In this paper, we develop a linearized fractional Crank-Nicolson-Galerkin FEM for Kirchhoff type quasilinear time-fractional integro-differential equation $\left(\mathcal{D}^{\alpha}\right)$. In general, the solutions to the time-fractional problems exhibit a weak singularity at time $t=0$. This singular behavior of the solutions is taken into account while deriving the convergence estimates of the developed numerical scheme. We prove that the proposed numerical scheme has an accuracy rate of $O(M^{-1}+N^{-2})$ in $L^{\infty}(0,T;L^{2}(\Omega))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(\Omega))$, where $M$ and $N$ are the degrees of freedom in the space and time directions respectively. A numerical experiment is presented to verify the theoretical results.

翻译：在本文中,我们为Kirchhoff型准线性时间折射性内分化方程式开发了一个线性分数分数的Crank-Nicolson-Galerkin FEM, 用于折射( mathcal{D ⁇ alpha ⁇ right)$left( mathcal{D ⁇ alpha ⁇ right) 。 一般来说, 时间折射问题的解决方案在时间上表现出微弱的单一性 $tt =0美元。 在计算所开发的数字办法的趋同估计值时,考虑到这一单一的解决方案行为。 我们证明, 拟议的数字方案精确率为$O( M ⁇ -1 ⁇ N ⁇ % 2}( 0. T;L ⁇ 2}( \\\\ omega)) $ 美元, 以及 $ ⁇ infty}( 0. 0. T;H ⁇ 1 ⁇ 0} (\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\