In this paper, we propose a class of methods to solve the parabolic Volterra integro-differential equations with bounded and unbounded domains. More precisely, we change the parabolic Volterra integro-differential equations to well-posed linear and nonlinear dynamical systems. Then, the obtained systems are solved by using a new class of algorithms consisting linear multi-step formulas in which these schemes are constructed through the hybrid of Gergory's formula, finite difference and multi-step methods. Error bounds are derived in both bounded and unbounded domains. Some numerical examples are then presented to illustrate the efficiency and accuracy of the proposed methods. Furthermore, stability and convergence of proposed methods are established and we denote the numerical simulations. Moreover, some tests are conducted on data with measurement noise to consider the performance of the proposed methods.

在本文中，我们提出了一类求解具有有界和无界域的抛物型 Volterra 积分微分方程的方法。更准确地说，我们将抛物线 Volterra 积分微分方程更改为适定线性和非线性动力系统。然后，使用由线性多步公式组成的一类新算法求解所获得的系统，其中这些方案是通过 Gorgory 公式、有限差分和多步方法的混合构建的。误差界限在有界和无界域中都有。然后给出了一些数值例子来说明所提出方法的效率和准确性。此外，建立了所提出方法的稳定性和收敛性，我们表示数值模拟。而且，