Well-posedness and mathematical analysis of linear evolution equations with a new parameter

Abstract

In this dissertation we apply linear evolution equations to the Newtonian derivative, Caputo time fractional derivative and $-time fractional derivative. It is notable that the most utilized fractional order derivatives for modelling true life challenges are Riemann- Liouville and Caputo fractional derivatives, however these fractional derivatives have the same weakness of not satisfying the chain rule, which is one of the most important elements of the match asymptotic method [2, 3, 16]. Furthermore the classical bounded perturbation theorem associated with Riemann-Liouville and Caputo fractional derivatives has con rmed not to be in general truthful for these models, particularly for solution operators of evolution systems of a derivative with fractional parameter ' that is less than one (0 < ' < 1) [29]. To solve this problem, we introduce the derivative with new parameter, which is de ned as a local derivative but has a fractional order called $-derivative and apply this derivative to linear evolution equation and to support what we have done in the theory, we utilize application to population dynamics and we provide the numerical simulations for particular cases.