Variable-Order Fractional Scale Calculus

General variable-order fractional scale derivatives are introduced and studied. Both the stretching and the shrinking cases are considered for definitions of the derivatives of the GL type and of the Hadamard type. Their properties are deduced and discussed. Fractional variable-order systems of autoregressive-moving-average type are introduced and exemplified. The corresponding transfer functions are obtained and used to find the corresponding impulse responses.

Automated summary

This paper introduces and investigates a new type of general variable-order fractional scale derivatives, considering both the stretching and shrinking cases for the definitions of the GL type and the Hadamard type derivatives. The properties of these derivatives are deduced and discussed. Additionally, fractional variable-order systems of autoregressive-moving-average type are introduced and exemplified, with the corresponding transfer functions obtained and used to find the corresponding impulse responses.