Fractional Cauchy Problem with Riemann-Liouville Derivative on Time Scales

Abstract

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