Abstract： In this paper， we study the following fractional differential equation：

Where with m and n being positive integers， f is

a smooth function and is a integrable function， through mean value theorem and linearize problem， we find the characteristic equation and its solution.

Keywords： Fractional different equation， characteristic equation.

1 Introduction

Fractional calculus includes fractional order integral and fractional derivative， because its valuable applications， fractional calculus has gained enough importance. See[1，2，3，4]

Liouville， Riemann， Leibniz have studied the earliest systematic and Caputo defines the fractional differential.

The paper will study the following equation：

（1.1）

Where with m and n being positive integers， f is

a smooth function and is a integrable function， through mean value theorem and linearize problem， we find the characteristic equation and its solution.

2 Main Results

Proposition 2.1： Suppose that is a smooth function and there is a such that ， then the linearization of （1.1） near the equilibrium

Subsitituting it into （2.7）， we have （2.5） and （2.6）

Reference

[1]Samko，S.G.，Kilbas，A.A.，Marivhev，O.I.： Fractional Integrals and Derivatives： Theory and Applications. Gordon and Breach，， Yverdon（1993）

[2]Rabei， E.M.， Nawafleh，K.I. Baleanu， D.： The Hamiltonian formalism with fractional derivatives. J.Math Anal.Appl.327，891-897（2007）

[3]Bhalekar， S.， Daftardar-Gejji， V.Baleanu， D.， Magin， R： Fractional Bloch equation with delay.Comput，Math.appl.61（5），1355-1365（2007）

[4] Kilbas， A.A.， Srivastava， H.M.， Trujillo， J.J.， Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies， vol. 204. Elsevier， Amsterdam， 2006