An ADI orthogonal spline collocation method for a new two-dimensional distributed-order fractional integro-differential equation

We consider an alternating direction implicit (ADI) scheme combined with an arbitrary-order orthogonal spline collocation (OSC) method in space for the numerical solution of the distributed-order fractional integro-differential equation. The weighted and shifted Grünwald formula is employed to discretize the distributed-order time-fractional derivative combined with the second-order fractional quadrature convolution rule introduced by Lubich for the Riemann-Liouville fractional integral. We prove the stability of the proposed ADI OSC scheme and derive error estimates. Finally, some numerical tests are given.