Fractional-order difference equations for physical lattices and some applications

2015 ◽  
Vol 56 (10) ◽  
pp. 103506 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Order Difference

scholarly journals Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
Mary Jacintha ◽  
Abdullah Özbekler
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Fractional Operators ◽  
Damping Term ◽  
Fractional Difference ◽  
Riccati Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Positive Integers

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.

scholarly journals On the Stability of Linear Incommensurate Fractional-Order Difference Systems

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1754
Author(s):  
Noureddine Djenina ◽  
Adel Ouannas ◽  
Iqbal M. Batiha ◽  
Giuseppe Grassi ◽  
Viet-Thanh Pham
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Equivalent System ◽  
Stability Of Solutions ◽  
Extended Version ◽  
Transform Method ◽  
Order Difference ◽  
Difference Systems ◽  
The Stability

To follow up on the progress made on exploring the stability investigation of linear commensurate Fractional-order Difference Systems (FoDSs), such topic of its extended version that appears with incommensurate orders is discussed and examined in this work. Some simple applicable conditions for judging the stability of these systems are reported as novel results. These results are formulated by converting the linear incommensurate FoDS into another equivalent system consists of fractional-order difference equations of Volterra convolution-type as well as by using some properties of the Z-transform method. All results of this work are verified numerically by illustrating some examples that deal with the stability of solutions of such systems.

scholarly journals Fractional Order Difference Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
J. Jagan Mohan ◽  
G. V. S. R. Deekshitulu
Keyword(s):  
Difference Equation ◽  
Fractional Order ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Order Difference ◽  
Fractional Difference Equations ◽  
Independent Variable

A difference equation is a relation between the differences of a function at one or more general values of the independent variable. These equations usually describe the evolution of certain phenomena over the course of time. The present paper deals with the existence and uniqueness of solutions of fractional difference equations.

The Time-Varying Fractional Order Difference Equations

2003 ◽  
Author(s):  
Piotr W. Ostalczyk
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
Difference Equations ◽  
Control Strategy ◽  
Closed Loop ◽  
Time Varying ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Numerical Example ◽  
Fractional Orders

In this paper we explore the linear difference equations with fractional orders being a function of time. A description of closed-loop dynamical systems described by such equations is proposed. In the numerical example a simple control strategy based on time-varying fractional orders is presented.

scholarly journals Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1751
Author(s):  
Oana Brandibur ◽  
Eva Kaslik ◽  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Asymptotic Stability ◽  
Fractional Order ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Discrete Version ◽  
Model Parameters ◽  
Step Size ◽  
Order Difference ◽  
Stability And Instability ◽  
Fractional Orders

Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders of the considered Caputo-type differences, as well as the elements of the linear system’s matrix and the discretization step size. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Furthermore, numerical simulations are undertaken in order to substantiate the theoretical findings, showing that the membrane potential may exhibit complex bursting behavior for suitable choices of the model parameters and fractional orders of the Caputo-type differences.

On the asymptotic stability of linear system of fractional-order difference equations

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Raghib Abu-Saris ◽  
Qasem Al-Mdallal
Keyword(s):  
Asymptotic Stability ◽  
Linear System ◽  
Fractional Order ◽  
Difference Equations ◽  
Equilibrium Solution ◽  
Initial Condition ◽  
System Of Difference Equations ◽  
Order Difference ◽  
Order Linear ◽  
The Stability

AbstractIn this paper we investigate the stability of the equilibrium solution of the νth order linear system of difference equations $(\Delta _{a + \nu - 1}^\nu y)(t) = \Lambda y(t + \nu - 1);t \in \mathbb{N}_a ,a \in \mathbb{R},and\Lambda \in \mathbb{R}^{p \times p} ,$ subject to the initial condition $y(a + \nu - 1) = y - 1,$, where 0 < ν < 1 and y−1 ∈ ℝp.

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