In this article we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in ${\mathbb{R}}^1$. The analysis is performed in the weighted Sobolev spaces, $H_{(a,b)}^s(\mathrm{I})$. Three different characterizations of $H_{(a,b)}^s(\mathrm{I})$ are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters α and r defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.

在本文中，我们研究了有限域上分数扩散，对流，反应方程解的正则性 ${\mathbb{[R}}^{\mathrm{1个}}$。分析是在加权Sobolev空间中进行的，$H_{（\mathrm{一种}，b）}^s（\mathrm{一世}）$。三种不同的表征$H_{（\mathrm{一种}，b）}^s（\mathrm{一世}）$介绍了这些空间以及所需的嵌入定理。分析表明，解决方案的规律性受解决方案的端点行为限制，端点行为由定义分数扩散算子的参数α和r决定。此外，分析表明，对于足够光滑的右手侧函数，分数阶扩散反应方程的解的正则性低于分数阶扩散方程的解的正则性。另外，分数扩散对流反应方程的解的正则性比分数扩散对流反应方程的正则性低两个数量级。