Qualitative analysis of Caputo fractional delayed difference system: a novel delayed discrete fractional sine and cosine-type function
Authors
N. I. Mahmudov
- Department of Mathematics, Eastern Mediterranean University, Famagusta 99628 T. R. Northern Cyprus, Mersin 10, Turkey.
- Research Center of Econophysics, Azerbaijan State University of Economics (UNEC), Istiqlaliyyat Str. 6, Baku 1001, Azerbaijan.
M. Aydin
- Department of Medical Services and Techniques, Muradiye Vocational School, Van Yuzuncu Yil University, Van, Turkey.
Abstract
In this paper, we provide an explicit solution for the homogeneous fractional
delay oscillation difference equation with an order \(2{\delta}\) ranging from 1 to
2. This solution is achieved through the construction of discrete sine and
cosine-type delayed matrix functions. Subsequently, we employ the discrete
Laplace transform technique, a powerful method for handling nonhomogeneous
terms, to investigate the solution of the corresponding nonhomogeneous
equation. The study then delves into the Ulam-Hyers-type stabilities of the
homogeneous equation, leveraging the representation of the solution. To
validate the stability theory, we illustrate a numerical example. Finally, we
extend our analysis by presenting an exact solution for the nonhomogeneous
fractional difference equation with \(1<2{\delta}<2\), utilizing the discrete
two-parameter delayed sine and cosine-type function.
Share and Cite
ISRP Style
N. I. Mahmudov, M. Aydin, Qualitative analysis of Caputo fractional delayed difference system: a novel delayed discrete fractional sine and cosine-type function, Journal of Nonlinear Sciences and Applications, 18 (2025), no. 1, 43--63
AMA Style
Mahmudov N. I., Aydin M., Qualitative analysis of Caputo fractional delayed difference system: a novel delayed discrete fractional sine and cosine-type function. J. Nonlinear Sci. Appl. (2025); 18(1):43--63
Chicago/Turabian Style
Mahmudov, N. I., Aydin, M.. "Qualitative analysis of Caputo fractional delayed difference system: a novel delayed discrete fractional sine and cosine-type function." Journal of Nonlinear Sciences and Applications, 18, no. 1 (2025): 43--63
Keywords
- Linear system
- fractional difference
- time-delay
- nabla sine cosine
- discrete delayed perturbation
MSC
- 33E12
- 34Kxx
- 39Axx
- 39A06
- 44A55
References
-
[1]
T. Abdeljawad, Fractional difference operators with discrete generalized Mittag-Leffler kernels, Chaos Solitons Fractals, 126 (2019), 315–324
-
[2]
T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Difference Equ., 2016 (2016), 18 pages
-
[3]
J. Alzabut, T. Abdeljawad, A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system, Appl. Anal. Discrete Math., 12 (2018), 36–48
-
[4]
F. M. Atıcı, M. Atıcı, N. Nguyen, T. Zhoroev, G. Koch, A study on discrete and discrete fractional pharmacokineticspharmacodynamics models for tumor growth and anti-cancer effects, Comput. Math. Biophys., 7 (2019), 10–24
-
[5]
F. M. Atıcı, P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., 2009 (2009), 12 pages
-
[6]
F. M. Atici, P. W. Eloe, Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 41 (2011), 353–370
-
[7]
F. M. Atıcı, P.W. Eloe, Gronwall’s inequality on discrete fractional calculus, Comput. Math. Appl., 64 (2012), 3193–3200
-
[8]
M. Aydin, N. I. Mahmudov, Iterative learning control for impulsive fractional order time-delay systems with nonpermutable constant coefficient matrices, Int. J. Adapt. Control Signal Process., 36 (2022), 1419–1438
-
[9]
M. Aydin, N. I. Mahmudov, On a study for the neutral Caputo fractional multi-delayed differential equations with noncommutative coefficient matrices, Chaos Solitons Fractals, 161 (2022), 11 pages
-
[10]
M. Aydin, N. I. Mahmudov, ψ-Caputo type time-delay Langevin equations with two general fractional orders, Math. Methods Appl. Sci., 46 (2023), 9187–9204
-
[11]
D. Baleanu, G.-C. Wu, Some further results of the Laplace transform for variable-order fractional difference equations, Fract. Calc. Appl. Anal., 22 (2019), 1641–1654
-
[12]
B. Bonilla, M. Rivero, J. J. Trujillo, On systems of linear fractional differential equations with constant coefficients, Appl. Math. Comput., 187 (2007), 68–78
-
[13]
X. Cao, J. Wang, Finite-time stability of a class of oscillating systems with two delays, Math. Methods Appl. Sci., 41 (2018), 4943–4954
-
[14]
J. Cˇ ermák, T. Kisela, L. Nechvátal, DiscreteMittag-Leffler functions in linear fractional difference equations, Abstr. Appl. Anal., 2011 (2011), 21 pages
-
[15]
J. Cheng, Fractional difference equation theory, Xiamen University Press, Xiamen (2011)
-
[16]
C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 12 (2003), 692–703
-
[17]
J. Diblík, M. Feˇckan, M. Pospíšil, Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices, Ukrainian Math. J., 65 (2013), 64–76
-
[18]
J. Diblík, D. Y. Khusainov, Representation of solutions of linear discrete systems with constant coefficients and pure delay, Adv. Difference Equ., 2006 (2006), 13 pages
-
[19]
J. Diblík, D. Y. Khusainov, Representation of solutions of discrete delayed system x(k + 1) = Ax(k) + Bx(k −m) + f(k) with commutative matrices, J. Math. Anal. Appl., 318 (2006), 63–76
-
[20]
J. Diblík, D. Y. Khusainov, J. Lukáˇcová, M. Ružicková, Control of oscillating systems with a single delay, Adv. Difference Equ., 2010 (2010), 15 pages
-
[21]
J. Diblík, B. Morávková, Discrete matrix delayed exponential for two delays and its property, Adv. Difference Equ., 2013 (2013), 18 pages
-
[22]
J. Diblík, B. Morávková, Representation of the solutions of linear discrete systems with constant coefficients and two delays, Abstr. Appl. Anal., 2014 (2014), 19 pages
-
[23]
K. Diethelm, The analysis of fractional differential equations, Springer-Verlag, Berlin (2010)
-
[24]
F. Du, B. Jia, Finite-time stability of a class of nonlinear fractional delay difference systems, Appl. Math. Lett., 98 (2019), 233–239
-
[25]
F. Du, J.-G. Lu, Exploring a new discrete delayed Mittag-Leffler matrix function to investigate finite-time stability of Riemann-Liouville fractional-order delay difference systems, Math. Methods Appl. Sci., 45 (2022), 9856–9878
-
[26]
P. Eloe, Z. Ouyang, Multi-term linear fractional nabla difference equations with constant coefficients, Int. J. Difference Equ., 10 (2015), 91–106
-
[27]
C. Goodrich, A. C. Peterson, Discrete fractional calculus, Springer, Cham (2015)
-
[28]
N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann- Liouville fractional derivatives, Rheol. Acta, 45 (2006), 765–771
-
[29]
L.-L. Huang, G.-C. Wu, D. Baleanu, H.-Y. Wang, Discrete fractional calculus for interval–valued systems, Fuzzy Sets and Systems, 404 (2021), 141–158
-
[30]
B. Jia, L. Erbe, A. Peterson, Comparison theorems and asymptotic behavior of solutions of discrete fractional equations, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), 18 pages
-
[31]
B. Jia, L. Erbe, A. Peterson, Comparison theorems and asymptotic behavior of solutions of Caputo fractional equations, Int. J. Difference Equ., 11 (2016), 163–178
-
[32]
D. Y. Khusainov, J. Diblík, M. Ružicková, J. Lukácová, Representation of a solution of the Cauchy problem for an oscillating system with pure delay, Nonlinear Oscill., 11 (2008), 276–285
-
[33]
D. Y. Khusainov, G. V. Shuklin, Linear autonomous time-delay system with permutation matrices solving, Stud. Univ. Žilina Math. Ser., 17 (2003), 101–108
-
[34]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam (2006)
-
[35]
C. Li, D. Qian, Y. Chen, On Riemann-Liouville and Caputo derivatives, Discrete Dyn. Nat. Soc., 2011 (2011), 15 pages
-
[36]
M. Li, J. Wang, Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput., 324 (2018), 254–265
-
[37]
C. Liang, J. Wang, Analysis of iterative learning control for an oscillating control system with two delays, Trans. Inst. Meas. Control, 40 (2018), 1757–1765
-
[38]
C. Liang, J. Wang, D. O’Regan, Controllability of nonlinear delay oscillating systems, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 18 pages
-
[39]
C. Liang, J. Wang, D. O’Regan, Representation of a solution for a fractional linear system with pure delay, Appl. Math. Lett., 77 (2018), 72–78
-
[40]
C. Liang, W.Wei, J.Wang, Stability of delay differential equations via delayed matrix sine and cosine of polynomial degrees, Adv. Difference Equ., 2017 (2017), 17 pages
-
[41]
N. I. Mahmudov, Representation of solutions of discrete linear delay systems with non permutable matrices, Appl. Math. Lett., 58 (2018), 8–14
-
[42]
N. I. Mahmudov, Delayed perturbation of Mittag-Leffler functions and their applications to fractional linear delay differential equations, Math. Methods Appl. Sci., 42 (2019), 5489–5497
-
[43]
N. I. Mahmudov, , Appl. Math. Lett., 92 (2019), 41–48
-
[44]
N. I. Mahmudov, Delayed linear difference equations: the method of Z-transform, Electron. J. Qual. Theory Differ. Equ., 2020 (2020), 12 pages
-
[45]
N. I. Mahmudov, M. Aydın, Representation of solutions of nonhomogeneous conformable fractional delay differential equations, Chaos Solitons Fractals, 150 (2021), 8 pages
-
[46]
A. D. Obembe, M. Enamul Hossain, S. A. Abu-Khamsin, Variable-order derivative time fractional diffusion model for heterogeneous porous media, J. Pet. Sci. Eng., 152 (2017), 391–405
-
[47]
M. Pospíšil, Representation of solutions of delayed difference equations with linear parts given by pairwise permutable matrices via Z-transform, Appl. Math. Comput., 294 (2017), 180–194
-
[48]
N. H. Sweilam, S. M. Al-Mekhlafi, Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives, J. Adv. Res., 7 (2016), 271–283
-
[49]
V. E. Tarasov, Handbook of fractional calculus with applications,, De Gruyter, Berlin (2019)
-
[50]
G.-C. Wu, T. Abdeljawad, J. Liu, D. Baleanu, K.-T. Wu, Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique, Nonlinear Anal. Model. Control, 24 (2019), 919–936
-
[51]
G.-C. Wu, D. Baleanu, W.-H. Luo, Lyapunov functions for Riemann-Liouville-like fractional difference equations, Appl. Math. Comput., 314 (2017), 228–236
-
[52]
G.-C. Wu, D. Baleanu, S.-D. Zeng, Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion, Commun. Nonlinear Sci. Numer. Simul., 57 (2018), 299–308
-
[53]
G.-C. Wu, Z.-G. Deng, D. Baleanu, D.-Q. Zeng, New variable-order fractional chaotic systems for fast image encryption, Chaos, 29 (2019), 11 pages
