Approximate solutions of fractional dynamical systems based on the invariant exponential functions with an application. A novel double-kernel fractional derivative
H. I. Abdel‐Gawad, M. Tantawy, Abdelazeem M. Abdelwahab
Abstract
The systems with fractional time derivative (FTD) and fractional space derivative (FSD), with different versions, have occupied a vast area of research in the literature. It is worth mentioning that in the literature when establishing the solutions of fractional systems, the only function dealt with is the Mittage-Leffler function. In the present work, our objectives are; First is to the reversibility identity (RI) of FTD. Second, is to construct the fractional exponential function invariant under a fractional derivative. In consequence, the trigonometric and hyperbolic fractional functions are constructed. So, exact solutions of linear fractional systems can be obtained. When RI holds then a FD establishes a calculus analog the ordinary ones. Although, these concepts are simple, they were not considered in the literature. Here, various FDs are considered. Furthermore, an approach for approximate analytic solutions for the fractional-prey-predator dynamical system with harvesting is presented via an iteration scheme and the convergence theorem is proved. Finally, a novel double kernel FD is introduced and as a particular case the Hilfer FD is established.