Research article

Numerical simulation of the space fractional $ (3+1) $-dimensional Gray-Scott models with the Riesz fractional derivative

  • Received: 20 January 2022 Revised: 12 March 2022 Accepted: 16 March 2022 Published: 22 March 2022
  • MSC : 65M30, 65D20

  • The reaction-diffusion process always behaves extremely magically, and any a differential model cannot reveal all of its mechanism. Here we show the patterns behavior can be described well by the fractional reaction-diffusion model (FRDM), which has unique properties that the integer model does not have. Numerical simulation is carried out to elucidate the attractive properties of the fractional (3+1)-dimensional Gray-Scott model, which is to model a chemical reaction with oscillation. The Fourier transform for spatial discretization and fourth-order Runge-Kutta method for time discretization are employed to illustrate the fractal reaction-diffusion process.

    Citation: Dan-Dan Dai, Wei Zhang, Yu-Lan Wang. Numerical simulation of the space fractional $ (3+1) $-dimensional Gray-Scott models with the Riesz fractional derivative[J]. AIMS Mathematics, 2022, 7(6): 10234-10244. doi: 10.3934/math.2022569

    Related Papers:

  • The reaction-diffusion process always behaves extremely magically, and any a differential model cannot reveal all of its mechanism. Here we show the patterns behavior can be described well by the fractional reaction-diffusion model (FRDM), which has unique properties that the integer model does not have. Numerical simulation is carried out to elucidate the attractive properties of the fractional (3+1)-dimensional Gray-Scott model, which is to model a chemical reaction with oscillation. The Fourier transform for spatial discretization and fourth-order Runge-Kutta method for time discretization are employed to illustrate the fractal reaction-diffusion process.



    加载中


    [1] M. Abbaszadeh, M. Dehghan, A reduced order finite difference method for solving space-fractional reaction-diffusion systems: The Gray-Scott model, Eur. Phy. J. Plus, 2019. https://doi.org/10.1140/epjp/i2019-12951-0 doi: 10.1140/epjp/i2019-12951-0
    [2] X. L. Zhang, W. Zhang, Y. L. Wang, T. T. Ban, The space spectral interpolation collocation method for reaction-diffusion systems, Therm. Sci., 25 (2021), 1269–1275. https://doi.org/10.2298/TSCI200402022Z doi: 10.2298/TSCI200402022Z
    [3] A. Mmm, B. Jar, C. Kpab, Dynamical behavior of reaction diffusion neural networks and their synchronization arising in modeling epileptic seizure: A numerical simulation study, Comput. Math. Appl., 80 (2020), 1887–1927. https://doi.org/10.1016/j.camwa.2020.08.020 doi: 10.1016/j.camwa.2020.08.020
    [4] A. Atangana, R. T. Alqahtani, New numerical method and application to Keller-Segel model with fractional order derivative, Chaos Soliton. Fract., 116 (2018), 14–21. https://doi.org/10.1016/j.chaos.2018.09.013 doi: 10.1016/j.chaos.2018.09.013
    [5] X. J. Yang, General fractional derivatives: Theory, methods and applications, New York: CRC Press, 2019. https://doi.org/10.1201/9780429284083
    [6] Y. L. Wang, M. J. Du, F. G. Tan, Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions, Appl. Math. Comput., 219 (2013), 5918–5925. https://doi.org/10.1016/j.amc.2012.12.009 doi: 10.1016/j.amc.2012.12.009
    [7] N. Anjum, C. H. He, J. H. He, Two-scale fractal theory for the population dynamics, Fractals, 29 (2021), 2150182. https://doi.org/10.1142/S0218348X21501826 doi: 10.1142/S0218348X21501826
    [8] W. Zhang, Y. L. Wang, M. C. Wang, The reproducing kernel for the reaction-diffusion model with a time variable fractional order, Therm. Sci., 24 (2020), 2553–2559. https://doi.org/10.2298/TSCI2004553Z doi: 10.2298/TSCI2004553Z
    [9] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications, Academic Press, 1998.
    [10] J. H. He, W. F. Hou, C. H. He, T. Saeed, T. Hayat, Variational approach to fractal solitary waves, Fractals, 29 (2021), 2150199. https://doi.org/10.1142/S0218348X21501991 doi: 10.1142/S0218348X21501991
    [11] D. Tian, Q. T. Ain, N. Anjum, C. H. He, B. Cheng, Fractal N/MEMS: From the pull-in instability to pull-in stability, Fractals, 29 (2020), 2150030. https://doi.org/10.1142/S0218348X21500304 doi: 10.1142/S0218348X21500304
    [12] D. Tian, C. H. He, A fractal micro-electromechanical system and its pull-in stability, J. Low Freq. Noise V. A., 40 (2021), 1380–1386. https://doi.org/10.1177/1461348420984041 doi: 10.1177/1461348420984041
    [13] C. H. He, K. A. Gepreel, Low frequency property of a fractal vibration model for a concrete beam, Fractals, 29 (2021), 2150117. https://doi.org/10.1142/S0218348X21501176 doi: 10.1142/S0218348X21501176
    [14] D. Baleanu, X. J. Yang, H. M. Srivastava, Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett., 47 (2015), 54–60. https://doi.org/10.1016/j.aml.2015.02.024 doi: 10.1016/j.aml.2015.02.024
    [15] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Netherlands, 2006. https://doi.org/10.1016/S0304-0208(06)80001-0
    [16] T. M. Atanackovic, S. Pilipovic, B. Stankovic, D. Zorica, T. M. Atanackovic, Fractional calculus with applications in mechanics: Vibrations and diffusion processes, Drug Dev. Ind. Pharm., 2014. https://doi.org/10.1002/9781118577530 doi: 10.1002/9781118577530
    [17] P. I. Butzer, U. Westphal, An introduction to fractional calculus, in applications of fractional calculus in physics, World Scientific, Singapore, 2000, 1–85. https://doi.org/10.1142/9789812817747-0001
    [18] S. Samko, A. Kilbas, O. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, Amsterdam, 1993.
    [19] V. V. Uchaikin, Fractional derivatives for physicists and engineers, Higher Education Press and Springer Verlag, Beijing/Berlin, 2013.
    [20] E. E. Sel'Kov, Self-oscillations in glycolysis, Fed. Eur. Biochem. Soc. J., 4 (1968), 79–86.
    [21] P. Gray, S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability, Chem. Eng. Sci., 38 (1983), 29–43.
    [22] Y. Liu, E. Y. Fan, B. L. Yin, H. Li, J. F. Wang, TT-M finite element algorithm for a two-dimensional space fractional Gray-Scott model, Comput. Math. Appl., 80 (2020), 1793–1809. https://doi.org/10.1016/j.camwa.2020.08.011 doi: 10.1016/j.camwa.2020.08.011
    [23] A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT, 54 (2014), 937–954. https://doi.org/10.1007/s10543-014-0484-2 doi: 10.1007/s10543-014-0484-2
    [24] G. H. Lee, A second-order operator splitting Fourier spectral method for fractional-in-space reaction-diffusion equations, J. Comput. Appl. Math., 33 (2018), 395–403. https://doi.org/10.1016/j.cam.2017.09.007 doi: 10.1016/j.cam.2017.09.007
    [25] T. T. Wang, F. Y. Song, H. Wang, G. E. Karniadakis, Fractional gray-scott model: Well-posedness, discretization, and simulations, Comput. Method. Appl. M., 347 (2019), 1030–1049. https://doi.org/10.1016/j.cma.2019.01.002 doi: 10.1016/j.cma.2019.01.002
    [26] M. Abbaszadeh, M. Dehghan, I. M. Navon, A pod reduced-order model based on spectral Galerkin method for solving the space-fractional Gray-Scott model with error estimate, Eng. Comput., 2020, 1–24. https://doi.org/10.1007/s00366-020-01195-5 doi: 10.1007/s00366-020-01195-5
    [27] K. M. Owolabi, B. Karaagac, Dynamics of multi-pulse splitting process in one-dimensional Gray-Scott system with fractional order operator, Chaos Soliton. Fract., 136 (2020). https://doi.org/10.1016/j.chaos.2020.109835 doi: 10.1016/j.chaos.2020.109835
    [28] W. Wang, Y. Lin, Y. Feng, L. Zhang, Y. J. Tan, Numerical study of pattern formation in an extended Gray-Scott model, Commun. Nonlinear Sci., 16 (2011), 2016–2026. https://doi.org/10.1016/j.cnsns.2010.09.002 doi: 10.1016/j.cnsns.2010.09.002
    [29] T. Chen, S. M. Li, J. Llibr, Phase portraits and bifurcation diagram of the Gray-Scott model-sciencedirect, J. Math. Anal. Appl., 496 (2020), 124840. https://doi.org/10.1016/j.jmaa.2020.124840 doi: 10.1016/j.jmaa.2020.124840
    [30] V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158–182. https://doi.org/10.1016/j.amc.2014.06.097 doi: 10.1016/j.amc.2014.06.097
    [31] 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
    [32] Y. Zhang, J. Cao, W. Bu, A. Xiao, A fast finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction-diffusion equation, Int. J. Model. Simul. SC, 11 (2020), 2050016. https://doi.org/10.1142/S1793962320500166 doi: 10.1142/S1793962320500166
    [33] Y. L. Wang, L. N. Jia, H. L. Zhang, Numerical solution for a class of space-time fractional equation in reproducing, Int. J. Comput. Math., 96 (2019), 2100–2111. https://doi.org/10.1080/00207160.2018.1544367 doi: 10.1080/00207160.2018.1544367
    [34] F. Z. Geng, X. Y. Wu, Reproducing kernel function-based Filon and Levin methods for solving highly oscillatory integral, Appl. Math. Comput., 397 (2021), 125980. https://doi.org/10.1016/j.amc.2021.125980 doi: 10.1016/j.amc.2021.125980
    [35] X. Y. Li, B. Y. Wu, A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations, J. Comput. Appl. Math., 311 (2017), 387–393. https://doi.org/10.1016/j.cam.2016.08.010 doi: 10.1016/j.cam.2016.08.010
    [36] X. Y. Li, B. Y. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math., 243 (2013), 10–15. https://doi.org/10.1016/j.cam.2012.11.002 doi: 10.1016/j.cam.2012.11.002
    [37] F. Z. Geng, X. Y. Wu, Reproducing kernel functions based univariate spline interpolation, Appl. Math. Lett., 122 (2021), 107525. https://doi.org/10.1016/j.aml.2021.107525 doi: 10.1016/j.aml.2021.107525
    [38] X. Y. Li, B. Y. Wu, Superconvergent kernel functions approaches for the second kind Fredholm integral equations, Appl. Numer. Math., 67 (2021), 202–210. https://doi.org/10.1016/j.apnum.2021.05.004 doi: 10.1016/j.apnum.2021.05.004
    [39] X. Y. Li, H. L. Wang, B. Y. Wu, An accurate numerical technique for fractional oscillation equations with oscillatory solutions, Math. Method. Appl. Sci., 45 (2022), 956–966. https://doi.org/10.1002/mma.7825 doi: 10.1002/mma.7825
    [40] D. D. Dai, T. T. Ban, Y. L. Wang, W. Zhao, The piecewise reproducing kernel method for the time variable fractional order advection-reaction-diffusion equations, Therm. Sci., 25 (2021), 1261–1268. https://doi.org/10.2298/TSCI200302021D doi: 10.2298/TSCI200302021D
    [41] I. Podlubny, Matrix approach to discrete fractional calculus, Fract. Calc. Appl. Anal., 3 (2000), 359–386. https://doi.org/10.1016/j.jcp.2009.01.014 doi: 10.1016/j.jcp.2009.01.014
    [42] C. Han, Y. L. Wang, Z. Y. Li, A high-precision numerical approach to solving space fractional Gray-Scott model, Appl. Math. Lett., 125 (2022), 107759. https://doi.org/10.1016/j.aml.2021.107759 doi: 10.1016/j.aml.2021.107759
    [43] C. Han, Y. L. Wang, Z. Y. Li, Numerical solutions of space fractional variable-coefficient KdV-modified KdV equation by Fourier spectral method, Fractals, 29 (2021), 2150246. https://doi.org/10.1142/S0218348X21502467 doi: 10.1142/S0218348X21502467
    [44] X. Y. Li, C. Han, Y. L. Wang, Novel patterns in fractional-in-space nonlinear coupled fitzhugh-nagumo models with Riesz fractional derivative, Fractal Fract., 6 (2022), 136. https://doi.org/10.3390/fractalfract6030136 doi: 10.3390/fractalfract6030136
    [45] J. S. Duan, D. C. Hu, M. Li, Comparison of two different analytical forms of response for fractional oscillation equation, Fractal Fract., 5 (2021), 188. https://doi.org/10.3390/fractalfract5040188 doi: 10.3390/fractalfract5040188
    [46] Q. T. Ain, J. H. He, N. Anjum, M. Ali, The fractional complex transform: A novel approach to the time-fractional Schr$\ddot{o}$dinger equation, Fractals, 28 (2020), 2050141. https://doi.org/10.1142/S0218348X20501418 doi: 10.1142/S0218348X20501418
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1554) PDF downloads(121) Cited by(1)

Article outline

Figures and Tables

Figures(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog