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 + α convergent in time with 0 < α < 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.

中文翻译：

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基于双指数变换的具有弱奇异核的Volterra积分微分方程的形式二阶反向微分公式Sinc搭配方法

本文提出了一种形式上的二阶反向微分公式 (BDF2) Sinc 搭配方法，用于求解具有弱奇异核的 Volterra 积分微分方程。在时间方向上，时间导数通过 BDF2 离散化，并使用二阶卷积求积法则逼近 Riemann-Liouville 分数积分项。然后基于空间中的双指数变换通过 Sinc 近似建立一个完全离散的方案。通过能量法推导出收敛性和稳定性分析。提供了数值例子来说明所提出方法的有效性，可以发现我们的方案在空间上是超指数收敛的，并且在时间上 1 +  α阶收敛且 0 <  α < 1，分别。同时，将基于单指数变换的数值结果与所提出的方法进行了比较，以说明我们方法的高精度。