×

Caputo and related fractional derivatives in singular systems. (English) Zbl 1427.34007

Summary: By using the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo-Fabrizio (CF) and the Atangana-Baleanu (AB) fractional derivative, firstly we focus on singular linear systems of fractional differential equations with constant coefficients that can be non-square matrices, or square & singular. We study existence of solutions and provide formulas for the case that there do exist solutions. Then, we study the existence of unique solution for given initial conditions. Several numerical examples are given to justify our theory.

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
39A12 Discrete version of topics in analysis
93C05 Linear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] L. Abadias, C. Lizama, P. J. Miana, M. P. Velasco, On well-posedness of vector-valued fractional differential - difference equations, 2016. arXiv:1606.05237; L. Abadias, C. Lizama, P. J. Miana, M. P. Velasco, On well-posedness of vector-valued fractional differential - difference equations, 2016. arXiv:1606.05237 · Zbl 1405.47003
[2] Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction- diffusion equation, Appl. Math. Comput., 273, 948-956 (2016) · Zbl 1410.35272
[3] Atangana, A.; Nieto, J. J., Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7, 10, 1-7 (2015)
[4] Atangana, A.; Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Sci. Int. Sci. J. (2016)
[5] Baleanu, D.; Burhanettin, G. Z.; Machado, J. A.T., New trends in Nanotechnology and Fractional Calculus Applications (2010), Springer: Springer New York, NY, USA · Zbl 1196.65021
[6] Bonilla, B.; Rivero, M.; Trujillo, J. J., On systems of linear fractional differential equations with constant coefficients., Appl. Math. Comput., 187.1, 68-78 (2007) · Zbl 1121.34006
[7] Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Progress Fract. Differ. Appl., 1, 2 (2015)
[8] Caputo, M.; Fabrizio, M., Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, 2, 2 (2016)
[9] Dai, L., Singular control systems, (Thomas, M.; Wyner, A., Lecture Notes in Control and Information Sciences (1988))
[10] Dassios, I. K., Optimal solutions for non-consistent singular linear systems of fractional nabla difference equations, Circuits Syst. Signal Process., 34, 6, 1769-1797 (2015) · Zbl 1341.93038
[11] Dassios, I. K.; Baleanu, D. I., Duality of singular linear systems of fractional nabla difference equations., Appl. Math. Model., 39, 14, 4180-4195 (2015) · Zbl 1443.65436
[12] Dassios, I. K.; Baleanu, D., On a singular system of fractional nabla difference equations with boundary conditions,, Bound. Value Probl., 2013, 148 (2013) · Zbl 1296.39002
[13] Dassios, I.; Baleanu, D.; Kalogeropoulos, G., On non-homogeneous singular systems of fractional nabla difference equations,, Appl. Math. Comput., 227, 112-131 (2014) · Zbl 1364.39006
[14] Dassios, I., Geometric relation between two different types of initial conditions of singular systems of fractional nabla difference equations, Math. Methode Appl. Sci., 40, 17, 6085-6095 (2017) · Zbl 1391.39009
[15] Dassios, I., Stability and robustness of singular systems of fractional nabla difference equations, Circuits Syst. Signal Process., 36, 1, 49-64 (2017) · Zbl 1368.93592
[16] Dassios, I., A practical formula of solutions for a family of linear non-autonomous fractional nabla difference equations,, J. Comput. Appl. Math., 339, 317-328 (2018) · Zbl 1464.65284
[17] Gantmacher, R. F., The Theory of Matrices I, II (1959), Chelsea, New York · Zbl 0085.01001
[18] Ghorbanian, V.; Rezapour, S., On a system of fractional finite difference inclusions, Adv. Differ. Equ., 2017, 1, 325 (2017) · Zbl 1444.39008
[19] R. Hilfe (Ed.), Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. 463; R. Hilfe (Ed.), Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. 463 · Zbl 0998.26002
[20] Ionescu, C.; Lopes, A.; Copot, D.; Machado, J. A.T.; Bates, J. H.T., The role of fractional calculus in modeling biological phenomena: a review, Commun. Nonlinear Sci. Numer. Simul., 51, 141-159 (2017) · Zbl 1467.92050
[21] Kalogeropoulos, G. I., Matrix pencils and linear systems (1985), City University, London, (Ph.D. thesis)
[22] Kaczorek, T., Fractional continuous-time linear systems., Selected Problems of Fractional Systems Theory, 27-52 (2011), Springer: Springer Berlin Heidelberg
[23] Li, W. N.; Sheng, W., Sufficient conditions for oscillation of a nonlinear fractional nabla difference system, SpringerPlus, 5, 1, 1178 (2016)
[24] Liu, Y.; Wang, J.; Gao, C.; Gao, Z.; Wu, X., On stability for discrete-time non-linear singular systems with switching actuators via average dwell time approach, Trans. Inst. Measure. Control, 39, 12, 1771-1776 (2017)
[25] Lizama, C., The poisson distribution, abstract fractional difference equations, and stability. Lizama, c., Proc. Am. Math. Society, 145, 9, 3809-3827 (2017) · Zbl 1368.39001
[26] Lizama, C.; Murillo?Arcila, M.; Leal, C., Lebesgue regularity for differential difference equations with fractional damping, Math. Methods Appl. Sci., 41, 7, 2535-2545 (2018) · Zbl 1391.35405
[27] Lv, W., Existence and uniqueness of solutions for a discrete fractional mixed type sum-difference equation boundary value problem., Discrete Dyn. Nature Soc., 501, 376261 (2015) · Zbl 1418.39016
[28] Milano, F.; Dassios, I., Primal and dual generalized eigenvalue problems for power systems small-signal stability analysis, IEEE Trans. Power Syst., 32, 6, 4626-4635 (2017)
[29] Milano, F.; Dassios, I., Small-signal stability analysis for non-index 1 Hessenberg form systems of delay differential-algebraic equations, IEEE Trans. Circuits Syst. I Regul. Pap., 63, 9, 1521-1530 (2016) · Zbl 1468.94741
[30] MingLiang, Z., Noether symmetry theory of fractional order constrained hamiltonian systems based on a fractional factor, Karbala Int. J. Modern Sci., 4, 1, 180-186 (2018)
[31] Ozturk, O., A study of ∇-discrete fractional calculus operator on the radial Schrödinger equation for some physical potentials, Quaest. Math., 40, 7, 879-889 (2017) · Zbl 1423.34015
[32] O. Ozturk, Discrete fractional solutions of a physical differential equation via ∇-DFC operator., 2018. arXiv:1803.05016; O. Ozturk, Discrete fractional solutions of a physical differential equation via ∇-DFC operator., 2018. arXiv:1803.05016
[33] Rahmat, M. R.; Noorani, M. S., Caputo type fractional difference operator and its application on discrete time scales., Adv. Differ. Equ., 2015, 1, 1-15 (2015) · Zbl 1422.26008
[34] Rezapour, S.; Salehi, S., On the existence of solution for a \(k\)-dimensional system of three points nabla fractional finite difference equations, Bull. Iran. Math. Soc., 41, 6, 1433-1444 (2015) · Zbl 1373.39007
[35] Rugh, W. J., Linear System Theory (1996), Prentice Hall International (Uk): Prentice Hall International (Uk) London · Zbl 0892.93002
[36] Wei, Y.; Peter, W. T.; Yao, Z.; Wang, Y., The output feedback control synthesis for a class of singular fractional order systems, ISA Trans., 69, 1-9 (2017)
[37] Wu, G. C.; Baleanu, D.; Luo, W. H., Lyapunov functions for Riemann-Liouville like fractional difference equations, Appl. Math. Comput., 314, 228-236 (2017) · Zbl 1426.39010
[38] Yin, C.; Zhong, S. M.; Huang, X.; Cheng, Y., Robust stability analysis of fractional-order uncertain singular nonlinear system with external disturbance., Appl. Math. Comput., 269, 2015, 351-362 (2015) · Zbl 1410.93100
[39] Xin, B.; Liu, L.; Hou, G.; Ma, Y., Chaos synchronization of nonlinear fractional discrete dynamical systems via linear control, Entropy, 19, 7, 351 (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.