A time-fractional diffusion problem is considered on a spherically symmetric domain in ${\mathbb{R}}^d$ for $d=1,2,3$. The solution of such a problem is shown in general to have a weak singularity near the initial time $t=0$, and bounds on certain derivatives of this solution are derived. To solve the problem numerically, spatial polar coordinates are used; a finite difference method is constructed on a mesh that is graded in time and spherical in space. The discretisation uses the L1 scheme in time and a polar-coordinate discretisation of the diffusion term. Its convergence is analysed and error bounds are derived that are robust in α, the order of the time derivative, as $\alpha \to 1^{-}$. Numerical experiments show that our results are sharp.

在球对称域上考虑时间分数扩散问题 ${\mathbb{电阻}}^d$ 为了 $d=1,2,3$. 此类问题的解一般在初始时刻附近具有弱奇异性$吨=0$，并推导出该解的某些导数的界限。为了数值解决问题，使用空间极坐标；有限差分法是在时间分级、空间球形的网格上构建的。离散化使用 L1 时间方案和扩散项的极坐标离散化。分析其收敛性并推导出在时间导数的阶数α中具有鲁棒性的误差界限，如$\alpha \to 1^{-}$. 数值实验表明，我们的结果是尖锐的。