• Fouad Mohammad Salama school of mathematical sciences, Universiti Sains Malaysia, Penang, Malaysia
  • Norhashidah Hj. Mohd Ali school of mathematical sciences, Universiti Sains Malaysia, Penang, Malaysia
Keywords: Fractional cable equation, Caputo fractional derivative, Laplace transform, Finite difference scheme, Stability.


It is time and memory consuming when numerically solving time fractional differential equations as it requires O(N^2) computational cost and O(NM) memory complexity. N and M are the total number of time levels and space grid points, respectively. In this paper, we present an efficient hybrid method with O(N) computational cost and O(M) memory complexity in solving the two-dimensional time fractional cable equation. The Laplace transform method and implicit finite difference scheme are used to derive the hybrid method. The stability of the numerical scheme has been carried out. Numerical results show that the hybrid method compares well with the exact solution and performs faster compared to a standard finite difference scheme.


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K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, (1993).

S. G. Samko, A. A. Kilbas, O. I. Marichey, Fractional integrals and derivatives: theory and applications, Gordon & Breach, Yverdon, (1993).

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

R. L. Bagley, P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, Journal of applied mechanics, 51, No. 2, (1984), 294–298.

F. Mainardi, Fractals and Fractional Calculus Continuum Mechanics, Springer, Bologna, (1997).

B. I. Henry, S. L. Wearne, Fractional reaction–diffusion, Physica A: Statistical Mechanics and its Applications, 276, No. 3-4, (2000), 448-455.

R. Metzler, E. Barkai, J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Physical review letters, 82, No. 18, (1999), 3563-3567.

R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A: Mathematical and General, 37, No. 31, (2004), 161-208.

Magin, R.L., Fractional Calculus in Bioengineering, Begell House Publishers, Danbury, (2006).

D. A. Benson, S. W. Wheatcraft, Application of a fractional advection-dispersion equation, Water resources research, 36, No. 6, (2000), 1403–1412.

R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous-time finance III: the diffusion limit, in mathematical finance, Trends in Mathematics, (2001), 171-180.

M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A: Statistical Mechanics and its Applications, 314, No. 1-4, (2002), 749-755.

L. Sabatelli, S. Keating, J. Dudley, P. Richmond, Waiting time distributions in financial markets, The European Physical Journal B-Condensed Matter and Complex Systems, 27, No. 2, (2002), 273-275.

T. A. M. Langlands, B. I. Henry, S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions, SIAM Journal on Applied Mathematics, 71, No. 4, (2011), 1168-1203.

B. I. Henry, T. A. M. Langlands, S. L. Wearne, Fractional cable models for spiny neuronal dendrites, Physical review letters, 100, No. 12, (2008), 128103-4.

N. H. Sweilam, M. M. Khader, M. Adel, Numerical simulation of fractional Cable equation of spiny neuronal dendrites, Journal of advanced research, 5, No. 2, (2014), 253-259.

R. Du, W. R. Cao, Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Applied Mathematical Modelling, 34, No. 10, (2010), 2998–3007.

M. Cui, Compact finite difference method for the fractional diffusion equation, Journal of Computational Physics, 228, No. 20, (2009), 7792–7804.

J. Huang, Y. Tang, L. Vázquez, J. Yang, Two finite difference schemes for time fractional diffusion-wave equation, Numerical Algorithms, 64, No. 4, (2013), 707–720.

Z. Liu, X. Li, A Crank–Nicolson difference scheme for the time variable fractional mobile–immobile advection–dispersion equation, Journal of Applied Mathematics and Computing, 56, No. 1-2, (2018), 391–410.

E. Sousa, Finite difference approximations for a fractional advection diffusion problem, Journal of Computational Physics, 228, No. 11, (2009), 4038–4054.

F. Liu, Q. Yang, I. Turner, Stability and convergence of two new implicit numerical methods for the fractional cable equation, Proceedings of the ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, ASME, (2009), San Diego, USA, 1015-1024.

J. Quintana-Murillo, S. B. Yuste, An explicit numerical method for the fractional cable equation, International journal of differential equations, 2011, (2011), 57-69.

C. M. Chen, F. Liu, K. Burrage, Numerical analysis for a variable-order nonlinear cable equation, Journal of Computational and Applied Mathematics, 236, No. 2, (2011), 209-224.

X. Hu, L. Zhang, Implicit compact difference schemes for the fractional cable equation, Applied Mathematical Modelling, 36, No. 9, (2012), 4027-4043.

I. Karatay, N. Kale, A new difference scheme for fractional cable equation and stability analysis, International Journal of Applied Mathematics Research, 4, No. 1, (2015), 52-57.

H. R. Ghehsareh, A. Zaghian, S. M. Zabetzadeh, The use of local radial point interpolation method for solving two-dimensional linear fractional cable equation, Neural Computing and Applications, 29, No. 10, (2016), 745-754.

B. Yu, X. Jiang, Numerical identification of the fractional derivatives in the two-dimensional fractional cable equation, Journal of Scientific Computing, 68, No. 1, (2016), 252-272.

A. T. Balasim, N. H. M. Ali, A comparative study of the point implicit schemes on solving the 2D time fractional cable equation, Proceedings of the 24th National Symposium on Mathematical Sciences, AIP Conf. Proc., 1870, 040050-1–040050-7. doi: 10.1063/1.4995882.

M. Z. Li, L. J. Chen, Q. Xu, X. H. Ding, An efficient numerical algorithm for solving the two-dimensional fractional cable equation, Advances in Difference Equations, 2018, No. 1, (2018), 424.

S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Communications in Computational Physics, 21, No. 3, (2017), 650-678.

C. Gong, W. Bao, G. Tang, Y. Jiang, J. Liu, A parallel algorithm for the two-dimensional time fractional diffusion equation with implicit difference method, The Scientific World Journal, 2014, (2014), 1-8.

C. Gong, W. Bao, G. Tang, Y. Jiang, J. Liu, Computational challenge of fractional differential equations and the potential solutions: a survey, Mathematical Problems in Engineering, 2015, (2015), 1-13.

K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations, Fractional Calculus and Applied Analysis, 14, No. 3, (2011), 475–490.

W. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations, Journal of Computational and Applied Mathematics, 206, No. 1, (2007), 174–188.

S. L. Lei, H. W. Sun, A circulant preconditioner for fractional diffusion equations, Journal of Computational Physics, 242, (2013), 715–725.

J. Ren, Z. Z. Sun, W. Dai, New approximations for solving the Caputo-type fractional partial differential equations, Applied Mathematical Modelling, 40, No. 4, (2016), 2625-2636.

M. Bishehniasar, S. Salahshour, A. Ahmadian, F. Ismail, D. Baleanu, An accurate approximate-analytical technique for solving time-fractional partial differential equations, Complexity, 2017, (2017), 1-12.

Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225 No. 2, (2007), 1533-155

How to Cite
Salama, F. M., & Ali, N. H. M. (2019). FAST O(N) HYBRID METHOD FOR THE SOLUTION OF TWO DIMENSIONAL TIME FRACTIONAL CABLE EQUATION. COMPUSOFT: An International Journal of Advanced Computer Technology, 8(11). Retrieved from