On source identification problem for a parabolic equation with memory term

Recently, parabolic equations with non-local term have received a lot of interest. In this article, we study an inverse source identification problem from final time measurement associated with a parabolic equation involving memory term. The problem is transformed to solving a linear operator equation, where the associated operator is positive, self-adjoint and compact operator of infinite rank, which implies that the problem is ill-posed. The stable approximations are obtained by three methods: Lavrentiev method, Tikhonov regularization method and truncation method. Hölder rate of convergence is obtained for both apriori and aposteriori parameter choice strategies by considering source sets to be subset of some Sobolev space for all the three methods. We also show that for certain source set, the obtained rates are order optimal in the sense of worst case error for both the apriori and aposteriori case. Moreover, we also present a numerical study to implement the theory.