一个分数阶积分微分方程的Cauchy问题及其解
An Integro-Differential Equation of the Fractional Form: Cauchy Problem and Solution
摘要：
本文讨论了由分数阶偏微分方程$[\partial_{tt}-κ\mathbb{D}]u=0$定义的Cauchy问题，其中$\mathbb{D}$是一阶伪微分Riesz算子，初始条件为$u(x,0)=μ(\sqrtπx_0)^{-1}e^{-(x/x_0)^2}$，$u_t(x,0)=0$ .作为推论，得到了用狄拉克δ分布$φ(x)=μδ(x)$代替高斯脉冲$u(x,0)$所产生的柯西问题的解。
英文摘要：
We solve the Cauchy problem defined by the fractional partial differential equation $[\partial_{tt}-κ\mathbb{D}]u=0$, with $\mathbb{D}$ the pseudo-differential Riesz operator of first order, and the initial conditions $u(x,0)=μ(\sqrtπx_0)^{-1}e^{-(x/x_0)^2}$, $u_t(x,0)=0$. The solution of the Cauchy problem resulting from the substitution of the Gaussian pulse $u(x,0)$ by the Dirac delta distribution $φ(x)=μδ(x)$ is obtained as corollary.
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