Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform

TL;DR

This paper develops a new fractional calculus framework on arbitrary time scales by defining fractional derivatives and integrals via the inverse Laplace transform, extending classical fractional calculus to dynamic equations on time scales.

Contribution

It introduces fractional derivatives and integrals on time scales using inverse Laplace transform, providing foundational properties and extending fractional calculus to dynamic equations.

Findings

Defined fractional derivatives and integrals on time scales.

Proved key properties of the new fractional operators.

Extended fractional calculus to dynamic equations on arbitrary time scales.

Abstract

We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the inverse Laplace transform on time scales. Useful properties of the new fractional operators are proved.