Analysis of recent fractional evolution equations and applications

Abstract It is believed that Mittag-Leffler function and its variants govern the solutions of most fractional evolution equations. Is it always true? We establish some unknown and useful properties that characterize some aspects of fractional calculus above power-law. These include, the correspondence relations, the equality of mixed partials, the boundedness, the continuity and the eigen-property for the derivative with non singular kernel. Some related evolution equations are studied using those properties and they unexpectedly show that solutions are not governed by Mittag-Leffler function nor its variants. Rather, solutions are governed by unusual functions of type exponential (37), easier to handle and more friendly than power series like Mittag-Leffler functions. Numerical approximations of a second order nonhomogeneous fractional Cauchy problem are performed and show regularity in the dynamics. An application to a geoscience model, the Rossler system of equations is performed where existence and uniqueness results are provided. The system appears to be chaotic and the properties of the derivative with no singular kernel do not interrupt this chaos. We conclude by a similar application to heat model where we solve the model numerically and graphically reveal solutions comparable with expected exact ones.

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