Fundamental fractional exponential matrix: New computational formulae and electrical applications

It is well known that the fundamental fractional exponential matrix (FFEM) is closely related to the formal solution of the homogeneous and non-homogeneous linear time-conformable fractional dynamical differential system (LT-CFDDS) with delay in control and it also plays a central role in the solution of any other (matrix) fractional differential system (FDS). In this paper, we present several new theoretical results and computational formulae of the FFEM e(At alpha/alpha) for any arbitrary square matrix A. Moreover, some attractive and interesting special cases for FFEM are also derived and discussed. In addition, three important and interesting physical applications related to linear fractional electrical circuits (ECs) in 2 and 3-dimentions are considered and solved by means of FFEM. These applications are the fractional RC, RL and RLC-ECs. The method exists by converting the data of a given linear fractional EC into homogeneous or non-homogeneous LT-CFDDS for which solutions can be easily computed. For further and simplicity analysis, we provide an illustrative example for each problem and present the general exact solution (GES) in a simple form based on our new approach. Moreover, graphical results are created and discussed in order to ensure that the solutions of specific problems are stable at different values of alpha and assure that our suggestion technique is a simple, efficient, accurate, powerful analytic tool and can be applied successfully to solve many other conformable fractional differential problems in various fields. Finally, the classical behaviors of physical problems are recovered when the fractional order alpha is equal to 1.

Automated summary

The paper introduces new theoretical results and computational formulae of the FFEM for any arbitrary square matrix A, and discusses attractive special cases for FFEM. It also applies FFEM to solve important physical applications related to linear fractional electrical circuits, and shows that the solutions are stable at different values of alpha and recover classical behaviors when alpha equals 1.