IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v41y2009i3p1095-1104.html
   My bibliography  Save this article

Analysis of the solutions of coupled nonlinear fractional reaction–diffusion equations

Author

Listed:
  • Gafiychuk, V.
  • Datsko, B.
  • Meleshko, V.
  • Blackmore, D.

Abstract

This paper is concerned with analysis of coupled fractional reaction–diffusion equations. As an example, the reaction–diffusion model with cubic nonlinearity and Brusselator model are considered. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. Computer simulation and analytical methods are used to analyze possible solutions for a linearized system. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. It is shown that an increase of the fractional derivative index leads to periodic solutions which become stochastic as the index approaches the value of 2. It is established by computer simulation that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary condition. The characteristic features of these solutions consist in the transformation of the steady state dissipative structures to homogeneous oscillations or spatio-temporal structures at certain values of the fractional index.

Suggested Citation

  • Gafiychuk, V. & Datsko, B. & Meleshko, V. & Blackmore, D., 2009. "Analysis of the solutions of coupled nonlinear fractional reaction–diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1095-1104.
  • Handle: RePEc:eee:chsofr:v:41:y:2009:i:3:p:1095-1104
    DOI: 10.1016/j.chaos.2008.04.039
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077908002178
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2008.04.039?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Yu, Rui & Zhang, Hongqing, 2006. "New function of Mittag–Leffler type and its application in the fractional diffusion-wave equation," Chaos, Solitons & Fractals, Elsevier, vol. 30(4), pages 946-955.
    2. Gafiychuk, V.V. & Datsko, B.Yo., 2006. "Pattern formation in a fractional reaction–diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(2), pages 300-306.
    3. Momani, Shaher & Odibat, Zaid, 2007. "Numerical comparison of methods for solving linear differential equations of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1248-1255.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Caicedo, Alejandro & Cuevas, Claudio & Mateus, Éder & Viana, Arlúcio, 2021. "Global solutions for a strongly coupled fractional reaction-diffusion system in Marcinkiewicz spaces," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    2. Heydari, M.H. & Atangana, A., 2019. "A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 339-348.
    3. Zeeshan Ali & Poom Kumam & Kamal Shah & Akbar Zada, 2019. "Investigation of Ulam Stability Results of a Coupled System of Nonlinear Implicit Fractional Differential Equations," Mathematics, MDPI, vol. 7(4), pages 1-26, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. He, Ji-Huan, 2009. "Nonlinear science as a fluctuating research frontier," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2533-2537.
    2. Gafiychuk, V. & Datsko, B. & Meleshko, V., 2008. "Analysis of fractional order Bonhoeffer–van der Pol oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(2), pages 418-424.
    3. Xu, Lan, 2009. "The variational iteration method for fourth order boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1386-1394.
    4. Roman Parovik, 2020. "Mathematical Modeling of Linear Fractional Oscillators," Mathematics, MDPI, vol. 8(11), pages 1-26, October.
    5. Deng, Hongmin & Li, Tao & Wang, Qionghua & Li, Hongbin, 2009. "A fractional-order hyperchaotic system and its synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 962-969.
    6. Odibat, Zaid M., 2009. "Exact solitary solutions for variants of the KdV equations with fractional time derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1264-1270.
    7. Marwan Abukhaled, 2013. "Variational Iteration Method for Nonlinear Singular Two-Point Boundary Value Problems Arising in Human Physiology," Journal of Mathematics, Hindawi, vol. 2013, pages 1-4, February.
    8. Mossa Al-sawalha, M. & Noorani, M.S.M., 2009. "A numeric–analytic method for approximating the chaotic Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1784-1791.
    9. Soliman, A.A., 2009. "On the solution of two-dimensional coupled Burgers’ equations by variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1146-1155.
    10. Macías-Díaz, J.E., 2018. "A numerically efficient Hamiltonian method for fractional wave equations," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 231-248.
    11. Tuan Hoang, Manh & Nagy, A.M., 2019. "Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 24-34.
    12. Goh, S.M. & Noorani, M.S.M. & Hashim, I., 2009. "Efficacy of variational iteration method for chaotic Genesio system – Classical and multistage approach," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2152-2159.
    13. Tomar, Saurabh & Singh, Mehakpreet & Vajravelu, Kuppalapalle & Ramos, Higinio, 2023. "Simplifying the variational iteration method: A new approach to obtain the Lagrange multiplier," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 640-644.
    14. Chen, Juan & Zhou, Hua-Cheng & Zhuang, Bo & Xu, Ming-Hua, 2023. "Active disturbance rejection control to stabilization of coupled delayed time fractional-order reaction–advection–diffusion systems with boundary disturbances and spatially varying coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    15. Yu, Yongguang & Li, Han-Xiong, 2008. "The synchronization of fractional-order Rössler hyperchaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(5), pages 1393-1403.
    16. Kateryna Marynets, 2021. "Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem," Mathematics, MDPI, vol. 9(7), pages 1-19, March.
    17. Odibat, Zaid M., 2009. "Computational algorithms for computing the fractional derivatives of functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(7), pages 2013-2020.
    18. Rehman, Mujeeb ur & Idrees, Amna & Saeed, Umer, 2017. "A quadrature method for numerical solutions of fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 38-49.
    19. Lenzi, E.K. & Menechini Neto, R. & Tateishi, A.A. & Lenzi, M.K. & Ribeiro, H.V., 2016. "Fractional diffusion equations coupled by reaction terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 9-16.
    20. Das, S., 2009. "A note on fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2074-2079.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:41:y:2009:i:3:p:1095-1104. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.