A formally second-order BDF finite difference scheme for the integro-differential equations with the multi-term kernels

In this article, a formally second-order backward differentiation formula (BDF) finite difference scheme is presented for the integro-differential equations with the multi-term kernels. In the time direction, the time derivative is approximated by a second-order BDF scheme and the Riemann-Liouville (R-L) fractional integral terms are discretized by the second-order convolution quadrature rule. We construct a fully discrete difference scheme with the space discretization by the standard central difference formula. The and -norms stability, and convergence in -norm are derived by the discrete energy method. In the numerical experiments, the results are consistent with the theoretical analysis.