A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives established reformulated. new similar to Grünwald–Letnikov’s deduced. Tempered versions are also introduced. Scale-invariant systems described exemplified. For solving corresponding differential equations, a logarithmic Mittag-Leffler series proposed.

ABSTRACT Fractional cosmology modifies the standard derivative to Caputo’s fractional of order ?, generating changes in General Relativity. Friedmann equations are modified, and evolution species densities depends on ? age Universe tU. We estimate stringent constraints using cosmic chronometers, Type Ia supernovae, joint analysis. obtain $\mu =2.839^{+0.117}_{-0.193}$ within 1? confidence level...

In this paper, a class of fractional order systems is considered and simple fractional order observers have been proposed to estimate the system’s state variables. By introducing a fractional calculus into the observer design, the developed fractional order observers guarantee the estimated states reach the original system states. Using the fractional order Lyapunov approach, the stability (zer...