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In this work, we focus on the final value problem of an inverse problem for both linear and nonlinear biharmonic equations. The aim of this study is to provide a regularized method for the bi-harmonic equation, once the observed data are obtained at a terminal time in $ L^{q}(\Omega) $. We obtain an approximated solution using the Fourier series truncation method and the terminal input data in $ L^{q}(\Omega) $ for $ q \ne 2 $. In comparision with previous studies, the most highlight of this study is the error between the exact and regularized solutions to be estimated in $ L^{q}(\Omega) $; wherein an embedding between $ L^{q}(\Omega) $ and Hilbert scale spaces $ \mathcal{H}^{\rho}(\Omega) $ is applied.