Fractional Cauchy Problem with Riemann-Liouville Fractional Delta Derivative on Time Scales

Abstract

The $\mathrm{\Delta }$-power function and fractional $\mathrm{\Delta }$-integrals and fractional $\mathrm{\Delta }$-differential are defined, and then the definitions and properties of $\mathrm{\Delta }$-Mittag-Leffler function are given. The properties of fractional $\mathrm{\Delta }$-integrals and fractional $\mathrm{\Delta }$-differential on time scales are discussed in detail. After that, the existence of the solution and the dependency of the solution upon the initial value for Cauchy type problem with fractional $\mathrm{\Delta }$-derivative are studied. Also the explicit solutions to homogeneous fractional $\mathrm{\Delta }$-differential equations and nonhomogeneous fractional $\mathrm{\Delta }$-differential equations are derived by using Laplace transform method.