Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?

Abstract

The concept of a fractional derivative is not at all intuitive, starting with not having a clear geometrical interpretation. Many different definitions have appeared, to the point that the need for order has arisen in the field. The diversity of potential applications is even more overwhelming. When modeling a problem, one must think carefully about what the introduction of fractional derivatives in the model can provide that was not already adequately covered by classical models with integer derivatives. In this work, we present some examples from control theory where we insist on the importance of the non-local character of fractional operators and their suitability for modeling non-local phenomena either in space (action at a distance) or time (memory effects). In contrast, when we encounter completely different nonlinear phenomena, the introduction of fractional derivatives does not provide better results or further insight. Of course, both phenomena can coexist and interact, as in the case of hysteresis, and then we would be dealing with fractional nonlinear models.

Erroneous Applications of Fractional Calculus: The Catenary as a Prototype

“…there is an improvement using fractional calculus, various authors engage in a series of theoretical developments or numerical manipulations that later make no sense (see [6]).…”

Coupled Fixed Point Theorem for the Generalized Langevin Equation with Four-Point and Strip Conditions

“…Inserting the condition xð0Þ = 0 in (17) gives c 2 = 0, and also inserting the boundary condition c Dxð0Þ = 0 in (16) gives c 0 = 0. Using the third boundary condition in (2) gives…”

“…The fractional calculus concept is not absolutely intuitive, where it has no clear geometrical interpretation. Several distinct forms have appeared, to the point that the necessity for order has developed in the field [1,2]. The variety of potential implementations is even more difficult.…”

Numerical Solution for Fuzzy Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells

“…These properties can be quantified using fractal dimensions and other fractional derivatives of the parameters, allowing a better understanding of the tumor growth and progression and the potential improvement of diagnosis and treatment planning. Some of such problems were discussed in detail in the recent literature by many researchers [ 7 , 8 , 9 , 10 ].…”