In this paper, we consider the Cauchy problem for the nonlinear modified Helmholtz equation (CMH) associated with a fractional Laplacian. The data in the problem including the final data and the source function are studied as random data. The CMH problem is severely ill-posed in Hadamard's sense so that the Fourier truncation method associated with some techniques in nonparametric regression is used to establish a stable solution. Moreover, we also obtain the expectation of the error estimates for the difference between the regularized solution and the exact solution in the $L^2-$norm. Finally, numerical experiments are presented for showing that this regularization method is flexible and stable.

在本文中，我们考虑了与分数阶拉普拉斯算子相关的非线性修正亥姆霍兹方程 (CMH) 的柯西问题。问题中的数据，包括最终数据和源函数，作为随机数据进行研究。CMH 问题在 Hadamard 的意义上是严重不适定的，因此使用与非参数回归中的某些技术相关的傅里​​叶截断方法来建立稳定的解决方案。此外，我们还获得了对正则化解与精确解之间差异的误差估计的期望${\mathrm{大号}}^{\mathrm{2个}}-$规范。最后通过数值实验证明了该正则化方法的灵活性和稳定性。