Abstract
In a recent work (Hoang in Math Comput Simul 199:359–373, 2022), a class of nonstandard finite difference (NSFD) schemes preserving the positivity and boundedness of the nonlinear Volterra integro-differential population growth model has been constructed. However, these NSFD schemes are only convergent of order one. In this work, we introduce a new class of second-order and dynamically consistent NSFD schemes derived from non-local approximations in combination with modified denominator functions. It is proved that the constructed NSFD schemes are not only convergent of order two but also dynamically consistent with respect to the positivity and boundedness of the Volterra’s population growth model. As an important application, we combine the second-order NSFD schemes with Richardson’s extrapolation method to generate higher-accuracy numerical solutions. Consequently, higher-accuracy numerical solutions for the population growth model can be obtained easily. Finally, a set of numerical examples is reported to support the theoretical findings.
Similar content being viewed by others
Data Availability
The author confirms that the data supporting the findings of this study are available within the article.
References
Al-Khaled K (2005) Numerical approximations for population growth models. Appl Math Comput 160:865–873
Ascher UM, Petzold LR (1998) Computer methods for ordinary differential equations and differential-algebraic equations. Society for Industrial and Applied Mathematics, Philadelphia
Burden RL, Douglas FJ (2015) Numerical analysis, 9th edn. Cengage Learning, Boston
Brunner H, van der Houwen PJ (1986) The numerical solution of Volterra equations. North-Holland, Amsterdam
Chen-Charpentier BM, Dimitrov DT, Kojouharov HV (2006) Combined nonstandard numerical methods for ODEs with polynomial right-hand sides. Math Comput Simul 73:105–113
Dang QA, Hoang MT, Dang QL (2018) Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses. J Comput Sci Cybern 32:171–185
Dang Quang A, Hoang Manh Tuan (2020) Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems. Int J Comput Math 97:2036–2054
El-shahed M (2005) Application of He’s Homotopy Perturbation Method to Volterra’s Integro-differential Equation. Int J Nonlinear Sci Numer Simul 6:163–167
Gonzalez-Parra G, Arenas AJ, Chen-Charpentier BM (2010) Combination of nonstandard schemes and Richardson’s extrapolation to improve the numerical solution of population models. Math Comput Model 52:1030–1036
Gupta M, Slezak JM, Alalhareth F, Roy S, Kojouharov HV (2020) Second-order Nonstandard Explicit Euler Method. AIP Conf Proc 2302:110003
Hoang MT (2022) Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model. Math Comput Simul 199:359–373
Joyce DC (1971) Survey of extrapolation processes in numerical analysis. SIAM Rev 13:435–490
Kojouharov HV, Roy S, Gupta M, Alalhareth F, Slezak JM (2021) A second-order modified nonstandard theta method for one-dimensional autonomous differential equations. Appl Math Lett 112:106775
Martin-Vaquero J, Martin del Rey A, Encinas AH, Hernandez Guillen JD, Queiruga-Dios A, Rodriguez Sanchez G (2017) Higher-order nonstandard finite difference schemes for a MSEIR model for a malware propagation. J Comput Appl Math 317:146–156
Martin-Vaquero J, Queiruga-Dios A, Martin del Rey A, Encinas AH, Hernandez Guillen JD, Rodriguez Sanchez G (2018) Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model. J Comput Appl Math 330:848–854
Marzban HR, Hoseini SM, Razzaghi M (2009) Solution of Volterra’s population model via block-pulse functions and Lagrange-interpolating polynomials. Math Methods Appl Sci 32:127–134
Mickens RE (1993) Nonstandard finite difference models of differential equations. World Scientific, Singapore
Mickens RE (2000) Applications of nonstandard finite difference schemes. World Scientific, Singapore
Mickens RE (2005) Advances in the applications of nonstandard finite difference schemes. World Scientific, Singapore
Mickens RE (2002) Nonstandard finite difference schemes for differential equations. J Differ Equ Appl 8:823–847
Mickens RE (2020) Nonstandard finite difference schemes: methodology and applications. World Scientific, Singapore
Mohyud-Din ST, Yildirim A, Gulkanat Y (2010) Analytical solution of Volterra’s population model. J King Saud Univ Sci 22:247–250
Parand K, Abbasbandy S, Kazem S, Rad JA (2011) A novel application of radial basis functions for solving a model of first-order integro-ordinary differential equation. Commun Nonlinear Sci Numer Simul 16:4250–4258
Patidar KC (2005) On the use of nonstandard finite difference methods. J Differ Equ Appl 11:735–758
Patidar KC (2016) Nonstandard finite difference methods: recent trends and further developments. J Differ Equ Appl 22:817–849
Richardson LF, Gaunt JA (1927) The deferred approach to the limit. Philos Trans R Soc Lond 226A:299–361
Scudo FM (1971) Vito Volterra and theoretical ecology. Theor Popul Biol 2:1–23
Small RD (1983) Population growth in a closed system. SIAM Rev 25:93–95
TeBeest KG (1997) Numerical and analytical solutions of Volterra’s population model. SIAM Rev 39:484–493
Wazwaz A (1999) Analytical approximations and Padé approximants for Volterra’s population model. Appl Math Comput 100:13–25
Wood DT, Kojouharov HV (2015) A class of nonstandard numerical methods for autonomous dynamical systems. Appl Math Lett 50:78–82
Acknowledgements
We would like to thank the editor and anonymous referees for useful and valuable comments that led to a great improvement of the paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We have no conflicts of interest to disclose.
Additional information
Communicated by Zhaosheng Feng.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hoang, M.T. A class of second-order and dynamically consistent nonstandard finite difference schemes for nonlinear Volterra’s population growth model. Comp. Appl. Math. 42, 85 (2023). https://doi.org/10.1007/s40314-023-02230-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02230-z
Keywords
- Second order
- NSFD schemes
- Dynamic consistency
- Volterra’s population growth model
- Positivity and boundedness