A numerical study of fractional population growth and nuclear decay model

Sara S. Alzaid; Pawan Kumar Shaw; Sunil Kumar; Sara S. Alzaid; Pawan Kumar Shaw; Sunil Kumar

doi:10.3934/math.2022637

AIMS Mathematics

2022, Volume 7, Issue 6: 11417-11442. doi: 10.3934/math.2022637

Previous Article Next Article

Research article Special Issues

A numerical study of fractional population growth and nuclear decay model

1.
Department of Mathematics, College of Science, King Saud University, P.O. Box 1142, Riyadh 11989, Saudi Arabia
2.
Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India
3.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
4.
Department of Mathematics, University Centre for Research and Development, Chandigarh University, Gharuan, Mohali, Punjab, India

Received: 04 February 2022 Revised: 11 March 2022 Accepted: 18 March 2022 Published: 12 April 2022
MSC : 26A33, 34A08, 93C10, 93C15, 78A70

This paper is devoted to solving the initial value problem (IVP) of the fractional differential equation (FDE) in Caputo sense for arbitrary order $ \beta\in(0, 1] $. Based on a few examples and application models, the main motivation is to show that FDE may model more effectively than the ordinary differential equation (ODE). Here, two cubic convergence numerical schemes are developed: the fractional third-order Runge-Kutta (RK3) scheme and fractional strong stability preserving third-order Runge-Kutta (SSRK3) scheme. The approximated solution is derived without taking any assumption of perturbations and linearization. The schemes are presented, and the convergence of the schemes is established. Also, a comparative study has been done of our proposed scheme with fractional Euler method (EM) and fractional improved Euler method (IEM), which has linear and quadratic convergence rates, respectively. Illustrative examples and application examples with the numerical comparison between the proposed scheme, the exact solution, EM, and IEM are given to reveal our scheme's accuracy and efficiency.
- fractional calculus,
- FDEs,
- fractional Euler method,
- fractional improved Euler method,
- IVP,
- Runge-Kutta method,
- strong stability preserving,
- population growth model
Citation: Sara S. Alzaid, Pawan Kumar Shaw, Sunil Kumar. A numerical study of fractional population growth and nuclear decay model[J]. AIMS Mathematics, 2022, 7(6): 11417-11442. doi: 10.3934/math.2022637

Related Papers:

Abstract

This paper is devoted to solving the initial value problem (IVP) of the fractional differential equation (FDE) in Caputo sense for arbitrary order $ \beta\in(0, 1] $. Based on a few examples and application models, the main motivation is to show that FDE may model more effectively than the ordinary differential equation (ODE). Here, two cubic convergence numerical schemes are developed: the fractional third-order Runge-Kutta (RK3) scheme and fractional strong stability preserving third-order Runge-Kutta (SSRK3) scheme. The approximated solution is derived without taking any assumption of perturbations and linearization. The schemes are presented, and the convergence of the schemes is established. Also, a comparative study has been done of our proposed scheme with fractional Euler method (EM) and fractional improved Euler method (IEM), which has linear and quadratic convergence rates, respectively. Illustrative examples and application examples with the numerical comparison between the proposed scheme, the exact solution, EM, and IEM are given to reveal our scheme's accuracy and efficiency.

References

[1]	I. Podlubny, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 198 (1999), 1–340.
[2]	R. L. Bagley, P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23 (1985), 918–925. https://doi.org/10.2514/3.9007 doi: 10.2514/3.9007
[3]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models, World Scientific, 2010. https://doi.org/10.1142/9781848163300
[4]	J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
[5]	C. A. Cruz-López, G. Espinosa-Parede, Fractional radioactive decay law and Bateman equations, Nucl. Eng. Technol., 54 (2022), 275–282. https://doi.org/10.1016/j.net.2021.07.026 doi: 10.1016/j.net.2021.07.026
[6]	C. A. Cruz-López, G. Espinosa-Paredes, J. Luis François, Development of the General Bateman Solution using fractional calculus: A theoretical and algorithmic approach, Comput. Phys. Commun., 273 (2022), 108268. https://doi.org/10.1016/j.cpc.2021.108268 doi: 10.1016/j.cpc.2021.108268
[7]	Z. Altawallbeh, M. Al-Smadi, I. Komashynska, A. Ateiwi, Numerical solutions of fractional systems of two-point BVPs by using the iterative reproducing kernel algorithm, Ukr. Math. J., 70 (2018), 687–701. https://doi.org/10.1007/s11253-018-1526-8 doi: 10.1007/s11253-018-1526-8
[8]	K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional adams method, Numer. Algorithms, 36 (2004), 31–52. https://doi.org/10.1023/B:NUMA.0000027736.85078.be doi: 10.1023/B:NUMA.0000027736.85078.be
[9]	V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677–2682. https://doi.org/10.1016/j.na.2007.08.042 doi: 10.1016/j.na.2007.08.042
[10]	R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal., 72 (2010), 2859–2862. https://doi.org/10.1016/j.na.2009.11.029 doi: 10.1016/j.na.2009.11.029
[11]	R. P. Agarwal, D. O'Regan, S. Staněk, Positive solutions for dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371 (2010), 57–68. https://doi.org/10.1016/j.jmaa.2010.04.034 doi: 10.1016/j.jmaa.2010.04.034
[12]	A. Yakar, M. E. Koksal, Existence results for solutions of nonlinear fractional differential equations, Abstr. Appl. Anal., 2012 (2012). https://doi.org/10.1155/2012/267108 doi: 10.1155/2012/267108
[13]	V. Daftardar-Gejji, A. Babakhani, Analysis of a system of fractional differential equations, J. Math. Anal. Appl., 293 (2004), 511–522. https://doi.org/10.1016/j.jmaa.2004.01.013 doi: 10.1016/j.jmaa.2004.01.013
[14]	G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501–544. https://doi.org/10.1016/0022-247X(88)90170-9 doi: 10.1016/0022-247X(88)90170-9
[15]	J. He, Variational iteration method for delay differential equations, Commun. Nonlinear Sci., 2 (1997), 235–236. https://doi.org/10.1016/S1007-5704(97)90008-3 doi: 10.1016/S1007-5704(97)90008-3
[16]	S. Momani, A. Freihat, M. AL-Smadi, Analytical study of fractional-order multiple chaotic FitzHugh-Nagumo neurons model using multistep generalized differential transform method, Abstr. Appl. Anal., 2014 (2014). https://doi.org/10.1155/2014/276279 doi: 10.1155/2014/276279
[17]	M. M. Khader, S. Kumar, An efficient computational method for solving a system of FDEs via fractional finite difference method, Appl. Appl. Math., 14 (2019).
[18]	A. Hemeda, Homotopy perturbation method for solving systems of nonlinear coupled equations, Appl. Math. Sci., 6 (2012), 4787–4800.
[19]	E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, A chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62 (2011), 2364–2373. https://doi.org/10.1016/j.camwa.2011.07.024 doi: 10.1016/j.camwa.2011.07.024
[20]	K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms, 16 (1997), 231–253. https://doi.org/10.1023/A:1019147432240 doi: 10.1023/A:1019147432240
[21]	I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional ivps, Commun. Nonlinear Sci., 14 (2009), 674–684. https://doi.org/10.1016/j.cnsns.2007.09.014 doi: 10.1016/j.cnsns.2007.09.014
[22]	S. S. Ray, R. Bera, Analytical solution of the bagley torvik equation by adomian decomposition method, Appl. Math. Comput., 168 (2005), 398–410. https://doi.org/10.1016/j.amc.2004.09.006 doi: 10.1016/j.amc.2004.09.006
[23]	S. Momani, K. Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162 (2005), 1351–1365. https://doi.org/10.1016/j.amc.2004.03.014 doi: 10.1016/j.amc.2004.03.014
[24]	J. H. He, Variational iteration method some recent results and new interpretations, J. Comput. Appl. Math., 207 (2007), 3–17. https://doi.org/10.1016/j.cam.2006.07.009 doi: 10.1016/j.cam.2006.07.009
[25]	C. Runge, Ü about the numerical solution ö solution of differential equations, Math. Ann., 46 (1895), 167–178.
[26]	W. Kutta, Contribution to approximate integration of total differential equations, Z. Math. Phys., 46 (1901), 435–453.
[27]	J. C. Butcher, A history of runge-kutta methods, Appl. Numer. Math., 20 (1996), 247–260. https://doi.org/10.1016/0168-9274(95)00108-5 doi: 10.1016/0168-9274(95)00108-5
[28]	D. Evans, New runge-kutta methods for initial value problems, Appl. Math. Lett., 2 (1989), 25–28. https://doi.org/10.1016/0893-9659(89)90109-2 doi: 10.1016/0893-9659(89)90109-2
[29]	M. S. Arshad, D. Baleanu, M. B. Riaz, M. Abbas, A novel 2-stage fractional Runge-Kutta method for a time-fractional logistic growth model, Discrete Dyn. Nat. Soc., 2020 (2020). https://doi.org/10.1155/2020/1020472 doi: 10.1155/2020/1020472
[30]	S. Kumar, P. K. Shaw, A. H. Abdel-Aty, E. E. Mahmoud, A numerical study on fractional differential equation with population growth model, Numer. Meth. Part. D. E., 2020, 1–22. https://doi.org/10.1002/num.22684 doi: 10.1002/num.22684
[31]	P. Tong, Y. Feng, H. Lv, Euler's method for fractional differential equations, WSEAS Trans. Math., 12 (2013), 1146–1153.
[32]	J. Patade, S. Bhalekar, A new numerical method based on daftardar-gejji and jafari technique for solving differential equations, World J. Model. Simul., 11 (2015), 256–271.
[33]	O. Y. Ababneh, New numerical methods for solving differential equations, J. Adv. Math., 16 (2019), 8384–8390.
[34]	A. Jhinga, V. Daftardar-Gejji, A new finite-difference predictor-corrector method for fractional differential equations, Appl. Math. Comput., 336 (2018), 418–432. https://doi.org/10.1016/j.amc.2018.05.003 doi: 10.1016/j.amc.2018.05.003
[35]	S. Kumar, An analytical algorithm for nonlinear fractional fornberg-whitham equation arising in wave breaking based on a new iterative method, Alex. Eng. J., 53 (2014), 225–231. http://dx.doi.org/10.1016/j.aej.2013.11.004 doi: 10.1016/j.aej.2013.11.004
[36]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.
[37]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 204 (2006).
[38]	G. M. Mittag-Leffler, Sur la nouvelle fonction $e_\alpha (x)$, CR Acad. Sci. Paris, 137 (1903), 554–558.
[39]	M. Al-Smadi, O. A. Arqub, Computational algorithm for solving Fredholm time-fractional partial integro differential equations of Dirichlet functions type with error estimates, Appl. Math. Comput., 342 (2019), 280–294. https://doi.org/10.1016/j.amc.2018.09.020 doi: 10.1016/j.amc.2018.09.020
[40]	M. Al-Smadi, A. Freihat, H. Khalil, S. Momani, R. A. Khan, Numerical multistep approach for solving fractional partial differential equations, Int. J. Comput. Methods, 14 (2017), 1750029. https://doi.org/10.1142/S0219876217500293 doi: 10.1142/S0219876217500293
[41]	S. Hasan, M. Al-Smadi, A. El-Ajou, S. Momani, S. Hadid, Z. Al-Zhour, Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system, Chaos Soliton. Fract., 143 (2021), 110506. https://doi.org/10.1016/j.chaos.2020.110506 doi: 10.1016/j.chaos.2020.110506
[42]	A. Wiman, Ü about the fundamental theorem in the theory of functions $E^a(x)$, Acta Math., 29 (1905), 191–201. https://doi.org/10.1007/BF02403202 doi: 10.1007/BF02403202
[43]	K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer Science & Business Media, 2010. https://doi.org/10.1007/978-3-642-14574-2
[44]	C. W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shockhcapturing schemes Ⅱ, J. Comput. Phys., 83 (1989), 32H78. https://doi.org/10.1016/0021-9991(89)90222-2 doi: 10.1016/0021-9991(89)90222-2
[45]	S. Gottlieb, On high order strong stability preserving runge-kutta and multi step time discretizations, J. Sci. Comput., 25 (2005), 105–128. https://doi.org/10.1007/s10915-004-4635-5 doi: 10.1007/s10915-004-4635-5
[46]	S. Gottlieb, D. I. Ketcheson, C. W. Shu, Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. https://doi.org/10.1142/7498
[47]	Z. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inform., 26 (2008), 15–27.
[48]	R. Almeida, N. R. Bastos, M. T. T. Monteiro, Modeling some real phenomena by fractional differential equations, Math. Method. Appl. Sci., 39 (2016), 4846–4855. https://doi.org/10.1002/mma.3818 doi: 10.1002/mma.3818
[49]	U. Nations, The world at six billion off site, World Population From Year 0 to Stabilization 5, 1999.
[50]	M. Awadalla, Y. Yameni, Modeling exponential growth and exponential decay real phenomena by $\psi$-caputo fractional derivative, J. Adv. Math. Comput. Sci., 28 (2018), 1–13. https://doi.org/10.9734/JAMCS/2018/43054 doi: 10.9734/JAMCS/2018/43054
[51]	A. E. Calik, H. Ertik, B. Öder, H. Şirin, A fractional calculus approach to investigate the alpha decay processes, Int. J. Modern. Phys. E, 22 (2013), 1350049. https://doi.org/10.1142/S0218301313500493 doi: 10.1142/S0218301313500493

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)