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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 thenorm. Finally, numerical experiments are presented for showing that this regularization method is flexible and stable.

Regularizing a final value problem for nonlinear modified Helmholtz equation with randomly perturbed data