We consider the fractional Klein-Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non-trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial ratefor 0 < < 2 and at some exponential rate when ≥ 2. Our approach is based on the asymptotic theory of 0 semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for the fractional Laplacian, which may be of independent interest. K E Y W O R D S damped wave equation, fractional derivative, geometric control condition M S C ( 2 0 1 0 ) 35B35, 35B40, 35G30
In this paper, a class of finite difference numerical technique is presented for the solution of the second-order linear inhomogeneous damped wave equation. The consistency, stability and convergences of these numerical schemes are discussed. The results obtained are compared to the exact solution as well as ordinary explicit, implicit finite difference methods, and the fourth-order compact method (FOCM) of [6]. The general idea of these methods is developed by using C 0 -semigroups operator theory. We also showed that the stability region for the explicit finite difference scheme depends on the damping coefficient.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.