Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T09:19:25.939Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory

Published online by Cambridge University Press:  28 May 2015

R. Nageshwar Rao  and
P. Pramod Chakravarthy
R. Nageshwar Rao*
Affiliation:
Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, 440010, India
P. Pramod Chakravarthy*
Affiliation:
Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, 440010, India
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

In this paper, a fitted Numerov method is constructed for a class of singularly perturbed one-dimensional parabolic partial differential equations with a small negative shift in the temporal variable. Similar boundary value problems are associated with a furnace used to process a metal sheet in control theory. Here, the study focuses on the effect of shift on the boundary layer behavior of the solution via finite difference approach. When the shift parameter is smaller than the perturbation parameter, the shifted term is expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable. When the shift parameter is larger than the perturbation parameter, a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted scheme is developed. This scheme is also unconditionally stable. The applicability of the proposed methods is demonstrated by means of two examples.

Keywords

65L11Singular perturbationsparabolic partial differential equationexponentially fitted methoddifferential-difference equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 1 , February 2014 , pp. 23 - 40
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Wu, J., Theory and Applications of Partial Functional Differential Equations, Springer, NewYork, 1996.Google Scholar
[2] Musila, M. and Lánský, P., Generalized Steinńs model for anatomically complex neurons, Biosystems, 25 (1991), pp. 179–191.CrossRefGoogle ScholarPubMed
[3] Arino, O., Hbid, M. L. and Ait Dads, E., Delay Differential Equations and Applications, Springer, Berlin, 2006.Google Scholar
[4] Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964.Google Scholar
[5] Murray, J. D., Mathematical Biology I: An Introduction, third edition, Springer-Verlag, Berlin, 2001.Google Scholar
[6] Buriea, J. B., Calonnec, A. and Ducrot, A., Singular perturbation analysis of travelling waves for a model in Phytopathology, Math. Modell. Natural Phenomena, 1(1) (2006), pp. 49–62.Google Scholar
[7] Ansari, A. R. and Hegarty, A. F., A note on iterative methods for solving singularly perturbed problems using non-monotone methods on Shishkin meshes, Comput. Methods Appl. Mech. Eng., 192 (2003), pp. 3673–3687.CrossRefGoogle Scholar
[8] Ansari, A. R., Hegarty, A. F. and Shishkin, G. I., A note on fitted operator methods for the classical laminar free jet problem, Appl. Numer. Math., 45 (2003), pp. 353–365.Google Scholar
[9] Ansari, A. R. and Hegarty, A. F., Numerical solution of a convection diffusion problem with Robin boundary conditions, J. Comput. Appl. Math., 156 (2003), pp. 221–238.Google Scholar
[10] Clavero, C., Jorge, J. C., Lisbona, F. and Shishkin, G. I., An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problem, IMA J. Numer. Anal., 20 (2000), pp. 263–280.Google Scholar
[11] Roos, H. G., Stynes, M. and Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems, Springer Series in Computational Mathematics, 24, Springer, Berlin, 1996.Google Scholar
[12] Miller, J. J. H., Óriordan, E. and 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] Farrell, P. A., Hegarty, A. F., Miller, J. J. H., Óriordan, E. and Shishkin, G. I., Robust Computational Techniques for Boundary Layers, Chapman & Hall, London, 2000.Google Scholar
[14] Hemker, P. W., Shishkin, G. I. and 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 (2003), pp. 387–404.Google Scholar
[15] Hemker, P. W., Shishkin, G. I. and Shishkina, L. P., The use of defect correction for the solution of parabolic singular perturbation problems, Z. Angew. Math. Mech., 76 (1997), pp. 59–74.Google Scholar
[16] Hemker, P. W., Shishkin, G. I. and Shishkina, L. P., ǫ-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems, IMA J. Numer. Anal., 20 (2000), pp. 99–121.Google Scholar
[17] Shishkin, G. I., Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems, Math. Model. Anal., 10 (2005), pp. 393–412.Google Scholar
[18] Ansari, A. R., Bakr, S. A. and Shishkin, G. I., A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math., 205 (2007), pp. 552–566.Google Scholar
[19] Wang, Y., An efficient computational method for a class of singularly perturbed delay parabolic partial differential equation, Int. J. Comput. Math., 88(16) (2011), pp. 3496–3506.Google Scholar
[20] Bashier, E. B. M. and Patidar, K. C., A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation, Appl. Math. Comput., 217 (2011), pp. 4728–4739.Google Scholar
[21] Tian, H., Numerical Treatment of Singularly Perturbed Delay Differential Equations, Ph.D. thesis, University of Manchester, 2000.Google Scholar
[22] Doolan, E. P., Miller, J. J. H. and Schilders, W. H. A., Uniform Numerical Methods for Problems with Initial and Boundary Layers, Doole Press, Dublin, 1980.Google Scholar
[23] Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, 1985.Google Scholar