This paper aims to construct an efficient and highly accurate numerical method to solve a class of parabolic integro-fractional differential equations, which is based on wavelets and $L2$-$1_\sigma$ scheme; specifically, the Haar wavelet decomposition is used for grid adaptation and efficient computations, while the high order $L2$-$1_\sigma$ scheme is taken into account to discretize the time-fractional operator. In particular, second-order discretizations are used to approximate the spatial derivatives to solve the one-dimensional problem. In contrast, a repeated quadrature rule based on trapezoidal approximation is employed to discretize the integral operator. On the other hand, we use the semi-discretization of the proposed two-dimensional 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 using the two-dimensional Haar wavelet. Here, we investigated theoretically and verified numerically the behavior of the proposed higher-order numerical method. In particular, the stability and convergence analysis of the proposed higher-order method has been studied. The obtained results are compared with some existing techniques through several graphs and tables, and it is shown that the proposed higher-order methods have better accuracy and produce less error compared with the $L1$ scheme.

翻译：本文旨在构建一种基于小波和$L2$-$1_σ$格式的高效高精度数值方法，用于求解一类抛物型积分-分数阶微分方程；具体而言，采用Haar小波分解进行网格自适应与高效计算，同时引入高阶$L2$-$1_σ$格式对时间分数阶算子进行离散。特别地，在求解一维问题时，采用二阶离散格式逼近空间导数；而积分算子则通过基于梯形逼近的重复求积法则进行离散。另一方面，对二维模型采用半离散化策略，其中分数阶算子基于$L2$-$1_σ$格式处理，积分部分则使用复合梯形逼近。随后，利用二维Haar小波近似空间导数。文中从理论分析与数值验证两方面研究了所提高阶数值方法的性能，并重点探讨了该方法解的稳定性与收敛性分析。通过多组图表与现有技术对比，结果表明：与$L1$格式相比，所提高阶方法具有更高的精度与更小的误差。