On Fractional Eulerian Numbers and Equivalence of Maps with Long Term Power-Law Memory (Integral Volterra Equations of the Second Kind) to Gr$\ddot{u}$nvald-Letnikov Fractional Difference (Differential) Equations

Abstract

In this paper we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the system's variable is a total (a convolution) of the generating functions of all previous values of the variable with weights, which are powers of the time passed. In discrete cases these systems can be described by difference equations in which a fractional difference on the left hand side is equal to a total (also a convolution) of the generating functions of all previous values of the system's variable with fractional Eulerian number weights on the right hand side. In the continuous limit the considered systems can be described by Gr$\ddot{u}$nvald-Letnikov fractional differential equations, which are equivalent to the Volterra integral equations of the second kind. New properties of fractional Eulerian numbers and possible applications of the results are discussed.

29 pages 3 figures

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• Fractional Integro-Differential Equations for Electromagnetic Waves in Dielectric Media
• Fractional Dissipative Standard Map
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Cited by in corpus (6)

• General Fractional Dynamics
• Stability of Fixed Points and Chaos in Fractional Systems
• Asymptotic cycles in fractional maps of arbitrary positive orders
• Singular points in the solution trajectories of fractional order dynamical systems
• Stability of Fixed Points in Generalized Fractional Maps of the Orders $0<\alpha <1$
• Asymptotic cycles in fractional generalizations of multidimensional maps