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

This work aims to construct an efficient and highly accurate numerical method to address the time singularity at $t=0$ involved in a class of time-fractional parabolic 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 is employed to discretize the Volterra operator. Subsequently, it helps to convert the proposed model into a second-order BVP in a semi-discrete form. The multi-dimensional Haar wavelets are then used for grid adaptation and efficient computations for the 2D problem, whereas the standard second-order approximations are employed to approximate the spatial derivatives for the 1D case. The stability analysis is carried out on an adaptive mesh in time. The convergence analysis leads to $O(N^{-2}+M^{-2})$ accurate solution in the space-time domain for the 1D problem having time singularity based on the $L^\infty$ norm for a suitable choice of the grading parameter. Furthermore, it provides $O(N^{-2}+M^{-3})$ accurate solution for the 2D problem having unbounded time derivative at $t=0$. The analysis also highlights a higher order accuracy for a sufficiently smooth solution resides in $C^3(\overline{\Omega}_t)$ even if the mesh is discretized uniformly. The truncation error estimates for the time-fractional operator, integral operator, and spatial derivatives are presented. Numerous tests are performed on several examples in support of the theoretical analysis. The advancement of the proposed methodology is demonstrated through the application of the time-fractional Fokker-Planck equation and the fractional-order viscoelastic dynamics having weakly singular kernels. It also confirms the superiority of the proposed method compared with existing approaches available in the literature.