This work deals with the resolution of an inverse Cauchy-type problem associated with the biharmonic equation such that the boundary conditions are known only on a part of the boundary of the domain. A problem often encountered in mechanics, especially in thin plate theory. Such problem is known to be ill-posed in the sense of Hadamard, i.e. the existence, uniqueness and stability of the solution are not always guaranteed. Therefore, it requires the employment of special techniques to solve it in a stable manner. To address this issue, we choose to employ the fading regularization method (Cimetière et al. (2000,2001), Delvare (2000)). Next, we investigate the numerical reconstruction of the missing boundary conditions on an inaccessible part of the boundary from the knowledge of over-prescribed exact or noisy data on the remaining and accessible part of the boundary. We present numerical implementations of this method using the method of fundamental solutions and the finite element method. We then propose to combine Discrete Kirchhoff plate elements with the fading regularization method to solve the Cauchy problem in thin plate theory.