A CRANK-NICOLSON <i>L</i>1/PI DIFFERENCE SCHEME FOR NONLINEAR TIME FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION ON GRADED MESHES

In this paper, a Crank-Nicolson $ L1 $/trapezoidal Product integration (PI) difference scheme is constructed to numerically solve a Volterra-type nonlinear integro-differential equation. Assuming the exact solution exhibits a weak singularity at $ t=0 $, the convergence order of the fully discrete scheme is $ \mathcal{O}\left(N^{-\min\{r\sigma, 2-\alpha, 2\}}+M^{-2}\right) $, and the stability is analyzed using an improved Gr$ \ddot{\text{o}} $nwall inequality in terms of the $ L^2 $-norm. Finally, the theoretical results are verified by numerical experiments.