The Conformable Ratio Derivative

This paper proposes a new definition for a conformable derivative. The strengths of the new derivative arise in its simplicity and similarity to fractional derivatives. An inverse derivative (integral) exists showing similar properties to fractional integrals. The derivative is scalable, and exhibits particular product and chain rules. When looked at as a function with a parameter, the ratio derivative K&alpha [f] of a function f converges pointwise to f as &alpha &rarr 0, and to the ordinary derivative as &alpha &rarr 1. The conformable derivative is nonlinear in nature, but a related operator behaves linearly within a power series and Fourier series. Furthermore, the related operator behaves completely fractionally when acting within an exponential-based Fourier series

Note on the generalized conformable derivative

We introduce a definition of a generalized conformable derivative of order alfa > 0 (where this parameter does not need to be integer), with which we overcome some deficiencies of known local derivatives, conformable or not. This definition allows us to compute fractional derivatives of functions defined on any open set on the real line (and not just on the positive half- line). Moreover, we extend some classical results to the context of fractional derivatives. Also, we obtain results for the case alfa > 1The research of José M. Rodríguez and José M. Sigarreta is supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI /10.13039/501100011033), Spain