4,655 research outputs found
Long Memory in a Linear Stochastic Volterra Differential Equation
In this paper we consider a linear stochastic Volterra equation which has a
stationary solution. We show that when the kernel of the fundamental solution
is regularly varying at infinity with a log-convex tail integral, then the
autocovariance function of the stationary solution is also regularly varying at
infinity and its exact pointwise rate of decay can be determined. Moreover, it
can be shown that this stationary process has either long memory in the sense
that the autocovariance function is not integrable over the reals or is
subexponential. Under certain conditions upon the kernel, even arbitrarily slow
decay rates of the autocovariance function can be achieved. Analogous results
are obtained for the corresponding discrete equation
Basics of Qualitative Theory of Linear Fractional Difference Equations
Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.This doctoral thesis concerns with the fractional calculus on discrete settings, namely in the frame of the so-called (q,h)-calculus and its special case h-calculus. First, foundations of the theory of linear fractional difference equations in (q,h)-calculus are established. In particular, basic properties, such as existence, uniqueness and structure of solutions, are discussed and a discrete analogue of the Mittag-Leffler function is introduced via eigenfunctions of a fractional difference operator. Further, qualitative analysis of a scalar and vector test fractional difference equation is performed in the frame of h-calculus. The results of stability and asymptotic analysis enable us to specify the connection to other mathematical disciplines, such as continuous fractional calculus, Volterra difference equations and numerical analysis. Finally, a possible generalization of the fractional calculus to more general settings is outlined.
On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations
In this paper we consider unbounded solutions of perturbed convolution
Volterra summation equations. The equations studied are asymptotically
sublinear, in the sense that the state--dependence in the summation is of
smaller than linear order for large absolute values of the state. When the
perturbation term is unbounded, it is elementary to show that solutions are
also. The main results of the paper are mostly of the following form: the
solution has an additional unboundedness property if and only if the
perturbation has property . Examples of property include monotone
growth, monotone growth with fluctuation, fluctuation on without
growth, existence of time averages. We also study the connection between the
times at which the perturbation and solution reach their running maximum, and
the connection between the size of signed and unsigned running maxima of the
solution and forcing term.Comment: 45 page
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
Relaxation oscillations, pulses, and travelling waves in the diffusive Volterra delay-differential equation
The diffusive Volterra equation with discrete or continuous delay is studied in the limit of long delays using matched asymptotic expansions. In the case of continuous delay, the procedure was explicitly carried out for general normalized kernels of the form Sigma/sub n=p//sup N/ g/sub n/(t/sup n//T/sup n+1/)e/sup -t/T/, pges2, in the limit in which the strength of the delayed regulation is much greater than that of the instantaneous one, and also for g/sub n/=delta/sub n2/ and any strength ratio. Solutions include homogeneous relaxation oscillations and travelling waves such as pulses, periodic wavetrains, pacemakers and leading centers, so that the diffusive Volterra equation presents the main features of excitable media
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