A Novel Theoretical Investigation of the Abu-Shady–Kaabar Fractional Derivative as a Modeling Tool for Science and Engineering
Francisco Martínez, Mohammed K. A. Kaabar
Abstract
A newly proposed generalized formulation of the fractional derivative, known as Abu-Shady–Kaabar fractional derivative, is investigated for solving fractional differential equations in a simple way. Novel results on this generalized definition is proposed and verified, which complete the theory introduced so far. In particular, the chain rule, some important properties derived from the mean value theorem, and the derivation of the inverse function are established in this context. Finally, we apply the results obtained to the derivation of the implicitly defined and parametrically defined functions. Likewise, we study a version of the fixed point theorem for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>α</a:mi> </a:math> -differentiable functions. We include some examples that illustrate these applications. The obtained results of our proposed definition can provide a suitable modeling guide to study many problems in mathematical physics, soliton theory, nonlinear science, and engineering.