Fractional Partial Derivatives Research Articles

Koksma–Hlawka type inequalities of fractional order

The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order \({\alpha > 0}\) is in \({\mathcal{L}_p}\) . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in \({\mathcal{L}_p}\) . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.

On the compressed elastic rod with rotary inertia on a viscoelastic

We study transversal vibrations of an elastic axially compressed rod on a fractional derivative type of viscoelastic foundation. We assume that the axial force has a constant and a time dependent part given by Dirac distributions. The dynamics of the system is described by a system of two partial differential equations, having integer and fractional derivatives. The solution of this system is obtained in the space of distributions and its asymptotic behavior is investigated. It is shown that the foundation increases the stability bound. AMS Mathematics Subject Classification (2000) 74H45.

Macrokinetics of Chemical Processes on Porous Catalysts Having Regard to Anomalous Diffusion

An analysis of the macrokinetics of chemical processes is undertaken on the condition that mass transfer is determined by anomalous diffusion of the reagents. In such a case the “reaction–diffusion” equation is written in the form of an equation with partial fractional derivatives. It is shown that in the internal diffusion region the rate of the process depends exponentially on the rate constant of the chemical reaction. Here, in contrast to the classical case of ordinary diffusion, the effective activation energy is defined as E k /γ, where 0 < γ < 2 and E k determines the activation energy of the chemical reaction. The classical case corresponds to γ = 2.