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A fractional analysis of Noyes–Field model for the nonlinear Belousov–Zhabotinsky reaction

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Abstract

Nonlinear phenomena play an essential role in various field of natural sciences and engineering. In particular, the nonlinear chemical reactions are observed in various domains, as, for instance, in biological and chemical physics. For this reason, it is important to investigate the solution to this nonlinear phenomenon. This article investigates numerical solutions to a nonlinear oscillatory system called the Belousov–Zhabotinsky with Caputo fractional-time derivative. The simplified Noyes–Field fractional model reads

$$\begin{aligned} {\mathcal {D}}_t^{\mu }p= & {} \xi _1p_{xx}+\beta \delta {w}+p-p^2-\delta {pw},\quad 0<\mu \le 1,\\ {\mathcal {D}}_t^{\mu }w= & {} \xi _2w_{xx}+ \gamma {w}-\lambda {pw}, \end{aligned}$$

where \(\xi _1\) and \(\xi _2\) are the diffusing constants for the concentration p and w respectively, \(\gamma \) and \(\beta \) are given constants, \(\lambda \ne 1\) and \(\delta \) are positive parameters. The two iterative techniques used in this work are the fractional reduced differential transform method and q-homotopy analysis transform method. The outcomes using these two methods reveal an efficient numerical solution with high accuracy and minimal computations. Furthermore, to better understand the effect of the fractional order, we present the solution profiles which demonstrate the behavior of the obtained results.

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References

  • Abedle-Rady AS, Rida SZ, Arafa AAM, Adedl-Rahim HR (2014) Approximate analytical solutions of the fractional nonlinear dispersive equations using homotopy perturbation Sumudu transform method. Int J Innov Sci Eng Technol 1(9):257–267

    Google Scholar 

  • Adomian G (1994) Solving Frontier problems of physics: the decomposition method. Kluwer, New York

    MATH  Google Scholar 

  • Akinyemi L (2019) q-Homotopy analysis method for solving the seventh-order time-fractional Lax’s Korteweg–de Vries and Sawada–Kotera equations. Comput Appl Math 38(4):1–22

    MathSciNet  MATH  Google Scholar 

  • Akinyemi L, Iyiola OS (2020) A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations. Adv Differ Equ 169:1–27. https://doi.org/10.1186/s13662-020-02625-w

    Article  MathSciNet  Google Scholar 

  • Akinyemi L, Iyiola OS, Akpan U (2020) Iterative methods for solving fourth- and sixth order time-fractional Cahn–Hillard equation. Math Methods Appl Sci 43(7):4050–4074. https://doi.org/10.1002/mma.6173

    Article  Google Scholar 

  • Ali M, Alquran M, Jaradat I (2019) Asymptotic-sequentially solution style for the generalized Caputo time-fractional Newell–Whitehead–Segel system. Adv Differ Equ 2019:1–9

    MathSciNet  MATH  Google Scholar 

  • Alquran M, Al-Khaled K, Chattopadhyay J (2015) Analytical solutions of fractional population diffusion model: residual power series. Nonlinear Stud. 22(1):31–9

    MathSciNet  MATH  Google Scholar 

  • Baleanu D, Guvenc ZB, Machado JT (2010) New trends in nanotechnology and fractional calculus applications. Springer, New York

    MATH  Google Scholar 

  • Baleanu D, Machado JAT, Luo AC (2011) Fract Dyn Control. Springer Science and Business Media, New York

    Google Scholar 

  • Baleanu D, Wu GC, Zeng SD (2017) Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fract 102:99–105

    MathSciNet  MATH  Google Scholar 

  • Caputo M (1969) Elasticita e dissipazione. Zanichelli, Bologna

    Google Scholar 

  • Das S (2009) Analytical solution of a fractional diffusion equation by variational iteration method. Comput Math Appl 57:483–487

    MathSciNet  MATH  Google Scholar 

  • El-Tawil MA, Huseen SN (2012) The Q-homotopy analysis method (q-HAM). Int J Appl Math Mech 8(15):51–75

    Google Scholar 

  • El-Tawil MA, Huseen SN (2013) On convergence of the q-homotopy analysis method. Int J Contemp Math Sci 8:481–497

    MathSciNet  Google Scholar 

  • Eltayeb H, Kilicman A (2012) Application of Sumudu decomposition method to solve nonlinear system of partial differential equations. Abstr Appl Anal 2012:1–13

    MathSciNet  MATH  Google Scholar 

  • Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7(4):355–369

    MATH  Google Scholar 

  • Gibbs RG (1980) Traveling waves in the Belousov–Zhabotinskii reaction. SIAM J Appl Math 38(3):422–444

    MathSciNet  MATH  Google Scholar 

  • He JH (1998) Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Methods Appl Mech Eng 167:57–68

    MathSciNet  MATH  Google Scholar 

  • He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178:257–62

    MathSciNet  MATH  Google Scholar 

  • He JH (2003) Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 135:73–9

    MathSciNet  MATH  Google Scholar 

  • Hilfer R, Anton L (1995) Fractional master equations and fractal time random walks. Phys Rev E 51:R848–R851

    Google Scholar 

  • Iyiola OS (2015) On the solutions of non-linear time-fractional gas dynamic equations: an analytical approach. Int J Pure Appl Math 98(4):491–502

    Google Scholar 

  • Iyiola OS (2016) Exact and approximate solutions of fractional diffusion equations with fractional reaction terms. Progr Fract Differ Appl 2(1):21–30

    Google Scholar 

  • Iyiola OS, Soh ME, Enyi CD (2013) Generalised homotopy analysis method (q-HAM) for solving foam drainage equation of time fractional type. Math Eng Sci Aerosp 4(4):105

    MATH  Google Scholar 

  • Jaradat A, Noorani MSM, Alquran M, Jaradat HM (2018a) Numerical investigations for time-fractional nonlinear model arise in physics. Results Phys 8:1034–1037

  • Jaradat I, Alquran M, Abdel-Muhsen R (2018b) An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering. Nonlinear Dyn 93(4):1911–1922

  • Jaradat I, Alquran M, Al-Khaled K (2018c) An analytical study of physical models with inherited temporal and spatial memory. Eur Phys J Plus 133:1–11

  • Joseph K (2012) Fractional dynamics: recent advances. World Scientific, Singapore

    MATH  Google Scholar 

  • Keskin Y, Oturanc G (2010) Reduced differential transform method: a new approach to fractional partial differential equations. Nonlinear Sci Lett A 1:61–72

    MATH  Google Scholar 

  • Khuri SA (2001) A Laplace decomposition algorithm applied to class of nonlinear differential equations. J Math Appl 1(4):141–155

    MathSciNet  MATH  Google Scholar 

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier Science B.V., Amsterdam

    MATH  Google Scholar 

  • Kumara D, Singha J, Baleanu D (2017) A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Math Methods Appl Sci 40:5642–5653

    MathSciNet  MATH  Google Scholar 

  • Kumar D, Seadawy AR, Joardar AK (2018a) Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chin J Phys 56(1):75–85

  • Kumar D, Singh J, Baleanu D (2018b) A new numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses. Nonlinear Dyn 91:307–317

  • Kurt A, Rezazadeh H, Şenol M, Neirameh A, Tasbozan O, Eslami M, Mirzazadeh M (2019) Two effective approaches for solving fractional generalized Hirota–Satsuma coupled KdV system arising in interaction of long waves. J Ocean Eng Sci 4(1):24–32

    Google Scholar 

  • Laskin N, Zaslavsky G (2006) Nonlinear fractional dynamics on a lattice with long range interactions. Phys A Stat Mech Appl 368(1):38–54

    Google Scholar 

  • Liao SJ (1995) An approximate solution technique not depending on small parameters: a special example. Int J Nonlinear Mech 30(3):371–380

    MathSciNet  MATH  Google Scholar 

  • Liao SJ (1998) Homotopy analysis method: a new analytic method for nonlinear problems. Appl Math Mech 19:957–962

    MathSciNet  MATH  Google Scholar 

  • Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147(2):499–513

    MathSciNet  MATH  Google Scholar 

  • Luchko YF, Srivastava HM (1995) The exact solution of certain differential equations of fractional order by using operational calculus. Comput Math Appl 29:73–85

    MathSciNet  MATH  Google Scholar 

  • Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London

    MATH  Google Scholar 

  • Miller KS, Ross B (1993) An introduction to fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  • Nasrolahpour H (2013) A note on fractional electrodynamics. Commun Nonlinear Sci Numer Simul 18:2589–2593

    MathSciNet  MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  • Pu YF (2007) Fractional differential analysis for texture of digital image. J Algorithm Comput Technol 1(3):357–380

    MathSciNet  Google Scholar 

  • Ray SS, Bera RK (2006) Analytical solution of a fractional diffusion equation by Adomian decomposition method. Appl Math Comput 174(1):329–336

    MathSciNet  MATH  Google Scholar 

  • Şenol M, Iyiola OS, Daei Kasmaei H, Akinyemi L (2019a) Efficient analytical techniques for solving time-fractional nonlinear coupled Jaulent–Miodek system with energy-dependent Schrödinger potential. Adv Differ Equ 2019:1–21

  • Şenol M, Tasbozan O, Kurt A (2019b) Numerical solutions of fractional Burgers’ type equations with conformable derivative. Chin J Phys 58:75–84

  • Singh J, Kumar D, Baleanu D, Rathore S (2018) An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation. Appl Math Comput 335:12–24

    MathSciNet  MATH  Google Scholar 

  • Soh ME, Enyi CD, Iyiola OS, Audu JD (2014) Approximate analytical solutions of strongly nonlinear fractional BBM-Burger’s equations with dissipative term. Appl Math Sci 8(155):7715–7726

    Google Scholar 

  • Sun HG, Zhang Y, Baleanu D, Chen W, Chen YQ (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231

    Google Scholar 

  • Sweilam NH, Hasan MMA, Baleanu D (2017) New studies for general fractional financial models of awareness and trial advertising decisions. Chaos Solitons Fract 104:772–784

    MathSciNet  MATH  Google Scholar 

  • Tarasov VE (2006) Gravitational field of fractal distribution of particles. Celest Mech Dyn Astron 94(1):1–15

    MathSciNet  MATH  Google Scholar 

  • Ullah A, Chen W, Sun HG, Khan MA (2017) An efficient variational method for restoring images with combined additive and multiplicative noise. Int J Appl Comput Math 3(3):1999–2019

    MathSciNet  MATH  Google Scholar 

  • Veeresha P, Prakasha DG, Baskonus HM (2019) Novel simulations to the time-fractional Fisher’s equation. Math Sci 13(1):33–42

    MathSciNet  MATH  Google Scholar 

  • Wazwaz AM, Gorguis A (2004) An analytic study of Fisher’s equation by using Adomian decomposition method. Appl Math Comput 154:609–620

    MathSciNet  MATH  Google Scholar 

  • Zhabotinsky Anatol M (2007) Scholarpedia 2(9):1435. https://doi.org/10.4249/scholarpedia

    Article  Google Scholar 

  • Zhang Y, Pu YF, Hu JR, Zhou JL (2012) A class of fractional-order variational image in-painting models. Appl Math Inf Sci 6(2):299–306

    MathSciNet  Google Scholar 

Download references

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  1. Department of Mathematics, Ohio University, Athens, USA

    Lanre Akinyemi

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  1. Lanre Akinyemi
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Correspondence to Lanre Akinyemi.

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Communicated by Agnieszka Malinowska.

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Akinyemi, L. A fractional analysis of Noyes–Field model for the nonlinear Belousov–Zhabotinsky reaction. Comp. Appl. Math. 39, 175 (2020). https://doi.org/10.1007/s40314-020-01212-9

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