An improved Euler method for time fractional nonlinear subdiffusion equations with initial singularity

As is known that many existing numerical methods for time fractional nonlinear subdiffusion equations (TFNSEs) often suffer from the phenomenon of order reduction, because the solution of TFNSEs usually has the initial singularity. To overcome this order reduction problem, in this paper, an improved Euler method is proposed for solving TFNSEs based on the technique of variable transformation in time. Then, it is proved that the temporal convergence order of the proposed method is the first order for any fractional order \(\alpha \in (0,1)\) $\alpha \in (0,1)$ , which achieves the optimal convergence order of the Euler method. Finally, numerical experiments are given to verify the correctness of our theoretical results.