A study of Ralston's cubic convergence with the application of population growth model

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

doi:10.3934/math.2022632

AIMS Mathematics

2022, Volume 7, Issue 6: 11320-11344. doi: 10.3934/math.2022632

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A study of Ralston's cubic convergence with the application of population growth 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: 15 March 2022 Accepted: 21 March 2022 Published: 11 April 2022
MSC : 26A33, 34A08, 93C10, 93C15, 78A70

This paper deals a new numerical scheme to solve fractional differential equation (FDE) involving Caputo fractional derivative (CFD) of variable order $ \beta \in ]0, 1] $. Based on a few examples and application models, the main objective is to show that FDE works more effectively than ordinary differential equations (ODEs). The proposed scheme is fractional Ralston's cubic method (RCM). The convergence analysis and stability analysis of the scheme is proved. The numerical scheme has been found without considering linearisation, perturbations, or any such assumptions. Finally, the efficiency of the proposed scheme will justify by solving a few examples of linear and non-linear FDEs with one application of FDE, world population growth (WPG) model of variable order $ \beta \in ]0, 1] $. Also, the comparison of fractional RCM scheme has been shown with the existing fractional Euler method (EM) and fractional improved Euler method (IEM).
- fractional calculus,
- FDEs,
- fractional Euler method,
- fractional improved Euler method,
- IVP, fractional Ralston's method,
- real-world problem,
- population growth model
Citation: Sara S. Alzaid, Pawan Kumar Shaw, Sunil Kumar. A study of Ralston's cubic convergence with the application of population growth model[J]. AIMS Mathematics, 2022, 7(6): 11320-11344. doi: 10.3934/math.2022632

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Abstract

This paper deals a new numerical scheme to solve fractional differential equation (FDE) involving Caputo fractional derivative (CFD) of variable order $ \beta \in ]0, 1] $. Based on a few examples and application models, the main objective is to show that FDE works more effectively than ordinary differential equations (ODEs). The proposed scheme is fractional Ralston's cubic method (RCM). The convergence analysis and stability analysis of the scheme is proved. The numerical scheme has been found without considering linearisation, perturbations, or any such assumptions. Finally, the efficiency of the proposed scheme will justify by solving a few examples of linear and non-linear FDEs with one application of FDE, world population growth (WPG) model of variable order $ \beta \in ]0, 1] $. Also, the comparison of fractional RCM scheme has been shown with the existing fractional Euler method (EM) and fractional improved Euler method (IEM).

References

[1]	I. Podlubny, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Math. Sci. Eng., 198 (1999), 1–340.
[2]	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
[3]	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
[4]	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
[5]	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
[6]	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
[7]	M. Khader, S. Kumar, An efficient computational method for solving a system of FDEs via fractional finite difference method, Appl. Appl. Math., 14 (2019).
[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]	A. Hemeda, Homotopy perturbation method for solving systems of nonlinear coupled equations, Appl. Math. Sci., 6 (2012), 4787–4800.
[10]	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
[11]	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
[12]	P. Tong, Y. Feng, H. Lv, Euler's method for fractional differential equations, WSEAS Trans. Math., 12 (2013), 1146–1153.
[13]	C. Milici, G. Drǎgǎnescu, J. T. Machado, Introduction to fractional differential equations, Springer, 25 (2018). https://doi.org/10.1007/978-3-030-00895-6 doi: 10.1007/978-3-030-00895-6
[14]	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
[15]	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), 8. https://doi.org/10.1155/2020/1020472 doi: 10.1155/2020/1020472
[16]	J. J. Nieto, Solution of a fractional logistic ordinary differential equation, Appl. Math. Lett., 123 (2022), 107568. https://doi.org/10.1016/j.aml.2021.107568 doi: 10.1016/j.aml.2021.107568
[17]	H. Günerhan, M. Yiǧider, J. Manafian, O. A. Ilhan, Numerical solution of fractional order logistic equations via conformable fractional differential transform method, J. Interdiscip. Math., 24 (2021), 1207–1220. https://doi.org/10.1080/09720502.2021.1918319 doi: 10.1080/09720502.2021.1918319
[18]	I. Area, J. J. Nieto, Fractional-order logistic differential equation with Mittag-Leffler-type kernel, Fractal Fract., 5 (2021), 273. https://doi.org/10.3390/fractalfract5040273 doi: 10.3390/fractalfract5040273
[19]	S. C. Chapra, R. P. Canale, Numerical methods for engineers, Boston: McGraw-Hill Higher Education, 2010.
[20]	A. Ralston, P. Rabinowitz, A first course in numerical analysis, 2 Eds., Mineola: Dover Publications, 2001.
[21]	T. Abdeljawad, A. Atangana, J. Gómez-Aguilar, F. Jarad, On a more general fractional integration by parts formulae and applications, Physica A, 536 (2019), 122494. https://doi.org/10.1016/j.physa.2019.122494 doi: 10.1016/j.physa.2019.122494
[22]	A. Khan, H. Khan, J. Gómez-Aguilar, T. Abdeljawad, Existence and hyers-ulam stability for a nonlinear singular fractional differential equations with mittag-leffler kernel, Chaos Soliton. Fract., 127 (2019), 422–427. https://doi.org/10.1016/j.chaos.2019.07.026 doi: 10.1016/j.chaos.2019.07.026
[23]	P. Bedi, A. Kumar, T. Abdeljawad, A. Khan, J. Gomez-Aguilar, Mild solutions of coupled hybrid fractional order system with caputo-hadamard derivatives, Fractals, 29 (2021). https://doi.org/10.1142/S0218348X21501589 doi: 10.1142/S0218348X21501589
[24]	H. Khan, T. Abdeljawad, J. Gomez-Aguilar, H. Tajadodi, A. Khan, Fractional order volterra integro-differential equation with mittag-leffler kernel, Fractals, 29 (2021). https://doi.org/10.1142/S0218348X21501541 doi: 10.1142/S0218348X21501541
[25]	O. Martínez-Fuentes, F. Meléndez-Vázquez, G. Fernández-Anaya, J. F. Gómez-Aguilar, Analysis of fractional-order nonlinear dynamic systems with general analytic kernels: Lyapunov stability and inequalities, Mathematics, 9 (2021), 2084. https://doi.org/10.3390/math9172084 doi: 10.3390/math9172084
[26]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North Holland Mathematics Studies: Elsevier Science B.V., 2006.
[27]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.
[28]	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
[29]	S. Hippolyte, A. K. Richard, Order of the runge-kutta method and evolution of the stability region, Ural Math. J., 5 (2019), 64–71. https://doi.org/10.15826/umj.2019.2.006 doi: 10.15826/umj.2019.2.006
[30]	S. Hippolyte, A. K. Richard, C. d'Ivoire, A new eighth order runge-kutta family method, J. Math. Res., 11 (2019), 190–199. https://doi.org/10.5539/jmr.v11n2p190 doi: 10.5539/jmr.v11n2p190
[31]	Z. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inform., 26 (2008), 15–27.
[32]	R. Almeida, N. R. Bastos, M. T. T. Monteiro, Modeling some real phenomena by fractional differential equations, Math. Methods Appl. Sci., 39 (2016), 4846–4855. https://doi.org/10.1002/mma.3818 doi: 10.1002/mma.3818
[33]	U. Nations, The world at six billion off site, World Population From Year 0 to Stabilization 5, 1999.
[34]	R. B. Albadarneh, M. Zerqat, I. M. Batiha, Numerical solutions for linear and non-linear fractional differential equations, Int. J. Pure Appl. Math., 106 (2016), 859–871. http://dx.doi.org/10.12732/ijpam.v106i3.12 doi: 10.12732/ijpam.v106i3.12

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