Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems

This paper proposes the alternating direction implicit (ADI) numerical approaches for computing the solution of multi-dimensional distributed-order fractional integrodifferential problems. The proposed method discretizes the unknown solution in two stages. First, the Riemann–Liouville fractional integral term and the distributed-order time-fractional derivative are discretized with the help of the second-order convolution quadrature and the weighted and shifted Grünwald formula, respectively. Second, the spatial discretization is obtained by the general centered finite difference (FD) technique. At the same time, the ADI algorithms are devised for reducing the computational burden. Additionally, the convergence analysis of proposed ADI FD schemes is analyzed in detail through the energy method. Finally, two numerical examples highlight the accuracy of the proposed method and verify the theoretical formulations.