Caputo Fractional Derivative Equation On Time Scales

In this paper, the properties of Caputo fractional differential on time scales are discussed in detail. After then, the existence and uniqueness of the solution and the dependency of the solution upon the initial value for Cauchy type problem with Caputo fractional derivative are studied. Also the explicit solutions to homogeneous equations are derived by using Laplace transform method.This paper is divided into four chap-ters.In the first chapter, we introduce background of this article, current situation of fractional differential and integral, major job and prior knowledge.In the second chapter, definitions and properties about Caputo fractional on time scales are given.In the third chapter, we discuss Cauchy type problem with Caputo fractional dif-ferential equation, including uniqueness of solution and dependency of solution upon the initial problem, and by applying transform method, we derive explicit solutions to two dimensional equation.In the fourth chapter, by applying the Laplace transform method, we derive explicit solutions to homogeneous and nonhomogeneous equations with constant coefficients.