Chaotic behavior and generation theorems of conformable time and space partial differential equations in specific Lebesgue spaces. (English) Zbl 07848751
Summary: This work aims to investigate the chaotic behavior of conformable partial differential equations. The main focus is on a specific conformable partial differential equation involving both time and space derivatives. To achieve this objective, the study begins by developing the theory of conformable Sobolev spaces, which provides a suitable framework for analyzing the operator associated with the proposed conformable partial differential equation. The investigation further utilizes the concept of conformable semigroups to establish a generation theorem for the solutions of the conformable partial differential equation. Drawing inspiration from previous research on chaos in semigroups, the work introduces a characterization of chaos specific to conformable equations. This characterization allows for a thorough analysis of the chaotic behavior associated with the semigroup generated by the proposed model.
MSC:
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
| 35R10 | Partial functional-differential equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 35P05 | General topics in linear spectral theory for PDEs |
Keywords:
chaos; conformable semigroup; conformable partial differential equation; conformable Sobolev spaceReferences:
| [1] | Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279, 57-66, 2015 · Zbl 1304.26004 · doi:10.1016/j.cam.2014.10.016 |
| [2] | Abdeljawad, T., Al Horani, M., Khalil, R.: Conformable fractional semigroups of operators. J. Semigroup Theory Appl., 2015 , Article-ID (2015) |
| [3] | Allahverdiev, BP; Tun, H.; Yaçlinkaya, Y., A completeness theorem for dissipative conformable fractional Sturm-Liouville operator in singular case, Filomat, 36, 7, 2461-2474, 2022 · doi:10.2298/FIL2207461A |
| [4] | Alaoui, AL; Azroul, E.; Hamou, AA, Monotone iterative technique for nonlinear periodic time fractional parabolic problems, Advances in the Theory of Nonlinear Analysis and its Application, 4, 3, 194-213, 2020 · doi:10.31197/atnaa.770669 |
| [5] | Al-Sharif, S., Al-Shorman, R.: New Results on the Generator of Conformable Semigroups |
| [6] | Atangana, A., Baleanu, D., Alsaedi, A.: New properties of conformable derivative. Open Mathematics 13, no. 1, 000010151520150081 (2015) · Zbl 1354.26008 |
| [7] | Atangana, A.; Khan, MA, Validity of fractal derivative to capturing chaotic attractors, Chaos. Solit. Fract., 126, 50-59, 2019 · Zbl 1448.34010 · doi:10.1016/j.chaos.2019.06.002 |
| [8] | Çenesiz, Y.; Kurt, A., The solutions of time and space conformable fractional heat equations with conformable Fourier transform, Acta Univ. Sapientiae Math., 7, 2, 130-140, 2015 · Zbl 1398.35264 |
| [9] | Chaudhary, M.; Kumar, R.; Singh, MK, Fractional convection-dispersion equation with conformable derivative approach, Chaos. Solit. Fract., 141, 2020 · Zbl 1496.35421 · doi:10.1016/j.chaos.2020.110426 |
| [10] | Desch, W.; Schappacher, W.; Webb, GF, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dyn. Syst., 17, 4, 793-819, 1997 · Zbl 0910.47033 · doi:10.1017/S0143385797084976 |
| [11] | Grosse-Erdmann, KG; Manguillot, AP, Linear Chaos, 2011, New York: Springer, New York · Zbl 1246.47004 · doi:10.1007/978-1-4471-2170-1 |
| [12] | Khalil, R.; Al Horani, M.; Youssef, A.; Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70, 2014 · Zbl 1297.26013 · doi:10.1016/j.cam.2014.01.002 |
| [13] | Lorenzo, C.F., Hartley, T.T., Qammar, H.K.: Chaos in a Fractional Order Chua System. No. NASA-TP-3543 (1996) |
| [14] | Martínez, F.; Martínez, I.; Kaabar, MK; Paredes, S., New results on complex conformable integral, AIMS Math., 5, 6, 7695-7710, 2020 · Zbl 1484.30002 · doi:10.3934/math.2020492 |
| [15] | Martínez, F.; Martínez, I.; Kaabar, MK; Paredes, S., Some new results on conformable fractional power series, Asia Pac. J. Math., 7, 31, 1-14, 2020 |
| [16] | Pazzy, A.: Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44 (1983) · Zbl 0516.47023 |
| [17] | Rabhi, L.; Al Horani, M.; Khalil, R., Existence results of mild solutions for nonlocal fractional delay integro-differential evolution equations via Caputo conformable fractional derivative, AIMS Math., 7, 7, 11614-11634, 2022 · doi:10.3934/math.2022647 |
| [18] | Sene, N., Solutions for some conformable differential equations, Progr. Fract. Differ. Appl, 4, 4, 493-501, 2018 |
| [19] | Wang, Y., Zhou, J., Li, Y.: Fractional Sobolev’s spaces on time scales via conformable fractional calculus and their application to a fractional differential equation on time scales. Adv. Math. Phys. (2016) · Zbl 1366.46023 |
| [20] | Yeh, J., Real Analysis: Theory of Measure and Integration, 2006, Singapore: World Scientific Publishing Company, Singapore · Zbl 1098.28002 · doi:10.1142/6023 |
| [21] | Zaslavsky, GM, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 6, 461-580, 2002 · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9 |
| [22] | Zhou, HW; Yang, S.; Zhang, SQ, Conformable derivative approach to anomalous diffusion, Phys. A, 491, 1001-1013, 2018 · Zbl 1514.60116 · doi:10.1016/j.physa.2017.09.101 |
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.
