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This work aims to construct an efficient and highly accurate numerical methodto address the time singularity at $t=0$ involved in a class of time-fractionalparabolic integro-partial differential equations in one and two dimensions. The$L2$-$1_\sigma$ scheme is used to discretize the time-fractional operator,whereas a modified version of the composite trapezoidal approximation isemployed to discretize the Volterra operator. Subsequently, it helps to convertthe proposed model into a second-order BVP in a semi-discrete form. Themulti-dimensional Haar wavelets are then used for grid adaptation and efficientcomputations for the 2D problem, whereas the standard second-orderapproximations are employed to approximate the spatial derivatives for the 1Dcase. The stability analysis is carried out on an adaptive mesh in time. Theconvergence analysis leads to $O(N^{-2}+M^{-2})$ accurate solution in thespace-time domain for the 1D problem having time singularity based on the$L^\infty$ norm for a suitable choice of the grading parameter. Furthermore, itprovides $O(N^{-2}+M^{-3})$ accurate solution for the 2D problem havingunbounded time derivative at $t=0$. The analysis also highlights a higher orderaccuracy for a sufficiently smooth solution resides in$C^3(\overline{\Omega}_t)$ even if the mesh is discretized uniformly. Thetruncation error estimates for the time-fractional operator, integral operator,and spatial derivatives are presented. Numerous tests are performed on severalexamples in support of the theoretical analysis. The advancement of theproposed methodology is demonstrated through the application of thetime-fractional Fokker-Planck equation and the fractional-order viscoelasticdynamics having weakly singular kernels. It also confirms the superiority ofthe proposed method compared with existing approaches available in theliterature.

Enhancing accuracy with an adaptive discretization for the non-local integro-partial differential equations involving initial time singularities