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

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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.

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Authors and Affiliations

  1. Scientific Research Computer Center, Lomonosov Moscow State University, Leninskiye Gory, Moscow, Russia, 119899

    Sergey Piskarev

  2. Institute for Analysis and Center for Dynamics, Faculty of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    Stefan Siegmund

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  1. Sergey Piskarev
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  2. Stefan Siegmund
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Correspondence to Stefan Siegmund.

Additional information

Sergey Piskarev: Research was supported by grants of the Russian Foundation for Basic Research \(15-01-00026\_a, 16-01-00039\_a\), \(17-51-53008\) and DAAD and partly by the German Research Foundation (DFG) within the Cluster of Excellence EXC 1056 ’Center for Advancing Electronics Dresden’ (CFAED)

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Piskarev, S., Siegmund, S. Approximations of stable manifolds in the vicinity of hyperbolic equilibrium points for fractional differential equations. Nonlinear Dyn 95, 685–697 (2019). https://doi.org/10.1007/s11071-018-4590-6

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