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Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations

Part of: Functional-differential and differential-difference equations Numerical approximation and computational geometry

Published online by Cambridge University Press:  10 November 2015

Devendra Kumar
Devendra Kumar*
Affiliation:
Department of Mathematics, Birla Institute of Technology & Science, Pilani, Rajasthan-333031, India
*
*Corresponding author. Email address:[email protected] (Devendra Kumar)
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Abstract

This paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter ε and the shifts depend on the small parameter ε has been considered. The fitted-mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm. The stability and parameter-uniform convergence analysis of the proposed method have been discussed. The method has been shown to have almost second-order parameter-uniform convergence. The effect of small parameters on the boundary layer has also been discussed. To demonstrate the performance of the proposed scheme, several numerical experiments have been carried out.

Keywords

Singular perturbationdifferential-difference equationsfitted-meshB-spline collocation methodboundary layer

MSC classification

Secondary: 34K10: Boundary value problems 34K28: Numerical approximation of solutions 65D05: Interpolation
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 496 - 514
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Bakhvalov, A. S., On the optimization of methods for solving boundary value problems with boundary layers, Zh. Vychisl. Mat. Mat. Fiz., vol. 9 (1969), pp. 841–859.Google Scholar
[2]Bellman, R. E., and Cooke, K. L., Differential Difference Equations, Academic Press, New York, 1963.CrossRefGoogle Scholar
[3]Cope, D. K., and Tuckwell, H. C., Firing rates of neurons with random excitation and inhibition, J. Theor. Biol., vol 80 (1979), pp. 1–14.CrossRefGoogle ScholarPubMed
[4]Derstine, M. W., Gibbs, H. M., and Kaplan, D. L., Bifurcation gap in a hybrid optical system, Phys. Rev. A, vol 26 (1982), pp. 3720–3722.CrossRefGoogle Scholar
[5]Doolan, E. P., Miller, J. J. H., and Schilders, W. H. A., Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.Google Scholar
[6]Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O'riordan, E., and Shishkin, G. I., Robust Computational Techniques for Boundary Layers, Chapman-Hall/CRC, New York, 2000.CrossRefGoogle Scholar
[7]Farrell, P. A., O'riordan, E., Miller, J. J. H., and Shishkin, G. I., Parameter-uniform fitted mesh method for quasilinear differential equations with boundary layers, Comput. Methods Appl. Math., vol 1 (2001), pp. 154–172.CrossRefGoogle Scholar
[8]Farrell, P. A., O'riordan, E., and Shishkin, G. I., A class of singularly perturbed semilinear differential equations with interior layers, Math. Comp., vol 74 (2005), pp. 1759–1776.CrossRefGoogle Scholar
[9]Fienberg, S. E., Stochastic models for a single neuron firing trains: A survey, Biometrics, vol 30 (1974), pp. 399–427.CrossRefGoogle ScholarPubMed
[10]Gartland, E. C., Graded-mesh difference schemes for singularly perturbed two point boundary value problems, Math. Comp., vol 51 (1988), pp. 631–657.CrossRefGoogle Scholar
[11]Hale, J. K., Functional Differential Equations, Springer-Verlag, New York, 1971.CrossRefGoogle Scholar
[12]Hall, C. A., On error bounds for spline interpolation, J. Approx. Theory, vol 1 (1968), pp. 209–218.CrossRefGoogle Scholar
[13]Holden, A. V., Models of the Stochastic Activity of Neurons, Springer-Verlag, New York, 1976.CrossRefGoogle Scholar
[14]Johannesma, P. I. M., Diffusion models of the stochastic activity of neurons. in: Caianello, E. R. (Ed.), Neural Networks Berlin: Springer.Google Scholar
[15]Kadalbajoo, M. K., and Aggarwal, V. K., Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems, Appl. Math. Comput., vol 161 (2005), pp. 973–987.Google Scholar
[16]Kadalbajoo, M. K., and Patidar, K. C., ε-Uniform fitted mesh finite difference methods for general singular perturbation problems, Appl. Math. Comput., vol 179 (2006), pp. 248–266.Google Scholar
[17]Kadalbajoo, M. K., Patidar, K. C., and Sharma, K. K., ε-uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs, Appl. Math. Comput., vol 182 (2006), pp. 119–139.Google Scholar
[18]Kadalbajoo, M. K., and Sharma, K. K., Numerical analysis of boundary value problems for singularly perturbed differential-difference equations: small shifts of mixed type with rapid oscillations, Comm. Numer. Methods Engrg., vol 20 (2004), pp. 167–182.CrossRefGoogle Scholar
[19]Kadalbajoo, M. K., and Sharma, K. K., Numerical treatment of a mathematical model arising from a model of neuronal variability, J. Math. Anal. Appl., vol 307 (2005), pp. 606–627.CrossRefGoogle Scholar
[20]Kellogg, R. B., and Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp., vol 32 (1978), pp. 1025–1039.CrossRefGoogle Scholar
[21]Lange, C. G., and Miura, R. M., Singular perturbation analysis of boundary-value problems for differential-difference equations V. Small shifts with layer behavior, SIAM J. Appl. Math., vol 54 (1994), pp. 249–272.CrossRefGoogle Scholar
[22]Lange, C. G., and Miura, R. M., Singular perturbation analysis of boundary value problems for differential difference equations. VI. Small shifts with rapid oscillations, SIAM J. Appl. Math., vol 54 (1994), pp. 273–283.CrossRefGoogle Scholar
[23]Longtin, A., and Milton, J., Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci., vol 90 (1988), pp. 183–199.CrossRefGoogle Scholar
[24]Mohapatra, J., and Natesan, S., Uniformly convergent numerical method for singularly perturbed differential-difference equation using grid equidistribution, Int. J. Numer. Meth. Biomed. Engng., vol 27 (2011), pp. 1427–1445.CrossRefGoogle Scholar
[25]Mohapatra, J., and Natesan, S., The parameter-robust numerical method based on defect-correction technique for singularly perturbed delay differential equations with layer behavior, Int. J. Comput. Methods, vol 7 (2010), pp. 573–594.CrossRefGoogle Scholar
[26]Mohapatra, J., and Natesan, S., Uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid, Numer. Math. Theor. Meth. Appl., vol 3 (2010), pp. 1–22.Google Scholar
[27]O'malley, R. E., Introduction to Singular Perturbations, Academic Press, New York, 1974.Google Scholar
[28]Miller, J. J. H., O'riordan, E., and Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.CrossRefGoogle Scholar
[29]Prenter, P. M., Spline and Variational Methods, John Wiley & Sons, New York, 1975.Google Scholar
[30]Roos, H. G., Stynes, M., and Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems, Springer-Verlag, Berlin, 1996.CrossRefGoogle Scholar
[31]Schoenberg, I. J., On Spline Functions. MRC Report 625, University of Wisconsin 1966.Google Scholar
[32]Segundo, J. P., Perkel, D. H., Wyman, H., Hegstad, H., and Moore, G. P., Input-output relations in computersimulated nerve cell: Influence of the statistical properties, strength, number and inter-dependence of excitatory pre-dependence of excitatory pre-synaptic terminals, Kybernetik, vol 4 (1968), pp. 157–171.CrossRefGoogle Scholar
[33]Shishkin, G. I., Difference schemes for singularly perturbed parabolic equation with discontinuous boundary condition, Zh. Vychisl. Mat. Mat. Fiz., vol 28 (1988), pp. 1679–1692.Google Scholar
[34]Smith, C. E., and Smith, M. V., Moments of voltage trajectories for Stein's model with synaptic reversal potentials, J. Theor. Neurobiol., vol 3 (1984), pp. 67–77.Google Scholar
[35]Stein, R. B., Some models ofneuronal variability, Biophys. J., vol 7 (1967), pp. 37–68.CrossRefGoogle Scholar
[36]Stein, R. B., A theoretical analysis of neuronal variability, Biophys. J., vol 5 (1965), pp. 173–194.CrossRefGoogle ScholarPubMed
[37]Tuckwell, H. C., Synaptic transmission in a model for stochastic neural activity, J. Theor. Biol., vol 77 (1979), pp. 65–81.CrossRefGoogle Scholar
[38]Varah, J. M., A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., vol 11 (1975), pp. 3–5.CrossRefGoogle Scholar
[39]Vulanović, R., Non-equidistant generalizations of the Gushchin-Shennikov scheme, Z. Angew. Math. Mech., vol 67 (1987), pp. 625–632.CrossRefGoogle Scholar
[40]Wilbur, W. J., and Rinzel, J., An analysis of Stein's model for stochastic neuronal excitation, Biological Cybernetics, vol 45 (1982), pp. 107–114.CrossRefGoogle Scholar