A formally second-order backward differentiation formula Sinc-collocation method for the Volterra integro-differential equation with a weakly singular kernel based on the double exponential transformation

This paper presents a formally second-order backward differentiation formula (BDF2) Sinc-collocation method for solving the Volterra integro-differential equation with a weakly singular kernel. In the time direction, the time derivative is discretized via the BDF2 and the second-order convolution quadrature rule is used to approximate the Riemann-Liouville fractional integral term. Then a fully discrete scheme is established via the Sinc approximation based on the double exponential transformation in space. The convergence and stability analysis are derived by the energy method. Numerical examples are provided to illustrate the effectiveness of proposed method and it can be found that our scheme is super-exponentially convergent in space and order 1 + alpha convergent in time with 0 < alpha < 1, respectively. Meanwhile, the numerical results based on the single exponential transformation are compared with the proposed method to illustrate the high accuracy of our method.