nonlinear Volterra integral equations of the third kind using collocation method based on radial basis functions

In this paper, we use a method based on radial basis functions and the collocation method for the numerical solution of a class of Volterra integral equations of the third kind, using zeros of the shifted Legendre polynomial as the collocation points. The principal benefit of this scheme is that it does not require any discretization and so it is independent of the geometry of the domains and can thus be applied to the solution of various kinds of integral equations. The procedure is more flexible for the majority of classes of Volterra integral equations of the third kind. The construction of the suggested technique has been introduced. The convergence analysis of the presented method is investigated. Finally, certain numerical examples are included to show the accuracy and efficiency of the new technique. The numerical results obtained and their comparison with other methods demonstrate the reliability of this method. Our proposed method gives acceptable accuracy with a small use of data, which also reduces the computational costs.