Study of composite fractional relaxation differential equation using fractional operators with and without singular kernels and special functions

Our aim in this article is to solve the composite fractional relaxation differential equation by using different definitions of the non-integer order derivative operator $D_{t}^{\alpha }$ , more specifically we employ the definitions of Caputo, Caputo–Fabrizio and Atangana–Baleanu of non-integer order derivative operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley’s generalised G function. Furthermore, the effects of the parameters involved in the model are graphically highlighted.