Department of Applied Mathematics and Statistics, Technological University of Cartagena, Cartagena 30203, Spain.
Gofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, Addis Ababa 26649, Ethiopia.
Comput Math Methods Med. 2022 Sep 26;2022:4119082. doi: 10.1155/2022/4119082. eCollection 2022.
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 -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.
一种新提出的分数导数的广义形式，即阿布-沙迪-卡巴尔分数导数，被用于简单地求解分数微分方程。对这个广义定义提出并验证了新的结果，从而完善了迄今为止所介绍的理论。特别是，建立了链法则、从中值定理推导出的一些重要性质以及反函数的求导法则。最后，我们将所得结果应用于隐式定义和参数定义函数的推导。同样，我们研究了适用于 -可微函数的不动点定理的一个版本。我们包含了一些示例来说明这些应用。我们提出的定义的结果可以为研究数学物理、孤子理论、非线性科学和工程中的许多问题提供合适的建模指导。
