## Abstract and Applied Analysis

### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 401596, 19 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393450242

Digital Object Identifier
doi:10.1155/2013/401596

Mathematical Reviews number (MathSciNet)
MR3147808

Zentralblatt MATH identifier
1295.26009

#### Citation

Zhu, Jiang; Zhu, Ying. Fractional Cauchy Problem with Riemann-Liouville Fractional Delta Derivative on Time Scales. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 401596, 19 pages. doi:10.1155/2013/401596. https://projecteuclid.org/euclid.aaa/1393450242

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