An iterative multistep kernel based method for nonlinear Volterra integral and integro-differential equations of fractional order.

An iterative multistep kernel based method is proposed for the nonlinear Volterra integral equations and nonlinear Volterra integro-differential equations of fractional order, which can produce reliable globally smooth numerical solutions. An error estimate of the positive definite kernel interpolation and, the convergence and error analysis of the proposed iterative scheme are investigated. Here, we focused on positive definite radial basis kernels and further, a new and applicable shape parameter selection strategy is proposed. The proposed multi-step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long-time simulations. In order to show the efficiency and versatility of the proposed method, some numerical experiments are reported. The comparison of the numerical results with the analytical solutions and the best-reported results in the literature confirm the good accuracy and applicability of the proposed method.