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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]
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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

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