The main objective of this paper is to establish a fractional-order operational matrix method based on Euler wavelets for solving linear Volterra–Fredholm integro-differential equations with weakly singular kernel. First, the one-dimensional Euler wavelet is introduced, and then by using it, the operational matrix of fractional integration is constructed. Using the operational matrix of integration of fractional order, the weakly singular linear fractional Volterra–Fredholm integro-differential equations are reduced to a system of algebraic equations. The convergence of the proposed method has been analyzed and the numerical convergence rate for the presented scheme is established. Also, the error analysis of the proposed technique has been investigated. Furthermore, some numerical examples are solved to illustrate the applicability and efficiency of the proposed approach. Finally, a comparison of absolute error values obtained by different wavelet methods has been presented to clarify the error analysis of the proposed method.

本文的主要目的是建立一种基于欧拉小波的分数阶运算矩阵方法，用于求解具有弱奇异核的线性 Volterra-Fredholm 积分微分方程。首先引入一维欧拉小波，然后利用它构造分数阶积分运算矩阵。使用分数阶积分的运算矩阵，弱奇异线性分数Volterra-Fredholm 积分微分方程简化为代数方程组。分析了所提出方法的收敛性，并确定了所提出方案的数值收敛率。此外，还研究了所提出技术的误差分析。此外，解决了一些数值例子，以说明所提出方法的适用性和效率。最后，比较了不同小波方法获得的绝对误差值，以阐明所提出方法的误差分析。