Control parameter & solutions to generalized evolution equations of stationarity, relaxation and diffusion

It has become a conjecture that power series like Mittag-Leffler functions and their variants naturally govern solutions to most of generalized fractional evolution models such as kinetic, diffusion or relaxation equations. Is this always true? In this article, three generalized evolution equations with additional fractional parameter are solved analytically with conventional techniques. They are processes related to stationary state system, relaxation and diffusion. In the analysis, we exploit the Sumudu transform to show that investigation on the stationary state system leads to results of invariability. However unlike other models, the generalized diffusion and relaxation models are proven not to be governed by Mittag-Leffler functions or any of their variants, but rather by a parameterized exponential function, new in the literature, more accurate and easier to handle. Graphical representations are performed and also show how that parameter, called β, can be used to control the stationarity of such generalized models. AMS subject classifications: 37L05, 26A33, 37N25, 60J60, Keywords: Generalized evolution models, Exponential with a parameter, Modified-Sumudu transform, Mittag-Leffler functio