This study 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 considered to discretize the time-fractional operator. Second-order discretizations are used to approximate the spatial derivatives to solve the one-dimensional problem, while a repeated quadrature rule based on trapezoidal approximation is employed to discretize the integral operator. In contrast, 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. The spatial derivatives are then approximated using two-dimensional Haar wavelets. In this study, we investigated theoretically and verified numerically the behavior of the proposed higher-order numerical methods. In particular, stability and convergence analyses are conducted. The obtained results are compared with those of 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 to the $L1$ scheme in favor of fractional-order integro-partial differential equations.

翻译：本研究旨在构建一种基于小波与$L2$-$1_\sigma$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小波分解实现网格自适应与高效计算，同时引入高阶$L2$-$1_\sigma$格式离散时间分数阶算子。针对一维问题，采用二阶离散逼近空间导数，并基于梯形近似的重复求积规则离散积分算子；而对于二维模型，则采用基于$L2$-$1_\sigma$格式的半离散化处理分数阶算子，积分部分使用复合梯形近似，空间导数通过二维Haar小波逼近。本研究从理论与数值两方面验证了所提高阶数值方法的性能，重点开展了稳定性与收敛性分析。通过若干图表与现有方法对比结果表明，相比$L1$格式，所提高阶方法在分数阶积分-偏微分方程求解中具有更高的精度与更低的误差。