An efficient Newton-ADI scheme for 2D time-fractional reaction–diffusion equations with weak initial singularity

A two-dimensional nonlinear reaction–diffusion equation involving a time fractional derivative of order α∈(0,1) is considered. The typical solution to such problems usually has an initial singularity at t=0. To capture the initial singularity, the Caputo time fractional derivative is approximated using the L2−1σ formula on the smoothly graded meshes. Spatial derivatives are approximated using standard central difference approximation. The computational cost is minimized by employing Newton’s linearization method in conjunction with the alternating direction implicit method. A comprehensive theoretical analysis including stability, solvability, and convergence of the discussed scheme, has been rigorously examined and it is shown that the method is convergent with convergence order O(M−min{3−α,rα,1+α,2+α}+hx2+hy2) where M is the temporal discretization parameter, hx, hy are the step sizes in the spatial direction and α∈(0,1) is the fractional order. The effectiveness of the proposed numerical scheme is demonstrated through two examples, one with a smooth solution and the other with a nonsmooth solution.