Abstract
The current article deals with the analysis and numerical solution of fractional order Schnakenberg (S-B) model. This model is a system of autocatalytic reactions by nature, which arises in many biological systems. This study is aiming at investigating the behavior of natural phenomena with a more realistic and practical approach. The solutions are obtained by applying the Grunwald–Letnikov (G–L) finite difference (FD) and the proposed G–L nonstandard finite difference (NSFD) computational schemes. The proposed formulation is explicit in nature, strongly structure preserving as well as it is independent of the time step size. One very important feature of our proposed scheme is that it preserves the positivity of the solution of continuous fractional order S-B model because the unknown variables involved in this system describe the chemical concentrations of different substances. The comparison of the proposed scheme with G–L FD method reflects the significance of the said method.
Similar content being viewed by others
References
Ahmed N, Rafiq M, Baleanu D, Rehman MA (2019) Spatio-temporal numerical modeling of auto-catalytic Brusselator model. Rom J Phys 64:110
Ahmed N, Rafiq M, Rehman MA, Iqbal MS, Ali M (2019) Numerical modelling of three dimensional Brusselator reaction diffusion system. AIP Adv 9:015205
Ahmed N, Tahira SS, Rafiq M, Rehman MA, Ali M, Ahmad MO (2019) Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model. Open Math 17:313–330
Almeida R (2017) What is the best fractional derivative to fil data? Appl Anal Discrete Math 11:358–368
Ameen I, Novati P (2017) The solution of fractional order epidemic model by implicit Adams methods. Appl Math Model 43:78–84
Arenas AJ, Gonzalez G, Chen Charpentier B (2016) Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order. Math Comput Simul 21:48–63
Baleanu D, Fernandez A (2019) On fractional operators and their classification. Mathematics 7(9):830
Baleanu D, Mustafa OG (2015) Asymptotic integration and stability: for ordinary, functional and discrete differential equations of fractional order. Series on complexity, nonlinearity and Chaos. World Scientific Publishing Company, Singapore
Baleanu D, Machado JAT, Guvenç ZB (2009) New trends in nanotechnology and fractional calculus applications. Springer, Dordrecht
Baleanu D, Machado JAT, Luo ACJ (2011) Fractional dynamics and control. Springer, New York
Baleanu D, Asad J, Petras I (2014) Fractional Bateman–Feshbach Tikochinsky oscillator. Commun Theor Phys 61(2):221–225
Baleanu D, Magin R, Bhalekar S, Daftardar-Gejji V (2015) Chaos in the fractional order nonlinear Bloch equation with delay. Commun Nonlinear Sci Numer Simul 25(1–3):41–49
Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2016) Fractional calculus: models and numerical methods. World Scientific, Singapore
Baleanu D, Jajarmi A, Sajjadi SS, Mozyrska D (2019) A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos Interdiscip J Nonlinear Sci 29(8):083127
Caputo M (1967) Linear model of dissipation whose Q is almost frequency independent—II. Geophys J Int 13(5):529–539
Caputo M (2014) The role of memory in modeling social and economic cycles of extreme events. A handbook of alternative theories of public economics. Edward Elgar Publishing, Cheltenham, pp 245–259
Cooper G, Cowan D (2003) The application of fractional calculus to potential field data. Explor Geophys 34:51–56
Fatima U, Ali M, Ahmed N (2018) Numerical modeling of susceptible latent breaking-out quarantine computer virus epidemic dynamics. Heliyon 4:e00631
Francisco Fernandez M (2009) On some approximate methods for nonlinear models. Appl Math Comput 215:168–174
Hajipour M, Jajarmi A, Baleanu D (2018) An efficient non-standard finite difference scheme for a class of fractional chaotic systems. J Comput Nonlinear Dynam 13(2):021013
Hammouch Z, Mekkaoui T, Belgacem FBM (2014) Numerical simulations for a variable order fractional Schnakenberg model. In: AIP conference proceeding, pp 1450–1637
Haq F, Shah K, ur Rahman G, Li Y, Shahzad M (2018) Computational analysis of complex population dynamical model with arbitrary order. Complexity 1–8
Khan MA, Hammouch Z, Baleanu D (2019) Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative. Math Model Nat Phenom 14(3):311
Mickens RE (1994) Nonstandard finite difference models of differential equations. World Scientific, Singapore
Ongun M, Arslan D, Garrappa R (2013) Nonstandard finite difference schemes for a fractional-order Brusselator system. Adv Differ Equ 2013:102. https://doi.org/10.1186/1687-1847-2013-102
Ortigueira M, Machado J (2017) Which derivative? Fractal Fract 1(1):3
Podlubny I (1999) Fractional differential equations, mathematics in science and engineering. Academic Press, New York
Scherer R, Kalla S, Tang Y, Huang J (2011) The Grunwald–Letnikov method for fractional differential equations. Comput Math Appl 62:902–917
Schnakenberg J (1979) Simple chemical reaction systems with limit cycle behaviour. J Theor Biol 81:389–400
Suryanto A, Darti I (2017) Stability analysis and nonstandard Grünwald–Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. In: AIP conference proceedings, vol 1913, p 020011
Sweilam NH, Nagy AM, Elpahri LE (2019) Nonstandard finite difference scheme for the fractional order Salmonella transmission model. J Fract Calc Appl 10(1):197–212
Veeresha P, Prakasha DG, Baskonus HM (2019) Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method. Math Sci. https://doi.org/10.1007/s40096-019-0284-6
Xie W, Xu J, Cai L, Lin Z (2017) Dynamic preserving method with changeable memory length of fractional-order chaotic system. Int J Nonlinear Mech 92:59–65
Yang XJ, Gao F, Srivastava HM (2017) New rheological models within local fractional derivative. Rom Rep Phys 69:113
Acknowledgements
We would like to express our sincere thanks to reviewers for the careful reading and helpful remarks of our paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by José Tenreiro Machado.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Iqbal, Z., Ahmed, N., Baleanu, D. et al. Structure preserving computational technique for fractional order Schnakenberg model. Comp. Appl. Math. 39, 61 (2020). https://doi.org/10.1007/s40314-020-1068-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-1068-1
Keywords
- Fractional order differential equations
- Schnakenberg model
- Grunwald–Letnikov approach
- Structure preserving method