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 $l^{r}(0,pi)$.