A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation

We propose and analyze a time-stepping Crank-Nicolson(CN) alternating direction implicit(ADI) scheme combined with an arbitrary-order orthogonal spline collocation (OSC) methods in space for the numerical solution of the fractional integro-differential equation with a weakly singular kernel. We prove the stability of the numerical scheme and derive error estimates. The analysis presented allows variable time steps which, as will be shown, can efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term. Finally, some numerical tests are given.