ABSTRACT
When fa = 0 and f=0 for k= 1,2, • • • M, backward Euler and second order Euler schemes in finite element settings were analyzed in [10], where certain exponential decay properties were obtained under some strong assumptions on the kernel. In this paper we study the exponential decay not only for the real continuous solutions but also for its numerical solutions. The exponential decay rate obtained is optimal. That is, it reduces to classical results for parabolic equations if the kernel K(t) = 0. In addition finite difference schemes of backward Euler methods also are considered and shown to decay exponentially.
