Fundamental results on weighted Caputo–Fabrizio fractional derivative. (September 2019) - Al-Refai, Mohammed; Jarrah, Abdulla M.
Highlights: The weighted Caputo–Fabrizio fractional derivative and integral are defined, and related properties are discussed. Solutions of linear and nonlinear fractional equations are obtained in closed forms, and by implementing Banach fixed point theorem. Applications to linear and nonlinear integro-differential equations are presented. Abstract: In this paper, we define the weighted Caputo–Fabrizio fractional derivative of Caputo sense, and study related linear and nonlinear fractional differential equations. The solution of the linear fractional differential equation is obtained in a closed form, and has been used to define the weighted Caputo–Fabrizio fractional integral. We study main properties of the weighted Caputo–Fabrizio fractional derivative and integral. We also, apply the Banach fixed point theorem to establish the existence of a unique solution to the nonlinear fractional differential equation. Two examples are presented to illustrate the efficiency of the obtained results.