The main aim of this paper is to construct an efficient highly accurate numerical scheme to solve a class of one and two-dimensional parabolic integro-fractional differential equations. The high order $L2$-$1_\sigma$ scheme is taken into account to discretize the time-fractional operator on a uniform mesh. To solve the one-dimensional problem, second-order discretizations are used to approximate the spatial derivatives whereas, a repeated quadrature rule based on trapezoidal approximation is employed to discretize the integral operator. In the case of two-dimensional problem, first, we make the semi-discretization of the proposed model based on the $L2$-$1_\sigma$ scheme for the fractional operator, and composite trapezoidal approximation for the integral part. Then, the spatial derivatives are approximated by the two-dimensional Haar wavelet. The stability and convergence analysis is carried out for both models. The experimental evidence proves the strong reliability of the present methods. Further, the obtained results are compared with some existing methods through several graphs and tables, and it is shown that the proposed methods not only have better accuracy but also produce less error in comparison with the $L1$ scheme.

翻译：本文的主要目的是构建一种高效高精度的数值格式，用于求解一类一维和二维抛物型积分-分数阶微分方程。采用高阶$L2$-$1_σ$格式在均匀网格上对时间分数阶算子进行离散化。对于一维问题，使用二阶离散化近似空间导数，同时采用基于梯形近似的重复求积法则对积分算子进行离散化。对于二维问题，首先基于分数阶算子的$L2$-$1_σ$格式和积分部分的复合梯形近似对模型进行半离散化，然后利用二维Haar小波对空间导数进行近似。对两种模型均进行了稳定性和收敛性分析。实验证据证明了本文方法的高度可靠性。此外，通过若干图表将所得结果与现有方法进行了比较，结果表明，与$L1$格式相比，本文方法不仅具有更高的精度，而且产生的误差更小。