| Abstract: |
| A two-dimensional nonlinear time fractional reaction--diffusion equation of order $\alpha\in (0, 1)$ is considered. The typical solution to such problems usually has an initial layer at $t=0.$ To capture the initial singularity, the Caputo time fractional derivative is approximated using the $L2-1_{\sigma}$ formula on the smoothly graded meshes. Spatial derivatives are approximated using standard central difference approximation. Computational cost is reduced using Newton`s linearized method and alternating direction implicit method. Theoretical analysis comprising Solvability, stability, and convergence of the finite difference scheme has been studied rigorously and it is shown that the method is convergent with convergence order $\mathcal{O}(M^{-\min\{3-\alpha, r\alpha, 1+\alpha, 2+\alpha\}}+h_{x}^{2}+h_{y}^{2})$ where $M$ is the temporal discretization parameter, $h_{x}$, $h_{y}$ are the step sizes in the spatial direction and $\alpha\in (0,1)$ is the fractional order. The applicability of the discussed numerical scheme is established by two illustrative examples having smooth and nonsmooth solutions. |
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