Skip to main content

Numerical Solution of the Time Fractional Cable Equation

  • 422 Accesses

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 333)


The time fractional diffusion equation has attracted the attention of many researchers in the last years due to its many applications in different domains. In this article we are concerned with one of these models, the time fractional cable equation:

$$\begin{aligned} \frac{\partial ^2 V(x,t)}{\partial x^2} -\frac{\partial ^{\alpha } V(x,t)}{\partial x^{\alpha }}-V(x,t)=0, \end{aligned}$$

which describes the spatial and temporal dependence of transmembrane potential V(xt) along the axial direction of a cylindrical nerve cell segment. Here \(\frac{\partial ^{\alpha } V(x,t)}{\partial x^{\alpha }}\) is a Caputo-type derivative, with \(0<\alpha <1\). We use a numerical scheme to solve Eq. (1), which is based on the L1-method and on a finite-difference scheme for the time and space discretization, respectively. In order to deal with the singularity at \(t=0\) we use non-uniform meshes. Numerical examples are presented which illustrate the efficiency of the method.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions


  1. Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett. 100(12), 128103 (2008)

    Google Scholar 

  2. Koch, C.: Biophysics of Computation: Information Processing in Single Neuron, vol. 298–300, pp. 43–44. Oxford University Press, New York (2004)

    Google Scholar 

  3. Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional derivative problem in two and three dimensions. Math. Comput. 88, 1–20 (2017)

    MathSciNet  Google Scholar 

  4. Li, C., Zeng, F.: Numerical Methods for Fractional Calculus, pp. 43–46. Chapman Hall Crcs, Boca Raton (2015)

    MATH  Google Scholar 

  5. Norman, R.S.: Cable theory for finite length dendritic cylinders with initial and boundary conditions. Biophys. J. 12(1), 25–45 (1972)

    Google Scholar 

  6. Stynes, M.: Too much regularity may force too much uniqueness. Fract. Calc. Appl. Anal. 19, 1554–1562 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on a graded meshes for a time-fraction diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Teka, W., Marinov, T.M.: Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Comput. Biol. 10(3), 1003526 (2014)

    Google Scholar 

  9. Tuckwell, H.C.: Introduction to Theoretical Neurobiology: Linear Cable Theory and Denditric Structure. Cambridge Studies in Mathematical Biology, vol. 1. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  10. Vitali, S., Castellani, G., Mainardi, F.: Time fractional cable equation and applications in neurophysiology. Chaos Solitons Fractals 102, 467–472 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pedro M. Lima .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Morgado, M.L., Lima, P.M., Mendes, M.V. (2020). Numerical Solution of the Time Fractional Cable Equation. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham.

Download citation