Fractional Reaction-Diffusion Equations for Modelling Complex Biological Patterns

Ku Azlina Ku Akil, Sithi V Muniandy, Einly Lim


We consider a system of nonlinear time-fractional reaction-diffusion equations (TFRDE) on a finite spatial domain x ∈ [0, L], and time t ∈ [0, T]. The system of standard reaction-diffusion equations with Neumann boundary conditions are generalized by replacing the first-order time derivatives with Caputo time-fractional derivatives of order α ∈ (0, 1). We solve the TFRDE numerically using Grünwald-Letnikov derivative approximation for time-fractional derivative and centred difference approximation for spatial derivative. We discuss the numerical results and propose the applications of TFRDE for the modelling of complex patterns in biological systems.


Fractional reaction-diffusion; Fractional derivative; Caputo derivative; Grünwald-Letnikov derivative;

Full Text:



R. Hifler, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.

D. A. Garzon-Alvarado, and A. M. Ramirez-Martinez, Theor. Biol. Med. Model., 8 (2011), 24.

V. Gafiychuk, B. Datsko, V. Meleshko, and D. Blackmore, Chaos, Soliton and Fractals, 41 (2009), 1095–1104.

V. Gafiychuk, B. Datsko, and V. Meleshko, J. Comput. Appl. Math., 220 (2008), 215–225.

B. I. Henry, T. A. M. Langlands, and S. L. Wearne, Phys. Rev. E, 72 (2005), 026101.

J. Buceta, and K. Lindenberg, Physica A, 325 (2003), 230–242. [7]I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.

K. B. Oldham, and S. Spanier, The fractional calculus. Theory and applications of differentiation and integration to arbitrary order, Academic Press, San Diego, 1974.

Y. Nikolova, and L. Boyadjiev, Fract. Calc. Appl. Anal., 13 (2010), 57–67.

F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, In K. Tas et al. (Eds.), Mathematical Methods in Engineering (pp. 23–55), Springer-Verlag, Dordrecht, 2007.

B. I. Henry, and S. L. Wearne, Physica A, 276 (2000), 448–455. [12]R. Metzler, and J. Klafter, Phys. Rep., 339 (2000), 1–77.

J. Q. Murillo, and S. B. Yuste, J. Comput. Nonlin. Dyn., 6 (2011), 021014-1–021014-6.

M. M. Meerschaert, and C. Tadjeran, Appl. Numer. Math., 56 (2006), 80-90.

T. A. M. Langlands, and B. I. Henry, J. Comput. Phys., 205 (2005), 719-736

M. Ciesielski, and J. Leszczynski, Computer Methods in Mechanics CMM–2003, (2003),

V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreire-Mejias, and H. R. Hicks, J. Comput. Phys., 192 (2003), 406–421.

I. Podlubny, A. Chechkin, T. Skovranek, Y. Q. Chen, and B. M. Vinagre Jara, J. Comput. Phys., 228 (2009), 3137–3153.

M. Garg, and P. Manohar, Fract. Calc. Appl. Anal., 13 (2010), 191–207.

B. Baeumer, M. Kovacs, and M. M. Meerschaert, Comput. Math. Appl., 55 (2008), 2212–2226.

M. M. Khader, Commun. Nonlinear Sci., 16 (2011), 2535–2542. [22]Z. Dahmani, A. Anber, and Y. K. Bouraoui, International Journal of Nonlinear Science, 9 (2010), 276–284.

Y. Liu, and B. Xin, Adv. Difference Equ., (2011), doi:10.1155/2011/190475.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C - The Art of Scientific Computing (2nd ed.), Cambridge University Press, New York, 1992.



  • There are currently no refbacks.

Copyright (c) 2014 Ku Azlina Ku Akil, Sithi V Muniandy, Einly Lim

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Copyright © 2005-2019 Penerbit UTM Press, Universiti Teknologi Malaysia. Disclaimer: This website has been updated to the best of our knowledge to be accurate. However, Universiti Teknologi Malaysia shall not be liable for any loss or damage caused by the usage of any information obtained from this website.