The Asymptotic Analysis for the Singularly Perturbed Subdiffusion Equations on Bounded Domain (Advances in Applied Mathematics and Mechanics, 2025)
In this paper, we study the asymptotic properties of the singularly perturbed subdiffusion equations in a bounded domain. First, we use the matched asymptotic expansion method to obtain the uniform asymptotic expansion for the solution $u(x,t)$ of the singularly perturbed subdiffusion equation. By asymptotic analysis, we can know that near the boundary configured with non-smooth boundary values, the solution $u(x,t)$ of the singularly perturbed subdiffusion equation has a boundary layer of thickness $\mathcal{O}(ε).$ By studying the asymptotic properties of the spatial partial derivatives $∂_xu(x,t)$ and $∂_{xx}u(x,t),$ we can know that the singularity is mainly concentrated in the boundary layers, and then the solution $u(x,t$) changes gently outside the boundary layers. Next, we introduce a new $\mathcal{L}1$-TFPM scheme to solve the singularly perturbed subdiffusion equations numerically. Some numerical experiments can demonstrate the correctness of the asymptotic analysis results.