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This study aims to construct an efficient and highly accurate numericalmethod to solve a class of parabolic integro-fractional differential equations,which is based on wavelets and $L2$-$1_\sigma$ scheme. Specifically, the Haarwavelet decomposition is used for grid adaptation and efficient computations,while the high order $L2$-$1_\sigma$ scheme is considered to discretize thetime-fractional operator. Second-order discretizations are used to approximatethe spatial derivatives to solve the one-dimensional problem, while a repeatedquadrature rule based on trapezoidal approximation is employed to discretizethe integral operator. In contrast, we use the semi-discretization of theproposed two-dimensional model based on the $L2$-$1_\sigma$ scheme for thefractional operator and composite trapezoidal approximation for the integralpart. The spatial derivatives are then approximated using two-dimensional Haarwavelets. In this study, we investigated theoretically and verified numericallythe behavior of the proposed higher-order numerical methods. In particular,stability and convergence analyses are conducted. The obtained results arecompared with those of some existing techniques through several graphs andtables, and it is shown that the proposed higher-order methods have betteraccuracy and produce less error compared to the $L1$ scheme in favor offractional-order integro-partial differential equations.

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