Approximations of stable manifolds in the vicinity of hyperbolic equilibrium points for fractional differential equations

Abstract

This paper is devoted to the numerical analysis of the abstract semilinear fractional problem $$D^\alpha u(t) = Au(t) + f(u(t)), u(0)=u^0,$$ in a Banach space E. We are developing a general approach to establish a semidiscrete approximation of stable manifolds. The phase space in the neighborhood of the hyperbolic equilibrium can be split in such a way that the original initial value problem is reduced to systems of initial value problems in the invariant subspaces corresponding to positive and negative real parts of the spectrum. We show that such a decomposition of the equation keeps the same structure under general approximation schemes. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and can be verified for finite element as well as finite difference methods.