A note on the fractional hyperbolic differential and difference equations

The initial boundary value problem for the fractional differential equation. {d(2)u(t)/dt(2) + D(t)(1/2)u(t) + Au(t) = f(t), 0 < t < 1, u(0) = 0, u'(0) = psi, in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem and its first and second order derivatives are established. The first order of accuracy difference scheme for the approximate solution of this problem is presented. The stability estimates for the solution of this difference scheme and its first and second order difference derivatives are established. In practice, the stability estimates for the solution of difference schemes for one dimensional fractional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional fractional hyperbolic equation with Dirichlet condition in space variables are obtained. (C) 2010 Elsevier Inc. All rights reserved.