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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
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.

scholarly journals Consensus of Multiagent Systems Described by Various Noninteger Derivatives

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
G. Nava-Antonio ◽  
G. Fernández-Anaya ◽  
E. G. Hernández-Martínez ◽  
J. J. Flores-Godoy ◽  
E. D. Ferreira-Vázquez
Keyword(s):  
Multiagent Systems ◽  
Directed Graphs ◽  
Lyapunov Stability Theory ◽  
Order Derivative ◽  
Variable Order ◽  
Conformable Derivative ◽  
External Disturbances ◽  
Recent Developments ◽  
Variable Order Derivative ◽  
Distributed Order

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.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

scholarly journals On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Abdon Atangana ◽  
Adem Kilicman
Keyword(s):  
Mass Transport ◽  
Fractional Derivative ◽  
Numerical Simulations ◽  
Transport Equation ◽  
Stability And Convergence ◽  
Hydrodynamic Dispersion ◽  
The Stability ◽  
Mass Transport Equation ◽  
Modified Equation ◽  
Convergence Of The Scheme

The hydrodynamic dispersion equation was generalized using the concept of variational order derivative. The modified equation was numerically solved via the Crank-Nicholson scheme. The stability and convergence of the scheme in this case were presented. The numerical simulations showed that, the modified equation is more reliable in predicting the movement of pollution in the deformable aquifers, than the constant fractional and integer derivatives.

Stability and Convergence of Two New Implicit Numerical Methods for the Fractional Cable Equation

2009 ◽  
Author(s):  
F. Liu ◽  
Q. Yang ◽  
I. Turner
Keyword(s):  
Numerical Methods ◽  
Partial Derivative ◽  
Stability And Convergence ◽  
Cable Equation ◽  
Neuronal Dynamics ◽  
Neuronal Dendrites ◽  
Space Step ◽  
The Stability ◽  
Fundamental Equations ◽  
Fractional Cable Equation

The cable equation is one the most fundamental equations for modeling neuronal dynamics. Cable equations with fractional order temporal operators have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper we consider the following fractional cable equation involving two fractional temporal derivatives: ∂u(x,t)∂t=0Dt1−γ1κ∂2u(x,t)∂x2−μ02Dt1−γ2u(x,t)+f(x,t), where 0 < γ1,γ2 < 1, κ > 0, and μ02 are constants, and 0Dt1−γu(x,t) is the Rieman-Liouville fractional partial derivative of order 1 − γ. Two new implicit numerical methods with convergence order O(τ + h2) and O(τ2 + h2) for the fractional cable equation are proposed respectively, where τ and h are the time and space step sizes. The stability and convergence of these methods are investigated using the energy method. Finally, numerical results are given to demonstrate the effectiveness of both implicit numerical methods. These techniques can also be applied to solve other types of anomalous subdiffusion problems.

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

Energies ◽  
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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