有限区间上的分数阶扩散波方程的解

Fractional diffusionwave equation on finite interval

考虑如下的分数阶扩散 波方程:Dαtu(t,x) = a2Dβxu(t,x), t >0,0< x < l,0<α≤2,0<β≤2,u(0,t) =0, u(l,t) =θ(t), t≥0,u(0,x) =φ(x), 0≤x≤ l(如果0<α≤1),ut(0,x) =0, 0≤x≤ l(如果1<α≤2).其中Dαt 和Dβx 分别为关于时间t 和空间x 的α次、β次 Caputo分数次算子, θ(t)为给定的函数. 利用 Dαt 和 Dβx 的变换, 给出该问题的解的表达式.

The following fractional diffusionwave equation on finite intervalD~α_tu(t,x)=a^2D~β_xu(t,x), t>0,0<x<l,0<α≤2,0<β≤2, u(0,t)=0,u(l,t)=θ(t), t≥0, u(0,x)=φ(x), 0≤x≤l(if 0<α≤1), u_t(0,x)=0, 0≤x≤l(if 1<α≤2).is discussed.Where D~α_t and D~β_x are Caputo fractional derivatives of order α and β with respect to t and x,respectively,θ(t)is a given function.Applying the Laplace transforms for D~α_t and D~β_x,the representness of solution is obtained.