Derivation, interpretation, and analog modelling of fractional variable order derivative definition

Abstract

The paper presents derivation and interpretation of one type of variable order derivative definitions. For mathematical modelling of considering definition the switching and numerical scheme is given. The paper also introduces a numerical scheme for a variable order derivatives based on matrix approach. Using this approach, the identity of the switching scheme and considered definition is derived. The switching scheme can be used as an interpretation of this type of definition. Paper presents also numerical examples for introduced methods. Finally, the idea and results of analog (electrical) realization of the switching fractional order integrator (of orders 0.5 and 1) are presented and compared with numerical approach. *

Analytical description and equivalence of additive-switching scheme for fractional variable-order continuous differ-integrals

“…It yields a common unified description for the variable-order differ-integral in discrete-time (Grunvald-Letnikov definition, [22]) and in continuous-time domain (R-L definition). Therefore, the scheme-description of different variable-order differintegral definitions allows to classify them with respect to their intrinsic properties reflected by switching schemes.…”

“…In the literature [20], [21], three general types of variable-order derivative definitions can be found, however, these definitions were given without interpretation and derivation. In [22], in the discrete-time domain, the additive-switching scheme, numerically identical to the 2nd type of Grunwald-Letnikov fractional derivative definition was introduced, both in difference and matrix form. Similar results for recursive type of fractional variable order G-L type derivative is presented in [23].…”

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

“…Similarly, the numerical approximation of VO has been developed by Zayernouri and Karniadakis [133] by using FDMs, and various FDMs for VO fractional diffusion equations have been proposed. Sierociuk et al [94] demonstrated a numerical scheme for a VO derivative based on matrix approach. Fu et al [38] applied the method of approximate particular solutions for fractional diffusion model.…”

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

“…Recently, some authors have considered the applications of derivatives of variable order in various sciences such as anomalous diffusion modeling, mechanical applications, multi-fractional Gaussian noises. Among these, there have been many works dealing with numerical methods for some class of variable order fractional differential equations, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”