A novel higher-order numerical method for parabolic integro-fractional differential equations based on wavelets and $L2$-$1_σ$ scheme

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.