In this paper, the numerical solutions of the linear fractional Fredholm–Volterra integro-differential equations have been investigated. For this purpose, Laguerre polynomials have been used to develop an approximation method. Precisely, using the suitable collocation points, a system of linear algebraic equations arises which is resulted by the transformation of the integro-differential equation. The fractional derivative is considered in the conformable sense, and the conformable fractional derivative of the Laguerre polynomials is obtained in terms of Laguerre polynomials. Additionally, for the first time in the literature, the exact matrix formula of the conformable derivatives of the Laguerre polynomials is established. Furthermore, the results of the proposed method have been given applying the method to some various examples.

本文研究了线性分数阶Fredholm-Volterra积分微分方程的数值解。为此，已使用Laguerre多项式开发一种近似方法。精确地，使用合适的搭配点，产生了线性代数方程组，这是由积分-微分方程式的变换产生的。分数导数是从一致的意义上考虑的，并且拉盖尔多项式的一致的分数导数是根据拉盖尔多项式获得的。另外，在文献中首次建立了Laguerre多项式的适形导数的精确矩阵公式。此外，已经将所提出的方法的结果应用到一些不同的例子。