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Lately introduced fractal fractional Caputo - Fabrizio operator. By substituting the single kernel at the classic derivative of fractal fractional Caputo with the ordinary kernel This modern operator was derived. We introduce some beneficial characteristics relied on the qualifier of fractal fractional Caputo - Fabrizio. Here, we extend Caputo-Fabrizio for nonlinear fractal fractional differential equations. We apply Legendre operational matrix relied on this modern operator and then, we employ it to solve the differential equations determined in the sense of fractal fractional Caputo-Fabrizio. To show the simplicity and precision of the suggested technicality Some numerical examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we derive the equivalent fractional integral system to the nonlinear fractional differential system involving Hadamard fractional derivative subject to coupled boundary conditions. The existence and uniqueness results of solution for proposed system have been obtained. Moreover, we give some sufficient conditions to guarantee that the solutions to such system are Ulam-Hyers stable and Ulam-Hyers-Rassias stable. Our investigations based on the nonlinear analysis and fixed point theorems of Banach and Schaefer. To justify our results, we provide pertinent illustrative examples. [ABSTRACT FROM AUTHOR]