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

TL;DR

This paper develops a mathematical framework for variable order derivatives, introduces a numerical matrix-based scheme, and demonstrates an electrical analog implementation, advancing fractional calculus modeling and applications.

Contribution

It provides a new interpretation and numerical scheme for variable order derivatives, including an electrical analog realization, which is novel in fractional calculus research.

Findings

Numerical scheme based on matrix approach effectively models variable order derivatives.

Electrical analog of fractional integrator successfully implemented and compared with numerical results.

Switching scheme offers a new interpretation of variable order derivatives.

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.