In this article, the product integration approach is extended to handle integrals involving derivatives of unknown functions. Using these extended formulas, a new approximation, called the PI approximation, is derived for the Caputo derivative of fractional order \(\nu \) such that \(m - 1< \nu < m\) , for some \(m \in \mathbb {N}\) . By combining the L1 and PI approximations of the Caputo derivative with second-order central difference schemes, two numerical methods are developed for solving multi-term time-fractional partial integro-differential equations involving the Volterra integral operator. The integrals in the PI method are approximated using Taylor expansions of the unknown function. These techniques are further extended to equations with weakly singular kernels in the Volterra term. The proposed methods transform the original problem into a system of algebraic equations that can be efficiently solved using standard numerical algorithms. We establish error bounds and provide stability estimates for Scheme-I. Finally, numerical experiments are performed to verify the accuracy and effectiveness of the PI approximation and the developed schemes, and comparative analyses with existing methods are presented through tables and graphs.
