Abstract

In this article, we define a fractional forward discrete operator. Then, for a family of linear nonautonomous fractional difference equations constructed by using this fractional discrete operator, we provide a practical formula of solutions. This family of problems covers several linear fractional difference equations that appear in the literature. Numerical examples are given to justify our theory.

References

1.
Dassios
,
I.
,
2012
, “
On Non-Homogeneous Generalized Linear Discrete Time Systems
,”
Circuits Syst. Signal Process
,
31
(
5
), pp.
1699
1712
.10.1007/s00034-012-9400-7
2.
Dassios
,
I.
,
Zimbidis
,
A.
, and
Kontzalis
,
C.
,
2014
, “
The Delay Effect in a Stochastic Multiplier–Accelerator Model
,”
J. Econ. Struct.
,
3
(
1
), p.
7
.10.1186/s40008-014-0007-y
3.
Dassios
,
I.
,
2015
, “
Optimal Solutions for Non-Consistent Singular Linear Systems of Fractional Nabla Difference Equations
,”
Circuits, Syst. Signal Process.
,
34
(
6
), pp.
1769
1797
.10.1007/s00034-014-9930-2
4.
DeGroot
,
M. H.
,
1974
, “
Reaching a Consensus
,”
J. Am. Stat. Assoc.
,
69
(
345
), pp.
118
121
.10.1080/01621459.1974.10480137
5.
Ogata
,
K.
,
1987
,
Discrete Time Control Systems
,
Prentice Hall
,
New York
.
6.
Abadias
,
L.
,
Lizama
,
C.
,
J. Miana
,
P.
, and
Velasco
,
M. P.
,
2019
, “
On Well–Posedness of Vector-Valued Fractional Differential-Difference Equations
,”
Discrete Contin. Dyn. Syst.
,
39
(
5
), pp.
2679
2708
.10.3934/dcds.2019112
7.
Dassios
,
I.
,
2018
, “
A Practical Formula of Solutions for a Family of Linear Non-Autonomous Fractional Nabla Difference Equations
,”
J. Comput. Appl. Math
,
339
, pp.
317
328
.10.1016/j.cam.2017.09.030
8.
Klamka
,
J.
,
2010
, “
Controllability and Minimum Energy Control Problem of Fractional Discrete-Time Systems
,”
New Trends in Nanotechnology and Fractional Calculus Applications
,
Springer
,
Dordrecht
, The Netherlands, pp.
503
509
.
9.
Rezapour
,
S.
, and
Salehi
,
S.
,
2015
, “
On the Existence of Solution for a k–Dimensional System of Three Points Nabla Fractional Finite Difference Equations
,”
Bull. Iranian Math. Soc.
,
41
(
6
), pp.
1433
1444
.http://bims.iranjournals.ir/article_705.html
10.
Dassios
,
I.
,
Kerci
,
T.
,
Baleanu
,
D.
, and
Milano
,
F.
,
2023
, “
Fractional–Order Dynamical Model for Electricity Markets
,”
Math. Methods Appl. Sci.
,
46
(
7
), pp. 8349–8361.10.1002/mma.7892
11.
Machado
,
J. A.
,
Mata
,
M. E.
, and
Lopes
,
A. M.
,
2015
, “
Fractional State Space Analysis of Economic Systems
,”
Entropy
,
17
(
12
), pp.
5402
5421
.10.3390/e17085402
12.
Wu
,
G. C.
, and
Baleanu
,
D.
,
2014
, “
Discrete Fractional Logistic Map and Its Chaos
,”
Nonlinear Dyn.
,
75
(
1–2
), pp.
283
287
.10.1007/s11071-013-1065-7
13.
Dassios
,
I.
,
Tzounas
,
G.
, and
Milano
,
F.
,
2023
,
Stability Criterion of a Class of Non-Causal Systems of Differential Equations
,
Circuits, Systems and Signal Processing, Vol. 42, Springer
,
Berlin
, pp.
2452
2467
.
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