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Fitted Numerical Methods for Singularly Perturbed One-Dimensional Parabolic Partial Differential Equations with Small Shifts Arising in the Modelling of Neuronal Variability

  • R. Nageshwar RaoEmail author
  • P. Pramod Chakravarthy
Original Research
  • 136 Downloads

Abstract

In this paper, we presented exponentially fitted finite difference methods for a class of time dependent singularly perturbed one-dimensional convection diffusion problems with small shifts. Similar boundary value problems arise in computational neuroscience in determination of the behavior of a neuron to random synaptic inputs. When the shift parameters are smaller than the perturbation parameter, the shifted terms are expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable and is convergent with order \(O(\Delta t+h^{2})\) where \(\Delta t\) and h respectively the time and space step-sizes. When the shift parameters are larger than the perturbation parameter a special type of mesh is used for the space variable so that the shifts lie on the nodal points and an exponentially fitted scheme is developed. This scheme is also unconditionally stable. By means of two examples, it is shown that the proposed methods provide uniformly convergent solutions with respect to the perturbation parameter. On the basis of the numerical results, it is concluded that the present methods offer significant advantage for the linear singularly perturbed partial differential difference equations.

Keywords

Singular perturbations Parabolic partial differential equation Convection–diffusion Exponentially fitted method 

References

  1. 1.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  2. 2.
    Tuckwell, H.C.: Introduction to Theoretical Neurobiology, vol. 1. Cambridge University Press, Cambridge (1988)CrossRefzbMATHGoogle Scholar
  3. 3.
    Tuckwell, H.C.: Introduction to Theoretical Neurobiology, vol. 2. Cambridge University Press, Cambridge (1988)CrossRefzbMATHGoogle Scholar
  4. 4.
    Tuckwell, H.C.: Stochastic Processes in the Neurosciences. SIAM, Philadelphia (1989)CrossRefzbMATHGoogle Scholar
  5. 5.
    Miroslav Musila, P.: Generalized Stein’s model for anatomically complex neurons. BioSystems 25, 179–191 (1991)CrossRefGoogle Scholar
  6. 6.
    Sampath, G., Srinivasan, S.K.: Stochastic models for spike trains of single neurons. Lecture Notes in Biomathematics. Springer, Berlin (1977)Google Scholar
  7. 7.
    Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: \(\varepsilon \)-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems. IMA J. Numer. Anal. 20, 99–121 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Clavero, C., Jorge, J.C., Lisbona, F., Shishkin, G.I.: An alternating direction scheme on a nonuniform mesh for reaction–diffusion parabolic problem. IMA J. Numer. Anal. 20, 263–280 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman & Hall, London (2000)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: The use of defect correction for the solution of parabolic singular perturbation problems. Z. Angew. Math. Mech. 76, 59–74 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: Novel defect-correction high-order in space and time accurate schemes for parabolic singularly perturbed convection-diffusion problems. Comput. Methods Appl. Math. 3, 387–404 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted numerical methods for singular perturbation problems. Error Estimate in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996)Google Scholar
  13. 13.
    Roos, H.G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations. Convection–Diffusion and Flow Problems, Springer Series in Computational Mathematics, vol. 24. Springer, Berlin (1996)Google Scholar
  14. 14.
    Shih, S.D.: On a class of singularly perturbed parabolic equations. Z. Angew. Math. Mech. 81, 337–345 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shishkin, G.I.: Robust novel high-order accurate numerical methods for singularly perturbed convection–diffusion problems. Math. Model. Anal. 10, 393–412 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Doole Press, Dublin (1980)zbMATHGoogle Scholar
  17. 17.
    Ramos, J.I.: A piecewise-analytical method for singularly perturbed parabolic problems. Appl. Math. Comput. 161, 501–512 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ramos, J.I.: An exponentially-fitted method for singularly perturbed, one-dimensional, parabolic problems. Appl. Math. Comput. 161, 513–523 (2005)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mohanty, R.K.: An implicit high accuracy variable mesh scheme for 1-D non-linear singular parabolic partial differential equations. Appl. Math. Comput. 186, 219–229 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Ramesh, V.P., Kadalbajoo, M.K.: Upwind and midpoint upwind difference methods for time-dependent differential difference equations with layer behavior. Appl. Math. Comput. 202, 453–471 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kumar, D., Kadalbajoo, M.K.: A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations. Appl. Math. Model. 35, 2805–2819 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nageshwar Rao, R., Pramod Chakravarthy, P.: A modified numerov method for solving singularly perturbed differential-difference equations arising in engineering and science. Results Phys. 2, 100–103 (2012)CrossRefGoogle Scholar
  23. 23.
    Nageshwar Rao, R., Pramod Chakravarthy, P.: A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior. Appl. Math. Model. 37, 5743–5755 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tian, H.: Numerical treatment of singularly perturbed delay differential equations. Ph.D. thesis, University of Manchester (2000)Google Scholar
  25. 25.
    Feldstein, M.A.: Discretization methods for retarded ordinary differential equations. Ph.D. thesis, University of California, Los Angeles (1964)Google Scholar
  26. 26.
    Bobisud, L.: Second-order linear parabolic equations with a small parameter. Arch. Rat. Mech. Anal. 27, 385–397 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)zbMATHGoogle Scholar
  28. 28.
    Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Clarendon Press, Oxford (1985)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2017

Authors and Affiliations

  1. 1.Department of Mathematics, School of Advanced SciencesVIT UniversityVelloreIndia
  2. 2.Department of MathematicsVisvesvaraya National Institute of TechnologyNagpurIndia

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