Abstract
We consider the Cauchy problem ( D ( k ) u ) ( t ) = λ u ( t ) , u ( 0 ) = 1 , where D ( k ) is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory 71 (2011), 583–600), λ > 0 . The solution is a generalization of the function t ↦ E α ( λ t α ) , where 0 < α < 1 , E α is the Mittag–Leffler function. The asymptotics of this solution, as t → ∞ , are studied.
Highlights
In several models of the dynamics of complex systems, the time evolution for observed quantities has exponential asymptotics of two possible types. These asymptotics are related with the solutions to the equations u1 ptq “ zuptq, t ą 0; up0q “ 1, where we will consider positive and negative z separately
For z ă 0, the solution decays to zero. In particular models such as, e.g., Glauber stochastic dynamics in the continuum, this corresponds to an exponential convergence to an equilibrium; see [1]
We study the asymptotic behavior of the solution of (3). Functions of this kind can be useful for fractional macroeconomic models with long dynamic memory; see [14] and references therein
Summary
In several models of the dynamics of complex systems, the time evolution for observed quantities has exponential asymptotics of two possible types. We study the asymptotic behavior of the solution of (3) Functions of this kind can be useful for fractional macroeconomic models with long dynamic memory; see [14] and references therein. If we assume the presence of effects of distributed lag (time delay) or fading memory in economic processes, the fractional generalization of the linear classical growth models can be described by the fractional differential equation Dα uptq “ λuptqf ptq with λ ą 0, α ą 0. We propose correct mathematical statements for growth models with memory in more general cases, for the general fractional derivative Dpkq with respect to the time variable Their application can be useful for mathematical economics for the description of processes with long memory and distributed lag. Note that the technique used below was developed initially in [12] for use in the study of intermittency in fractional models of statistical mechanics
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