Fractional Variable-Order Derivative and Difference Operators and Their Applications to Dynamical Systems Modelling

In this paper, we unify and extend recent developments in Lyapunov stability theory to present techniques to determine the asymptotic stability of six types of fractional dynamical systems. These differ by being modeled with one of the following fractional derivatives: the Caputo derivative, the Caputo distributed order derivative, the variable order derivative, the conformable derivative, the local fractional derivative, or the distributed order conformable derivative (the latter defined in this work). Additionally, we apply these results to study the consensus of a fractional multiagent system, considering all of the aforementioned fractional operators. Our analysis covers multiagent systems with linear and nonlinear dynamics, affected by bounded external disturbances and described by fixed directed graphs. Lastly, examples, which are solved analytically and numerically, are presented to validate our contributions.

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On a new definition of fractional variable-order derivative

Proceedings of the 14th International Carpathian Control Conference (ICCC) ◽

10.1109/carpathiancc.2013.6560566 ◽

2013 ◽

Cited By ~ 24

Author(s):

Dominik Sierociuk ◽

Wiktor Malesza ◽

Michal Macias

Keyword(s):

Order Derivative ◽

Variable Order ◽

Definition Of ◽

Variable Order Derivative

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Stability and Convergence of a Time-Fractional Variable Order Hantush Equation for a Deformable Aquifer

Abstract and Applied Analysis ◽

10.1155/2013/691060 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 8

Author(s):

Abdon Atangana ◽

S. C. Oukouomi Noutchie

Keyword(s):

Partial Derivative ◽

Mathematical Formulation ◽

Stability And Convergence ◽

Order Derivative ◽

Variable Order ◽

Leaky Aquifer ◽

The Stability ◽

Nicolson Scheme ◽

Variable Order Derivative ◽

Modified Equation

The medium through which the groundwater moves varies in time and space. The Hantush equation describes the movement of groundwater through a leaky aquifer. To include explicitly the deformation of the leaky aquifer into the mathematical formulation, we modify the equation by replacing the partial derivative with respect to time by the time-fractional variable order derivative. The modified equation is solved numerically via the Crank-Nicolson scheme. The stability and the convergence in this case are presented in details.

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Heat Transfer Analysis for Non-Contacting Mechanical Face Seals Using the Variable-Order Derivative Approach

Energies ◽

10.3390/en14175512 ◽

2021 ◽

Vol 14 (17) ◽

pp. 5512

Author(s):

Slawomir Blasiak

Keyword(s):

Heat Transfer ◽

Classical Equation ◽

Heat Transfer Analysis ◽

Order Derivative ◽

Variable Order ◽

Fractional Model ◽

Face Seals ◽

Fractional Models ◽

Characteristic Features ◽

Variable Order Derivative

This article presents a variable-order derivative (VOD) time fractional model for describing heat transfer in the rotor or stator in non-contacting mechanical face seals. Most theoretical studies so far have been based on the classical equation of heat transfer. Recently, constant-order derivative (COD) time fractional models have also been used. The VOD time fractional model considered here is able to provide adequate information on the heat transfer phenomena occurring in non-contacting face seals, especially during the startup. The model was solved analytically, but the characteristic features of the model were determined through numerical simulations. The equation of heat transfer in this model was analyzed as a function of time. The phenomena observed in the seal include the conduction of heat from the fluid film in the gap to the rotor and the stator, followed by convection to the fluid surrounding them. In the calculations, it is assumed that the working medium is water. The major objective of the study was to compare the results of the classical equation of heat transfer with the results of the equations involving the use of the fractional-order derivative. The order of the derivative was assumed to be a function of time. The mathematical analysis based on the fractional differential equation is suitable to develop more detailed mathematical models describing physical phenomena.

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