BDF ADI orthogonal spline collocation scheme for the fractional integro-differential equation with two weakly singular kernels

In this paper, a second-order backward differentiation formula (BDF) alternatingdirection implicit (ADI) orthogonal spline collocation (OSC) method is presented for the fractional integro-differential equation with two different weakly singular kernels. The integral terms are approximated by the second-order fractional quadrature rule suggested by Lubich. The convergence of the BDF ADI OSC method in L2 norm is proved. Besides, we provide the numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and the effectiveness of the BDF ADI OSC method. Numerical experiments also exhibit the optimal error estimates.