Fractional derivatives in electrical circuit theory – critical remarks

Abstract

A number of critical remarks related to the application of fractional derivatives in electrical circuit theory have been presented in this paper. Few cases have been pointed out that refer to observed in selected publications violations of dimensional uniformity of physical equation rules as well as to a potential impact on the Maxwell equations.

The use of fractional order derivatives for eddy current non-destructive testing

“…Nowadays this method is applied in various areas, e.g., in control systems, robotics, electrical engineering, physics, etc. [2][3][4][5][6][7][8]). Application of fractional derivatives faces many challenges; one may find correct formulation of fractional calculus in some publications [2], but there are erroneous applications as well [4][5][6].…”

“…Additionally, there are authors modifying known laws of physics by introducing to them fractional orders of derivatives instead of integers (for example in [6] where Maxwell's laws are transformed to incorrect, inadmissible forms). More comprehensive discussion of the above problems can be found in [7,[9][10]. Thus, one should be very attentive when using fractional derivatives in formulae to describe physical phenomena.…”

Analysis of an Electrical Circuit Using Two-Parameter Conformable Operator in the Caputo Sense

“…In the latest scientific publications, we may find the Caputo definition, which is used to describe many phenomena; for example, electrical circuits [11][12][13][14][15][16]. The use of fractional derivatives having different orders is a complicated process, and only a couple of illustrative implementations regarding this case are known [17][18][19].…”

Perona-Malik Model with Diffusion Coefficient Depending on Fractional Gradient via Caputo-Fabrizio Derivative

“…There are many definitions of fractional derivatives (three popular definitions were given by Grunwald-Letnikov (G-L), Riemann-Liouville (R-L), and Caputo). These have been used in numerous fields of science such as study of the anomalous diffusion phenomenon [24][25][26], circuit theory [27][28][29], and image processing [30,31], among other applications [11,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. Given the discussion above, we consider that using anisotropic diffusion models to eliminate noise in an image, preserving both strong and weak edges and without phenomena such as staircase, speckle, or any type of artifact, is a subject where much remains to be investigated.…”