In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

中文翻译：

---

具有弱奇异核的非线性分数阶积分微分方程的替代勒让德多项式方法

在本文中，我们提出了一种求解一类弱奇异非线性分数阶积分微分方程数值解的数值方案。该方法利用替代勒让德多项式。推导出基于替代勒让德多项式的运算矩阵来逼近此类方程的奇异核。积分和乘积的运算矩阵与导出的运算矩阵一起用于将非线性分数阶积分微分方程转换为非线性代数方程组。此外，还分析了所提出的方法的收敛性，特别是在误差分析的背景下。而且，