A new fractional derivative with a non-singular kernel involving exponential and trigonometric functions is proposed in this paper. The suggested fractional operator includes as a special case Caputo-Fabrizio fractional derivative. Theoretical and numerical studies of fractional differential equations involving this new concept are presented. Next, some applications to RC-electrical circuits are provided.

We consider certain classes of fractional differential equations and systems involving weighted fractional derivatives. We establish some comparison principles, and we study the uniqueness and the continuity with respect to the data.

We consider a certain class of coupled systems of fractional differential equations involving ?-Caputo fractional derivatives. A numerical approach is provided for solving this class of systems. The method is based on operational matrix of fractional integration of an arbitrary ?-polynomial basis. A theoretical study related to the numerical scheme and the convergence of the method is presented. Next, several numerical examples are given using different types of polynomials aiming to confirm the efficiency of our approach.