In the current work, we consider the nonlinear one-dimensional stochastic Sine-Gordon equation with appropriate initial and boundary conditions. The main goal of this work is presenting a numerical scheme based on radial basis functions (RBFs) and finite difference method to provide the approximate solution of mentioned equation. For approximating the solution, finite difference idea is used to overcome the time variable and then strictly positive definite RBFs such as Gaussian have been used to estimate the unknown function in time step n. Finally, several examples are given to check the accuracy and efficiency of the provided solution.

中文翻译：

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用新的无网格技术求解一维非线性随机 Sine-Gordon 方程

在当前的工作中，我们考虑具有适当初始和边界条件的非线性一维随机 Sine-Gordon 方程。这项工作的主要目标是提出一种基于径向基函数 (RBF) 和有限差分方法的数值方案，以提供上述方程的近似解。为了逼近解，使用有限差分思想来克服时间变量，然后使用严格正定 RBFs（例如 Gaussian）来估计时间步n 中的未知函数。最后，给出了几个例子来检查所提供解决方案的准确性和效率。