In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations (fies) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems (evps) associated with the two-dimensional fies. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fies in both $L^2$ and $L^{\infty }$ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in $L^2$ and $L^{\infty }$ norms. The numerical illustrations are introduced for the error of these methods.

本文考虑了混合傅立叶勒让德谱伽辽金(MFLSG)方法求解具有光滑核的Banach空间上的二维Fredholm积分方程( fie s)。也考虑使用相同的方法来找到与二维fie相关的特征值问题 ( evp s)的特征值。利用这些方法，我们建立了近似解以及迭代近似解与二维fie的精确解之间的误差。${升}^2$ 和 ${升}^{\infty }$规范。我们还通过 MFLSG 方法建立了近似特征值、特征向量和迭代特征向量和精确特征元素之间的误差${升}^2$ 和 ${升}^{\infty }$规范。对这些方法的误差进行了数值说明。