An approximate solution for a nonlinear biharmonic equation with discrete random data

In this paper, we study a problem of finding the solution for the nonlinear biharmonic equation Δ2u=f(x,t,u(x,t)) from the final data. By using a simple example, the ill-posedness of the present problem with random noise is demonstrated. The Fourier method is conducted in order to establish an estimator for the mild solution (called regularized solution) and the convergence results in some different cases are proposed. Finally, numerical experiments are presented for showing that this regularization method is flexible and stable.