Well-posedness of parabolic differential and difference equations with the fractional differential operator

The stable difference scheme for the approximate solution of the initial value problem du(t)/dt+d t 1/2u(t)+au(t) = f (t),0 <t<1,u(0) = 0 for the differential equation in a banach space e with the strongly positive operator a and fractional operator d t 1/2 is presented. the well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. in practice,the coercive stability estimates for the solution of difference schemes for the 2m-th order multi-dimensional fractional parabolic equation and the one-dimensional fractional parabolic equation with nonlocal boundary conditions in space variable are obtained.