Approximation of inverse source problem for time fractional pseudo-parabolic equation in $L^p$

In this work, we focus on the final value problem of an inverse problem for the pseudo-parabolic equation. This study aims to provide a regularization method for this equation, once the measurement data are obtained at the final time in $L^{r}(0,\pi)$. We obtain an approximated solution using the Fourier method and the final input data $L^{r}(0,\pi)$ for $r \neq 2$. Using embedding between $L^{r}(0,\pi)$ and Hilbert scales $\mathcal{H}^{\rho}(0,\pi)$, this study is the error between the exact and regularized solutions to be estimated in $L^{r}(0,\pi)$.