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

Abstract

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.

Keywords:

• Nonlinear space fractional modified Helmholtz equation
• Fourier truncation method
• nonparametric regression
• ill-posed problem

AMS Subject Classifications:

• 62G08
• 15A29
• 47A52

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2018.07.