Abstract In this article, an idea based on moving least squares (MLS) and spectral collocation method is used to estimate the solution of nonlinear stochastic Volterra–Fredholm integral equations (NSVFIEs). The main advantage of the suggested approach is that in some parts where interpolation and integration are necessary, this approach does not require any meshes. Therefore, it is independent of the geometry of the domains, and this advantage helps us to solve the problems on irregular domains with relatively fewer computations. Another advantage of our proposed method is that with a small number of points and base functions, we were able to obtain the results with acceptable accuracy, and this is very attractive and practical. Applying the proposed method leads to the conversion of the problem into a system of algebraic equations. It is worth noting, some examples and error estimations have been provided to illustrate the accuracy and applicability of this technique. Also, we present a convergence analysis of the proposed method.

中文翻译：

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移动最小二乘法和谱搭配法逼近随机Volterra-Fredholm积分方程的解

摘要 在本文中，基于移动最小二乘法（MLS）和谱搭配方法的思想被用于估计非线性随机Volterra-Fredholm 积分方程（NSVFIEs）的解。建议方法的主要优点是在某些需要插值和积分的部分，这种方法不需要任何网格。因此，它与域的几何形状无关，这一优势有助于我们以相对较少的计算来解决不规则域上的问题。我们提出的方法的另一个优点是使用少量的点和基函数，我们能够以可接受的精度获得结果，这是非常有吸引力和实用的。应用所提出的方法可以将问题转换为代数方程组。值得注意的是，已经提供了一些示例和误差估计来说明该技术的准确性和适用性。此外，我们还提出了所提出方法的收敛性分析。