Unifying fractional calculus with time scales

© 2012 Dr. Paul Anthony WilliamsFractional calculus, the study of integration and differentiation of fractional order, has recently been extended to include its discrete analogues of fractional difference calculus and fractional quantum calculus. Due to the similarities of the three theories there has been research on whether there exists a single theory that encapsulates them. One possible approach is via the theory of time scales calculus, however it is still an open problem as whether there exists a consistent theory for fractional calculus on time scales. We propose an axiomatic approach to defining a Riemann-Liouville type fractional integral on time scales. After evaluating the unifying properties of power functions on various time scales, we posit axioms that the functions would need to satisfy. We then use these functions as the kernel of an integral transform, such that if the time scale was the real numbers our definition is equivalent to the standard Riemann-Liouville fractional integral. Further if the time scale chosen was the integers we obtain backwards fractional summation. We incorporate fractional derivatives by composing the fractional integral with the standard time scales derivative. Our choices of axioms are simple but we demonstrate a large number of desirable properties of the fractional calculus. We show integrability and continuity conditions of the fractional integral and after proving a Dirichlet formula for nabla integration we also prove an index law. As for fractional derivatives, we prove a very general Taylor type theorem. This is then applied to consider Opial inequalities involving fractional derivatives. We start with a simple integral inequality involving the function and its fractional derivative, and then extend these results by considering positive powers of the factors and also weighted integrals. Opial inequalities have applications in studying fractional dynamic equations, and we demonstrate how they are used in finding bounds on solutions of these equations