Construction a distributed order smoking model and its nonstandard finite difference discretization

Mehmet Kocabiyik; Mevlüde Yakit Ongun; Mehmet Kocabiyik; Mevlüde Yakit Ongun

doi:10.3934/math.2022258

AIMS Mathematics

2022, Volume 7, Issue 3: 4636-4654. doi: 10.3934/math.2022258

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Research article

Construction a distributed order smoking model and its nonstandard finite difference discretization

Mehmet Kocabiyik ^{1,2
,
,},
Mevlüde Yakit Ongun ²

1.
Department of Mathematics, Burdur Mehmet Akif Ersoy University, Burdur, Turkey
2.
Department of Mathematics, Süleyman Demirel University, Isparta, Turkey

Received: 27 September 2021 Revised: 16 December 2021 Accepted: 20 December 2021 Published: 23 December 2021
MSC : 26A33, 65L07, 97M60

Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.
- distributed order fractional differential equation,
- nonstandard finite difference method,
- smoking model,
- numerical analysis,
- discretization
Citation: Mehmet Kocabiyik, Mevlüde Yakit Ongun. Construction a distributed order smoking model and its nonstandard finite difference discretization[J]. AIMS Mathematics, 2022, 7(3): 4636-4654. doi: 10.3934/math.2022258

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Abstract

Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.

References

[1]	A. Ahmad, M. Farman, F. Yasin, M. O. Ahmad, Dynamical transmission and effect of smoking in society, Int. J. Adv. Appl. Sci., 5 (2018), 71–75. https://doi.org/10.21833/ijaas.2018.02.012 doi: 10.21833/ijaas.2018.02.012
[2]	R. Amin, B. Alshahrani, M. Mahmoud, A. H. Abdel-Aty, K. Shah, W. Deebani, Haar wavelet method for solution of distributed order time-fractional differential equations, Alex. Eng. J., 60 (2021), 3295–3303. https://doi.org/10.1016/j.aej.2021.01.039 doi: 10.1016/j.aej.2021.01.039
[3]	H. Aminikhah, A. R. Sheikhani, H. Rezazadeh, Stability analysis of distributed order fractional Chen system, Sci. World J., 2013 (2013), 645080. https://doi.org/10.1155/2013/645080 doi: 10.1155/2013/645080
[4]	H. Aminikhah, A. H. R. Sheikhani, H. Rezazadeh, Approximate analytical solutions of distributed order fractional Riccati differential equation, Ain Shams Eng. J., 9 (2018), 581–588. https://doi.org/10.1016/j.asej.2016.03.007 doi: 10.1016/j.asej.2016.03.007
[5]	R. L. Bagley, P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-Part I, Int. J. Appl. Math., 2 (2000), 865–882.
[6]	R. L. Bagley, P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-Part II, Int. J. Appl. Math., 2 (2000), 965–988.
[7]	A. Boudaoui, Y. El hadj Moussa, Z. Hammouch, S. Ullah, A fractional-order model describing the dynamics of the novel coronavirus (COVID-19) with nonsingular kernel, Chaos Soliton. Fract., 146 (2021), 110859. https://doi.org/10.1016/j.chaos.2021.110859 doi: 10.1016/j.chaos.2021.110859
[8]	M. Caputo, Elasticita e dissipazione, Bologna: Zanichelli, 1969.
[9]	M. Caputo, Mean fractional-order-derivatives differential equations and filters, Annali dell'Universita di Ferrara, 41 (1995), 73–84.
[10]	M. Caputo, Distributed order differential equations modeling dielectric induction and diffusion, Fract. Calc. Appl. Anal., 4 (2001), 421–442.
[11]	C. Castillo-Garsow, G. Jordan-Salivia, A. Rodriguez-Herrera, Mathematical models for the dynamics of tobacco use, recovery and relapse, USA: Cornell University, 1997.
[12]	K. Diethelm, N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math., 225 (2009), 96–104. https://doi.org/10.1016/j.cam.2008.07.018 doi: 10.1016/j.cam.2008.07.018
[13]	D. T. Dimitrov, H. V. Kojouharovb, Nonstandard finite difference methods for predator prey models with general functional response, Math. Comput. Simulat., 78 (2008), 1–11. https://doi.org/10.1016/j.matcom.2007.05.001 doi: 10.1016/j.matcom.2007.05.001
[14]	L. Dorciak, Numerical models for simulation the fractional order control systems, The Academy of Sciences Institute of Experimental Physic, Kosiice, Slovak Republic, 1994.
[15]	V. S. Erturk, G. Zaman, S. Momani, A numeric analytic method for approximating a giving up smoking model containing fractional derivatives, Comput. Math. Appl., 64 (2012), 3065–3074. https://doi.org/10.1016/j.camwa.2012.02.002 doi: 10.1016/j.camwa.2012.02.002
[16]	N. J. Ford, M. L. Morgado, Distributed order equations as boundary value problems, Comput. Math. Appl., 64 (2012), 2973–2981. https://doi.org/10.1016/j.camwa.2012.01.053 doi: 10.1016/j.camwa.2012.01.053
[17]	T. T. Hartley, C. F. Lorenzo, Fractional system identification: an approach using continuous order distributions, National Aeronautics and Space Administration, 1999.
[18]	F. Haq, K. Shah, G. ur Rahman, M. Shahzad, Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alex. Eng. J., 57 (2018), 1061–1069. https://doi.org/10.1016/j.aej.2017.02.015 doi: 10.1016/j.aej.2017.02.015
[19]	T. Hussain, A. U. Awan, K. A. Abro, M. Ozair, M. Manzoor, A mathematical and parametric study of epidemiological smoking model: a deterministic stability and optimality for solutions, Eur. Phys. J. Plus, 136 (2021), 11. https://doi.org/10.1140/epjp/s13360-020-00979-4 doi: 10.1140/epjp/s13360-020-00979-4
[20]	J. T. Katsikadelis, Numerical solution of distributed order fractional differential equations, J. Comput. Phys., 259 (2014), 11–22. https://doi.org/10.1016/j.jcp.2013.11.013 doi: 10.1016/j.jcp.2013.11.013
[21]	A. A. Khan, R. Amin, S. Ullah, W. Sumelka, M. Altanji, Numerical simulation of a Caputo fractional epidemic model for the novel coronavirus with the impact of environmental transmission, Alex. Eng. J., 2021 (2021), In press. https://doi.org/10.1016/j.aej.2021.10.008 doi: 10.1016/j.aej.2021.10.008
[22]	X. Y. Li, B. Y. Wu, A numerical method for solving distributed order diffusion equations, Appl. Math. Lett., 53 (2016), 92–99. https://doi.org/10.1016/j.aml.2015.10.009 doi: 10.1016/j.aml.2015.10.009
[23]	Y. M. Li, S. Ullah, M. A. Khan, M. Y. Alshahrani, T. Muhammad, Modeling and analysis of the dynamics of HIV/AIDS with non-singular fractional and fractal-fractional operators, Phys. Scr., 96 (2021), 114008.
[24]	J. H. Lubin, N. E. Caporaso, Cigarette smoking, and lung cancer: Modeling total exposure and intensity, Cancer Epidemiol. Biomarkers Prev., 15 (2006), 517–523. https://doi.org/10.1158/1055-9965.EPI-05-0863 doi: 10.1158/1055-9965.EPI-05-0863
[25]	Y. Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12 (2009), 409–422.
[26]	M. M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77. https://doi.org/10.1016/j.cam.2004.01.033 doi: 10.1016/j.cam.2004.01.033
[27]	R. E. Mickens, Exact solutions to a finite difference model of a nonlinear reaction advection equation: implications for numerical analysis, Numer. Method. Part. Differ. Equ., 5 (1989), 313–325. https://doi.org/10.1002/num.1690050404 doi: 10.1002/num.1690050404
[28]	R. E. Mickens, Applications of nonstandard finite difference schemes, Atlanta, Ga, USA: World Scientific Publishing, 1999.
[29]	R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl., 8 (2002), 823–847. https://doi.org/10.1080/1023619021000000807 doi: 10.1080/1023619021000000807
[30]	L. P. Liu, D. P. Clemence, R. E. Mickens, A nonstandard finite difference scheme for contaminant transport with kinetic Langmuir sorption, Numer. Method. Part. Differ. Equ., 27 (2011), 767–785. https://doi.org/10.1002/num.20551 doi: 10.1002/num.20551
[31]	M. L. Morgado, M. Rebelo, Numerical approximation of distributed order reaction–diffusion equations, J. Comput. Appl. Math., 275 (2015), 216–227. https://doi.org/10.1016/j.cam.2014.07.029 doi: 10.1016/j.cam.2014.07.029
[32]	M. Y. Ongun, D. Arslan, R. Garrappa, Nonstandard finite difference schemes for a fractional-order Brusselator system, Adv. Differ. Equ., 2013 (2013), 102. https://doi.org/10.1186/1687-1847-2013-102 doi: 10.1186/1687-1847-2013-102
[33]	M. Y. Ongun, N. Ozdogan, A nonstandard numerical scheme for a predator-prey model with Allee effect, J. Nonlinear Sci. Appl., 10 (2017), 713–723. http://dx.doi.org/10.22436/jnsa.010.02.32 doi: 10.22436/jnsa.010.02.32
[34]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 1999.
[35]	A. Refahi, A. Ansari, H. S. Najafi, F. Mehrdoust, Analytic study on linear system of distributed order fractional differential equations, Le Matematiche, 67 (2012), 313. https://doi.org/10.4418/2012.67.2.1 doi: 10.4418/2012.67.2.1
[36]	H. Richter, The generalized Henon maps: examples for higher dimensional chaos, Int. J. Bifurcat. Chaos, 12 (2002), 1371–1384. https://doi.org/10.1142/S0218127402005121 doi: 10.1142/S0218127402005121
[37]	H. S. Najafi, A. R. Sheikhani, A. Ansari, Stability analysis of distributed order fractional differential equations, Abstr. Appl. Anal., 2011 (2011), 175323. https://doi.org/10.1155/2011/175323 doi: 10.1155/2011/175323
[38]	O. Sharomi, A. B. Gumel, Curtailing smoking dynamics: a mathematical modeling approach, Appl. Math. Comput., 195 (2008), 475–499. https://doi.org/10.1016/j.amc.2007.05.012 doi: 10.1016/j.amc.2007.05.012
[39]	X. Y. Shi, G. Li, X. Y. Zhou, X. Y. Song, Analysis of a differential equation model of HIV infection of $CD4^{+}$ T-cells with saturated reverse function, Turk. J. Math., 35 (2011), 649–666. http://doi.org/10.3906/mat-1006-333 doi: 10.3906/mat-1006-333
[40]	J. Singh, D. Kumar, M. A. Qurashi, D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Differ. Equ., 2017 (2017), 88. https://doi.org/10.1186/s13662-017-1139-9 doi: 10.1186/s13662-017-1139-9
[41]	S. Ucar, E. Ucar, N. Ozdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana Baleanu derivative, Chaos Soliton. Fract., 118 (2019), 300–306. https://doi.org/10.1016/j.chaos.2018.12.003 doi: 10.1016/j.chaos.2018.12.003
[42]	S. Ucar, Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives, DCDS-S, 14 (2021), 2571–-2589. http://doi.org/10.3934/dcdss.2020178 doi: 10.3934/dcdss.2020178
[43]	B. M. Vinagre, Y. Q. Chen, I. Petras, Two direct Tustin discretization methods for fractional order differentiator integrator, J. franklin I, 340 (2003), 349–362. http://doi.org/10.1016/j.jfranklin.2003.08.001 doi: 10.1016/j.jfranklin.2003.08.001
[44]	G. Zaman, Qualitative behavior of giving up smoking models, B. Malays. Math.l Sci. Soc., 34 (2011), 403–415.

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