This article deals with a fully discretized numerical scheme for solving fractional order Volterra integro-differential equations involving Caputo fractional derivative. Such problem exhibits a mild singularity at the initial time \(t=0\). To approximate the solution, the classical L1 scheme is introduced on a uniform mesh. For the integral part, the composite trapezoidal approximation is used. It is shown that the approximate solution converges to the exact solution. The error analysis is carried out. Due to presence of weak singularity at the initial time, we obtain the rate of convergence is of order \(O(\tau )\) on any subdomain away from the origin whereas it is of order \(O(\tau ^\alpha )\) over the entire domain. Finally, we present a couple of examples to show the efficiency and the accuracy of the numerical scheme.

本文讨论用于求解涉及 Caputo 分数阶导数的分数阶 Volterra 积分微分方程的完全离散化数值方案。这样的问题在初始时间\(t=0\)表现出轻微的奇异性。为了近似解，在均匀网格上引入了经典的 L1 方案。对于积分部分，使用复合梯形近似。结果表明，近似解收敛于精确解。进行误差分析。由于初始时刻存在弱奇异性，我们在远离原点的任何子域上得到收敛速度为\(O(\tau )\)而它是\(O(\tau ^\alpha )\)在整个域中。最后，我们提供了几个例子来展示数值方案的效率和准确性。