Recovering initial population density of fractional pseudo-parabolic problem associated with a nonlinear reaction

In this paper, we consider an inverse problem related to the fractional pseudo-parabolic equation with a nonlinear source term. Our investigation reveals the ill-posedness of the problem according to Hadamard’s definition. We present two improved variations of the optimal filtering method introduced by Seidman (SIAM J Numer Anal 33:162–170, 1996) to establish some optimal estimates under some an a priori assumptions on the regularity of the exact solution. Finally, the effectiveness of our algorithm is demonstrated through numerical examples.