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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)
Tuckwell, H.C.: Introduction to Theoretical Neurobiology, vol. 1. Cambridge University Press, Cambridge (1988)
Tuckwell, H.C.: Introduction to Theoretical Neurobiology, vol. 2. Cambridge University Press, Cambridge (1988)
Tuckwell, H.C.: Stochastic Processes in the Neurosciences. SIAM, Philadelphia (1989)
Miroslav Musila, P.: Generalized Stein’s model for anatomically complex neurons. BioSystems 25, 179–191 (1991)
Sampath, G., Srinivasan, S.K.: Stochastic models for spike trains of single neurons. Lecture Notes in Biomathematics. Springer, Berlin (1977)
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)
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)
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)
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)
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)
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)
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)
Shih, S.D.: On a class of singularly perturbed parabolic equations. Z. Angew. Math. Mech. 81, 337–345 (2001)
Shishkin, G.I.: Robust novel high-order accurate numerical methods for singularly perturbed convection–diffusion problems. Math. Model. Anal. 10, 393–412 (2005)
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)
Ramos, J.I.: A piecewise-analytical method for singularly perturbed parabolic problems. Appl. Math. Comput. 161, 501–512 (2005)
Ramos, J.I.: An exponentially-fitted method for singularly perturbed, one-dimensional, parabolic problems. Appl. Math. Comput. 161, 513–523 (2005)
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)
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)
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)
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)
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)
Tian, H.: Numerical treatment of singularly perturbed delay differential equations. Ph.D. thesis, University of Manchester (2000)
Feldstein, M.A.: Discretization methods for retarded ordinary differential equations. Ph.D. thesis, University of California, Los Angeles (1964)
Bobisud, L.: Second-order linear parabolic equations with a small parameter. Arch. Rat. Mech. Anal. 27, 385–397 (1968)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Clarendon Press, Oxford (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nageshwar Rao, R., Pramod Chakravarthy, P. Fitted Numerical Methods for Singularly Perturbed One-Dimensional Parabolic Partial Differential Equations with Small Shifts Arising in the Modelling of Neuronal Variability. Differ Equ Dyn Syst 27, 1–18 (2019). https://doi.org/10.1007/s12591-017-0363-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-017-0363-9