Analysis and Some Applications of a Regularized Ψ–Hilfer Fractional Derivative

The main purpose of this research is to present a generalization of Ψ–Hilfer fractional derivative, called as regularized Ψ–Hilfer, and study some of its basic characteristics. In this direction, we show that the ψ–Riemann–Liouville integral is the inverse operation of the presented regularized differentiation by means of the same function ψ. In addition, we consider an initial-value problem comprising this generalization and analyze the existence as well as the uniqueness of its solution. To do so, we first present an approximation sequence via a successive substitution approach; then we prove that this sequence converges uniformly to the unique solution of the regularized Ψ–Hilfer fractional differential equation (FDE). To solve this FDE, we suggest an efficient numerical scheme and show its accuracy and efficacy via some real-world applications. Simulation results verify the theoretical consequences and show that the regularized Ψ–Hilfer fractional mathematical system provides a more accurate model than the other kinds of integer- and fractional-order differential equations.