High-order orthogonal spline collocation ADI scheme for a new complex two-dimensional distributed-order fractional integro-differential equation with two weakly singular kernels

Abstract

We present an alternating direction implicit scheme for the numerical solution of the distributed-order fractional integro-differential equation with two weakly singular kernels. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. The weighted and shifted Grünwald difference formula is employed to discretize the distributed-order time-fractional derivative combined with the second-order quadrature convolution rule introduced by Lubich for the Riemann–Liouville fractional integral. We prove the stability of the proposed scheme and derive error estimates. We also describe an efficient implementation of the scheme and show the numerical results demonstrating the accuracy and convergence rates in norm.