On a Fractional Operator Combining Proportional and Classical Differintegrals

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function <mml:semantics>f(t)</mml:semantics>, by a fractional integral operator applied to the derivative <mml:semantics>f ' (t)</mml:semantics>. We define a new fractional operator by substituting for this <mml:semantics>f ' (t)</mml:semantics> a more general proportional derivative. This new operator can also be written as a Riemann-Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann-Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.