In this paper, the numerical technique with the help of the Lucas wavelets (LWs) and the Legendre–Gauss quadrature rule is presented to study the solution of fractional Fredholm–Volterra integro-differential equations. The modified operational matrices of integration and pseudo-operational of fractional derivative for the proposed wavelet functions are calculated. These matrices in comparison to operational matrices existing in other methods are more accurate. The Lucas wavelets and their operational matrices provide the precise numerical scheme to get the approximate solution. Also, we exhibit the upper bound of error based on the method. We illustrate the behavior of the new scheme in several numerical examples with the help of tables and figures. The results confirm the accuracy and applicability of the numerical approach.

在本文中，借助于Lucas小波（LWs）和Legendre-Gauss正交规则，提出了一种数值技术，以研究分数Fredholm-Volterra积分微分方程的解。计算了所提出的小波函数的修正的积分和伪导数运算矩阵。与其他方法中存在的运算矩阵相比，这些矩阵更准确。卢卡斯小波及其运算矩阵提供了精确的数值方案，以获得近似解。同样，我们展示了基于该方法的误差上限。我们借助表格和数字在几个数值示例中说明了新方案的行为。结果证实了数值方法的准确性和适用性。