含Caputo分数阶导数的分数阶微分方程

FRACTIONAL DIFFERENTIAL EQUATIONS WITH CAPUTO FRACTIONAL DERIVATIVES

求解了含Caputo分数阶导数的分数阶微分方程初值问题 d~αu/dtα+ω~αu(t;α)=h(t),t>0,0≤n-1<α≤n,ω>0, u^(k)(0^+;α)=u_k,k=0,1,…,n-1.利用Laplace变换方法和广义 Mittag-Leffler函数,得到其解为u(t;α)=integral from n=0 to t (r^(α-1)E_α,α(-(ωτ)~α))h(t-τ)dτ+sum from k=0 to n-1 u_kt^kE_(α,1+k)(-(ωt)~α)。

The initial value problem for the fractional differential equation with the Caputo fractional derivatives d~αu/dt~α+ω~αu(t;α)=h(t),t>0,0≤n-1<α≤n,ω> d^((k))(0^+;α)=u_k,k=0,1,…,n-1. is studied. Using the method of the Laplace transform its solution is given in terms of the generailized Mittag-Leffler functions