Fractional Cauchy Problem with Caputo Nabla Derivative on Time Scales

Jiang Zhu,Ling Wu

doi:10.1155/2015/486054

Abstract

The definition of Caputo fractional derivative is given and some of its properties are discussed in detail. After then, the existence of the solution and the dependency of the solution upon the initial value for Cauchy type problem with fractional Caputo nabla derivative are studied. Also the explicit solutions to homogeneous equations and nonhomogeneous equations are derived by using Laplace transform method.

Highlights

Fractional differential equation theory has gained considerable popularity and importance due to their numerous applications in many fields of science and engineering including physics, population dynamics, chemical technology, biotechnology, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems, and so on
Fractional generalized ∇-power functionhα(t, s) on time scales is defined as the shift ofhα(t, t0); that is
The Caputo fractional derivative of order α ≥ 0 is defined via Riemann-Liouville fractional derivative by m−1

Summary

Introduction

Fractional differential equation theory has gained considerable popularity and importance due to their numerous applications in many fields of science and engineering including physics, population dynamics, chemical technology, biotechnology, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems, and so on (see, e.g., [1,2,3,4], and the references therein). The delta fractional calculus and Laplace transform on some specific discrete time scales are discussed in [26,27,28]. In the light of the above work, we further studied the theory of fractional integral and derivative on general time scales in [29], where ∇-Laplace transform, fractional ∇-power function, ∇-Mittag-Leffler function, fractional ∇-integrals, and fractional ∇-differential on time scales are defined. Some of their properties are discussed in detail. We derive explicit solutions and fundamental system of solutions to homogeneous equations with constant coefficients and find particular solution and general solutions of the corresponding nonhomogeneous equations

Preliminaries

Definition and Properties of Caputo Fractional Derivative on Time Scales

The Cauchy Problem with Caputo Fractional Derivative