Approximations of Solutions of a Neutral Fractional Integro-Differential Equation

In the present work, we consider a fractional integro-differential equation in an arbitrary separable Hilbert space H. An associated integral equation and a sequence of approximate integral equations is studied. The existence and uniqueness of solutions to every approximate integral equation is obtained by using analytic semigroup and Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We show the convergence of the solutions using Faedo–Galerkin approximation and demonstrate some convergence results. Finally, an example is considered to show the effectiveness of the obtained theory. © 2016, Foundation for Scientific Research and Technological Innovation.