Approximate Solution of Volterra Integro-Fractional Differential Equations Using Quadratic Spline Function

In this paper, we suggest two new methods for approximating the solution to the Volterra integro-fractional differential equation (VIFDEs), based on the normal quadratic spline function and the second method used the Richardson Extrapolation technique the usage of discrete collocation points. The fractional derivatives are regarded in the Caputo perception. A new theorem for the Richardson Extrapolation points for using the finite difference approximation of Caputo derivative is introduced with their proof. New techniques using the first derivative at the initial point such that obtained by follow two cases the first using trapezoidal rule and the second using the first step of linear spline function using the Richardson Extrapolation method. Specifically, the program is given in examples analysis in Matlab (R2018b). Numerical examples are available to illuminate the productivity and trustworthiness of the methods, as well as, follow the Clenshaw Curtis rule for calculating the required integrals for those equations.