Regularization of an initial inverse problem for a biharmonic equation

In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous biharmonic equation. The problem is severely ill-posed in the sense of Hadamard. In order to obtain a stable numerical solution, we propose two regularization methods to solve the problem. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in $\mathcal{L}^{2}$ uniformly with respect to the space coordinate under some a priori assumptions on the solution. Finally, in order to increase the significance of the study, numerical results are presented and discussed illustrating the theoretical findings in terms of accuracy and stability.

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