ArticleOpen Access

ON NUMERICAL AND THEORETICAL FINDINGS FOR FRACTAL-FRACTIONAL ORDER GENERALIZED DYNAMICAL SYSTEM

HAIDONG QU

Department of Mathematics, Hanshan Normal University, Chaozhou 515041, P. R. China

E-mail Address: qhaidong@163.com

Search for more papers by this author

MUHAMMAD ARFAN

Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan

E-mail Address: arfan_uom@yahoo.com

Search for more papers by this author

KAMAL SHAH

Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan

Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia

E-mail Address: kamalshah408@gmail.com

Search for more papers by this author

AMAN ULLAH

Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan

E-mail Address: amanswat@hotmail.com

Search for more papers by this author

THABET ABDELJAWAD

Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia

Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea

Department of Computer Science and Technology, Hanshan Normal University, ChaoZhou, Guangdong 521041, P. R. China

E-mail Address: tabdeljawad@psu.edu.sa

Corresponding author.

Search for more papers by this author

, and

GENGZHONG ZHANG

Department of Computer Science and Technology, Hanshan Normal University, ChaoZhou, Guangdong 521041, P. R. China

E-mail Address: zhenggz@hstc.edu.cn

Search for more papers by this author

https://doi.org/10.1142/S0218348X23400194Cited by:3 (Source: Crossref)

This article is part of the issue:

Special Issue on Applications of Wavelets and Fractals in Engineering Sciences
Guest Editors: K. S. Nisar, F. A. Shah, S. K. Upadhyay and P. E. T. Jorgensen

Abstract

In this paper, we consider a general system of fractal-fractional order derivative in Atangana–Baleanu–Caputo sense. On the application of fixed point approach, we establish sufficient conditions regarding existence and uniqueness of solution. The said requirements are obtained via using Krasnoselkii’s and Banach fixed results. Further via nonlinear analysis, some interesting results for Hyers–Ulam (HU)-type stability are also derived. To compute numerical solution for the proposed nonlinear system, fractal-fractional order Adams–Bashforth method is used. To support our findings, we give some test problems. Also by Matlab, we also present their graphical interpretation. The analysis of this paper is in generalized format which can be applied to any real problem. Each equation is investigated separately for the said characteristics.

Keywords:

References

1. C. B. Boyer and U. C. Merzbach, A History of Mathematics (John Wiley & Sons, New York, 2011). Google Scholar
2. M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci. 4(21) (2010) 1021–1032. Google Scholar
3. R. L. Magin, Fractional Calculus in Bioengineering (Begell House Redding, 2006). Google Scholar
4. R. Hilfer, Threefold Introduction to Fractional Derivatives (Wiley Online Library, New York, 2008). Crossref, Google Scholar
5. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier Science, Amsterdam, 2006). Google Scholar
6. J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional Calculus, Commun. Nonlin. Sci. Numer. Simul. 16(3) (2010) 1140–1153. Crossref, Web of Science, Google Scholar
7. J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in Fractional Calculus (Springer, Dordrecht, 2007). Crossref, Google Scholar
8. C. Goodrich, Existence of positive solution to a class of fractional differential equations, J. Comp. Math. Appl. 59 (2010) 3889–3999. Web of Science, Google Scholar
9. R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, Singapore, 2011). Link, Google Scholar
10. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Springer, New York, 2010). Crossref, Google Scholar
11. M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Prog. Fract. Differ. Appl. 2(1) (2016) 1–11. Crossref, Google Scholar
12. M. Caputo, Linear model of dissipation whose Q is almost frequency independent, Int. J. Geo. Sci. 13(5) (1967) 529–539. Crossref, Web of Science, Google Scholar
13. A. Atangana and S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Solitons Fractals 123 (2019) 320–337, https://doi.org/10.1016/j.chaos.2019.04.020. Crossref, Web of Science, Google Scholar
14. C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin Inst. 356(3) (2019) 1535–1565. Crossref, Web of Science, Google Scholar
15. K. S. Nisar, K. Jothimani, K. Kaliraj and C. Ravichandran, An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos Solitons Fractals 146 (2021) 110915. Crossref, Web of Science, Google Scholar
16. S. Kumar, A. Kumar, Z. Odibat, M. Aldhaifallah and K. S. Nisar, A comparison study of two modified analytical approach for the solution of nonlinear fractional shallow water equations in fluid flow, AIMS Math. 5(4) (2020) 3035–3055. Crossref, Web of Science, Google Scholar
17. K. S. Nisar, K. Logeswari, V. Vijayaraj, H. M. Baskonus and C. Ravichandran, Fractional order modeling the gemini virus in Capsicum annuum with optimal control, Fractal Fract. 6(2) (2022) 61. Crossref, Web of Science, Google Scholar
18. G. Rahman, T. Abdeljawad, F. Jarad, A. Khan and K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Difference Equations 2019(1) (2019) 454. Crossref, Web of Science, Google Scholar
19. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, J. Fract. Calc. Appl. 5(4) (2002) 367–386. Google Scholar
20. C. L. Wu, Analysis of an HIV-AIDS treatment model with non-linear incidence rate, Chaos Solitons Fractals 41(1) (2009) 175–182. Crossref, Web of Science, Google Scholar
21. P. Georgesco and Y. Hsieh, Global stability for a virus dynamics model with non-linear incidence of infection and removal, SIAM 67(2) (2006) 337–353. Google Scholar
22. M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci. 4(21) (2010) 1021–1032. Google Scholar
23. M. Rahim, Applications of fractional differential equations, Appl. Math. Sci. 4(50) (2010) 2453–2461. Google Scholar
24. A. Naghipour and J. Manafian, Application of the Laplace Adomian decomposition method and implicit methods for solving Burger’s equation, J. Pure. Appl. Math. 6(1) (2015) 68–77. Google Scholar
25. K. Jothimani, N. Valliammal and C. Ravichandran, Existence result for a neutral fractional integro-differential equation with state dependent delay, J. Appl. Nonlinear Dynam. 7(14) (2018) 371–381. Crossref, Google Scholar
26. C. Ravichandran, K. Munusamy, K. S. Nisar and N. Valliammal, Results on neutral partial integro differential equations using Monch–Krasnosel’Skii fixed point theorem with nonlocal conditions, Fractal Fract. 6(2) (2022) 75. Crossref, Web of Science, Google Scholar
27. K. Kavitha, V. Vijayakumar, R. Udhayakumar and C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control 24(3) (2022) 1406–1415. Crossref, Web of Science, Google Scholar
28. N. Valliammal and C. Ravichandran, Results on fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Nonlinear Studies 25(1) (2018) 159–171. Google Scholar
29. K. Shah, M. Arfan, A. Ullah, Q. Al-Mdallal, K. J. Ansari and T. Abdeljawad, Computational study on the dynamics of fractional order differential equations with applications. Chaos Solitons Fractals 157 (2022) 111955. Crossref, Web of Science, Google Scholar
30. S. C. Brailsford et al., An analysis of the academic literature on simulation and modeling in health care, J. Simul. 3(3) (2009) 130–140. Crossref, Web of Science, Google Scholar
31. K. Shah et al., Fractal-fractional mathematical model addressing the situation of corona virus in Pakistan, Res. Phys. 19 (2020) 103560. Google Scholar
32. B. Esmail, H. S. Goghary and S. Abbasbandym, Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Appl. Math. Comput. 161(3) (2005) 733–744. Web of Science, Google Scholar
33. H. Singh, Operational matrix approach for approximate solution of fractional model of Bloch equation, J. King Saud Univ.-Sci. 29(2) (2017) 235–240. Crossref, Web of Science, Google Scholar
34. H. Singh, R. Pandey and H. Srivastava, Solving non-linear fractional variational problems using Jacobi polynomials, Mathematics 7(3) (2019) 224. Crossref, Web of Science, Google Scholar
35. H. Singh and H. M. Srivastava, Numerical investigation of the fractional-order liénard and duffing equations arising in oscillating circuit theory, Front. Phys. 8 (2020) 120. Crossref, Web of Science, Google Scholar
36. H. Singh, M. R. Sahoo and O. P. Singh, Numerical method based on Galerkin approximation for the fractional advection-dispersion equation, Int. J. Appl. Comput. Math. 3(3) (2017) 2171–2187. Crossref, Google Scholar
37. M. Arfan, K. Shah, A. Ullah, M. Shutaywi, P. Kumam and Z. Shah, On fractional order model of tumor dynamics with drug interventions under nonlocal fractional derivative, Res. Phys. 21 (2021) 103783. Google Scholar
38. M. Arfan, K. Shah, A. Ullah, S. Salahshour, A. Ahmadian and M. Ferrara, A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform, Discr. Contin. Dynam. Syst. S 15(2) (2022) 315, https://doi.org/10.3934/dcdss.2021011. Crossref, Google Scholar
39. M. Ur. Rahman, M. Arfan, Z. Shah, P. Kumam and M. Shutaywi, Nonlinear fractional mathematical model of tuberculosis (TB) disease with incomplete treatment under Atangana–Baleanu derivative, Alexandria Eng. J. 60(3) (2021) 2845–2856. Crossref, Web of Science, Google Scholar
40. M. Arfan et al., Investigation of fractal-fractional order model of COVID-19 in Pakistan under Atangana–Baleanu–Caputo (ABC) derivative, Res. Phys. 24 (2021) 104046. Google Scholar
41. H. Alrabaiah, M. Arfan, K. Shah, I. Mahariq and A. Ullah, A comparative study of spreading of novel corona virus disease by using fractional order modified SEIR model, Alexandria Eng. J. 60(1) (2020) 573–585. Crossref, Web of Science, Google Scholar
42. M. Arfan, K. Shah, A. Ullah and T. Abdeljawad, Study of fuzzy fractional order diffusion problem under the Mittag-Leffler Kernel Law, Phys. Scr. 96(7) (2021) 07400. Crossref, Web of Science, Google Scholar
43. J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015) 87–92. Google Scholar
44. W. Chen, H. Sun, X. Zhang and D. Korošak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl. 59 (2010) 1754–1758. Crossref, Web of Science, Google Scholar
45. R. Kanno, Representation of random walk in fractal space-time, Physica A 248 (1998) 165–175. Crossref, Web of Science, Google Scholar
46. B. Ghanbari and K. S. Nisar, Some effective numerical techniques for chaotic systems involving fractal-fractional derivatives with different laws, Front. Phys. 8 (2020) 192. Crossref, Web of Science, Google Scholar
47. J. F. Gomez-Aguilar, T. Cordova-Fraga, T. Abdeljawad, A. Khan and H. Khan, Analysis of fractal-fractional malaria transmission model, Fractals 28(8) (2020) 2040041, https://doi.org/10.1142/S0218348X20400411. Link, Web of Science, Google Scholar
48. S. Qureshi and A. Atangana, Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data, Chaos Solitons Fractals 136 (2020) 109812. Crossref, Web of Science, Google Scholar
49. M. Arfan, H. Alrabaiah, M. ur Rahman, Y. L. Sun, A. S. Hashim, B. A. Pansera, A. Ahmadian and S. Salahshour, Investigation of fractal-fractional order model of COVID-19 in Pakistan under Atangana–Baleanu–Caputo (ABC) derivative, Res. Phys. 24 (2021) 104046. Google Scholar
50. A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?, Chaos Solitons Fractals 136 (2020) 109860. Crossref, Web of Science, Google Scholar
51. J. Singh, D. Kumar, D. Baleanu and S. Rathore, An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation, Appl. Math. Comput. 335 (2018) 12–24, https://doi.org/10.1016/j.amc.2018.04.025. Crossref, Web of Science, Google Scholar
52. S. Qureshi, A. Yusuf, A. A. Shaikh, M. Inc and D. Baleanu, Fractional modeling of blood ethanol concentration system with real data application, Chaos 29 (2019) 013143, https://doi.org/10.1063/1.5082907. Crossref, Web of Science, Google Scholar
53. V. Gill, J. Singh and Y. Singh, Analytical solution of generalized space-time fractional advection-dispersion equation via coupling of Sumudu and Fourier transforms, Front Phys. 6 (2019) 151, https://doi.org/10.3389/fphy.2018.00151. Crossref, Web of Science, Google Scholar
54. S. Qureshi, A. Atangana and A. A. Shaikh, Strange chaotic attractors under fractal fractional operators using newly proposed numerical methods, Eur. Phys. J Plus 134 (2019) 523, https://doi.org/10.1140/epjp/i2019-13003-7. Crossref, Web of Science, Google Scholar
55. S. Kumar, R. Kumar, J. Singh, K. Nisar and D. Kumar, An efficient numerical scheme for fractional model of HIV-1 infection of $C D 4_{+}$ T-cells with the effect of antiviral drug therapy, Alexandria Eng. J. 59(4) (2020) 2053–2064, https://doi.org/10.1016/j.aej.2019.12.046. Crossref, Web of Science, Google Scholar
56. H. W. Berhe, S. Qureshi and A. A. Shaikh, Deterministic modeling of dysentery diarrhea epidemic under fractional Caputo differential operator via real statistical analysis, Chaos Solitons Fractals 131 (2020) 109536, https://doi.org/10.1016/j.chaos.2019.109536. Crossref, Web of Science, Google Scholar
57. J. Singh, A new analysis for fractional rumor spreading dynamical model in a social network with Mittag-Leffler law, Chaos 29 (2019) 013137, https://doi.org/10.1063/1.5080691. Crossref, Web of Science, Google Scholar
58. S. Qureshi, N. A. Rangaig and D. Baleanu, New numerical aspects of Caputo–Fabrizio fractional derivative operator, Mathematics 7 (2019) 374, https://doi.org/10.3390/math7040374. Crossref, Web of Science, Google Scholar
59. H. Srivastava, D. Kumar and J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model. 45 (2017) 192–204, https://doi.org/10.1016/j.apm.2016.12.008. Crossref, Web of Science, Google Scholar
60. S. Qureshi, Effects of vaccination on measles dynamics under fractional conformable derivative with Liouville–Caputo operator, Eur. Phys. J. Plus 135 (2020) 63, https://doi.org/10.1140/epjp/s13360-020-00133-0. Crossref, Web of Science, Google Scholar
61. J. Solís-Pérez, J. Gómez-Aguilar and A. Atangana, Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws, Chaos Solitons Fractals 114 (2018) 175–85, https://doi.org/10.1016/j.chaos.2018.06.032. Crossref, Web of Science, Google Scholar
62. K. M. Owolabi, Computational study of noninteger order system of predation, Chaos 29 (2019) 013120, https://doi.org/10.1063/1.5079616. Crossref, Web of Science, Google Scholar
63. M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Physica A 525 (2019) 373–393, https://doi.org/10.1016/j.physa.2019.03.069. Crossref, Web of Science, Google Scholar
64. A. Jajarmi, B. Ghanbari and D. Baleanu, A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence, Chaos 29 (2019) 093111, https://doi.org/10.1063/1.5112177. Crossref, Web of Science, Google Scholar
65. B. G. Ghanbari and J. Gómez-Aguilar, Analysis of two avian influenza epidemic models involving fractal-fractional derivatives with power and Mittag-Leffler memories, Chaos 29 (2019) 123113, https://doi.org/10.1063/1.5117285. Crossref, Web of Science, Google Scholar
66. R. Ganji, H. Jafari and D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos Solitons Fractals 130 (2020) 109405, https://doi.org/10.1016/j.chaos.2019.109405. Crossref, Web of Science, Google Scholar
67. N. Kadkhoda and H. Jafari, An analytical approach to obtain exact solutions of some space-time conformable fractional differential equations, Adv. Differ. Equ. 2019 (2019) 428, https://doi.org/10.1186/s13662-019-2349-0. Crossref, Web of Science, Google Scholar
68. H. Jafari, A. Babaei and S. Banihashemi, A novel approach for solving an inverse reaction–diffusion–convection problem, J. Optim. Theory Appl. 183 (2019) 688–704, https://doi.org/10.1007/s10957-019-01576-x. Crossref, Web of Science, Google Scholar
69. B. Liu, D. J. Hill and G. Chen, Synchronization errors and uniform synchronization with an error bound for chaotic systems, Int. J. Bifur. Chaos 18 (2008) 3341–3354, https://doi.org/10.1142/S021812740802241X. Link, Web of Science, Google Scholar
70. F. Sorrentino, G. Barlev, A. B. Cohen and E. Ott, The stability of adaptive synchronization of chaotic systems, Chaos 20 (2010) 013103, https://doi.org/10.1063/1.3279646. Crossref, Web of Science, Google Scholar
71. B. F. Kazemi and H. Jafari, Error estimate of the MQ-RBF collocation method for fractional differential equations with Caputo–Fabrizio derivative, Math. Sci. 11 (2017) 297–305, https://doi.org/10.1007/s40096-017-0232-2. Crossref, Google Scholar
72. A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals 102 (2017) 396–406, https://doi.org/10.1016/j.chaos.2017.04.027. Crossref, Web of Science, Google Scholar
73. J. F. Gómez et al., Fractional Derivatives with Mittag-Leffler Kernel (Springer, 2019). Crossref, Google Scholar

Vol. 31, No. 02

Metrics

Downloaded 195 times

History

Received 18 February 2022

Accepted 28 July 2022

Published: March 10, 2023

Information

This is an Open Access article in the “Special Issue on Applications of Wavelets and Fractals in Engineering Sciences”, edited by K. S. Nisar (Prince Sattam bin Abdulaziz University, Saudi Arabia), F. A. Shah (University of Kashmir, India), S. K. Upadhyay (Indian Institute of Technology, BHU, India), P. E. T. Jorgensen (University of Iowa, USA) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 (CC BY-NC) License which permits use, distribution and reproduction in any medium, provided that the original work is properly cited and is used for non-commercial purposes.

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

ON NUMERICAL AND THEORETICAL FINDINGS FOR FRACTAL-FRACTIONAL ORDER GENERALIZED DYNAMICAL SYSTEM

Abstract

Recommended