Some further results of the laplace transform for variable–order fractional difference equations

2019 ◽  
Vol 22 (6) ◽  
pp. 1641-1654 ◽  
Author(s):  
Dumitru Baleanu ◽  
Guo–Cheng Wu
Keyword(s):  
Laplace Transform ◽  
Exact Solutions ◽  
Difference Equations ◽  
Response Analysis ◽  
Variable Order ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Time Systems

Abstract The Laplace transform is important for exact solutions of linear differential equations and frequency response analysis methods. In comparison with the continuous–time systems, less results can be available for fractional difference equations. This study provides some fundamental results of two kinds of fractional difference equations by use of the Laplace transform. Some discrete Mittag–Leffler functions are defined and their Laplace transforms are given. Furthermore, a class of variable–order and short memory linear fractional difference equations are proposed and the exact solutions are obtained.

scholarly journals Solving Fractional Difference Equations Using the Laplace Transform Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Li Xiao-yan ◽  
Jiang Wei
Keyword(s):  
Laplace Transform ◽  
Difference Equations ◽  
Initial Value Problems ◽  
Initial Value ◽  
Laplace Transform Method ◽  
Fractional Difference ◽  
Transform Method ◽  
Fractional Difference Equations ◽  
The Laplace Transform ◽  
Linear And Nonlinear

We discuss the Laplace transform of the Caputo fractional difference and the fractional discrete Mittag-Leffer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.

Qualitative analysis for two fractional difference equations

2020 ◽  
Vol 9 (1) ◽  
pp. 265-272
Author(s):  
Mohammed B. Almatrafi ◽  
Marwa M. Alzubaidi
Keyword(s):  
Qualitative Analysis ◽  
Exact Solutions ◽  
Difference Equations ◽  
Global Attractivity ◽  
Fractional Difference ◽  
Rational Difference Equations ◽  
Fractional Difference Equations ◽  
Special Cases

AbstractSome difference equations are generally studied by investigating their long behaviours rather than their exact solutions. The proposed equations cannot be solved analytically. Hence, this article discusses the main qualitative behaviours of two rational difference equations. Some appropriate hypotheses are examined and given to show the local and global attractivity. Special cases from the considered equations are solved analytically. The periodicity is also proved in this work. We also illustrate the achieved results in some 2D figures.

The function and associated polynomials

1951 ◽  
Vol 47 (2) ◽  
pp. 309-314 ◽  
Author(s):  
D. F. Lawden
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Difference Equations ◽  
Linear Differential Equations ◽  
Linear Difference Equations ◽  
Transform Method ◽  
Associated Polynomials ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Image Position

A transform method for the solution of linear difference equations, analogous to the method of the Laplace transform in the field of linear differential equations, has been described by Stone (1). The transform u(z) of a sequence un is defined by the equation

scholarly journals Oscillatory behavior of nonlinear Hilfer fractional difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tuğba Yalçın Uzun
Keyword(s):  
Difference Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Oscillatory Behavior ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
All Solutions ◽  
Alpha Beta

AbstractIn this paper, we study the oscillation behavior for higher order nonlinear Hilfer fractional difference equations of the type $$\begin{aligned}& \Delta _{a}^{\alpha ,\beta }y(x)+f_{1} \bigl(x,y(x+\alpha ) \bigr) =\omega (x)+f_{2} \bigl(x,y(x+ \alpha ) \bigr),\quad x\in \mathbb{N}_{a+n-\alpha }, \\& \Delta _{a}^{k-(n-\gamma )}y(x) \big|_{x=a+n-\gamma } = y_{k}, \quad k= 0,1,\ldots,n, \end{aligned}$$ Δ a α , β y ( x ) + f 1 ( x , y ( x + α ) ) = ω ( x ) + f 2 ( x , y ( x + α ) ) , x ∈ N a + n − α , Δ a k − ( n − γ ) y ( x ) | x = a + n − γ = y k , k = 0 , 1 , … , n , where $\lceil \alpha \rceil =n$ ⌈ α ⌉ = n , $n\in \mathbb{N}_{0}$ n ∈ N 0 and $0\leq \beta \leq 1$ 0 ≤ β ≤ 1 . We introduce some sufficient conditions for all solutions and give an illustrative example for our results.

Asymptotic stability of fractional difference equations with bounded time delays

2020 ◽  
Vol 23 (2) ◽  
pp. 571-590
Author(s):  
Mei Wang ◽  
Baoguo Jia ◽  
Feifei Du ◽  
Xiang Liu
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Integral Inequality ◽  
Time Delays ◽  
Asymptotical Stability ◽  
Stability Conditions ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation

AbstractIn this paper, an integral inequality and the fractional Halanay inequalities with bounded time delays in fractional difference are investigated. By these inequalities, the asymptotical stability conditions of Caputo and Riemann-Liouville fractional difference equation with bounded time delays are obtained. Several examples are presented to illustrate the results.

scholarly journals Oscillation properties of solutions of fractional difference equations

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 185-192 ◽  
Author(s):  
Mustafa Bayram ◽  
Aydin Secer
Keyword(s):  
Difference Equations ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Further Development ◽  
Oscillation Properties

In this article, studied the properties of the oscillation of fractional difference equations, and we obtain some results. The results we obtained are an expansion and further development of highly known results. Then we showed them with examples.

Asymptotic behavior of mild solutions for nonlinear fractional difference equations

2018 ◽  
Vol 21 (2) ◽  
pp. 527-551 ◽  
Author(s):  
Zhinan Xia ◽  
Dingjiang Wang
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Difference Equations ◽  
Mild Solutions ◽  
Contraction Mapping Principle ◽  
Fractional Difference ◽  
Alternative Theorem ◽  
Fractional Difference Equations ◽  
Banach Contraction ◽  
Asymptotically Periodic

AbstractIn this paper, we establish some sufficient criteria for the existence, uniqueness of discrete weighted pseudo asymptotically periodic mild solutions and asymptotic behavior for nonlinear fractional difference equations in Banach space, where the nonlinear perturbation is Lipschitz type, or non-Lipschitz type. The results are a consequence of application of different fixed point theorems, namely, the Banach contraction mapping principle, Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

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