Volume 7, Issue 1
A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory

R. Nageshwar Rao & P. Pramod Chakravarthy

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 23-40.

Published online: 2014-07

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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

Singular perturbations, parabolic partial differential equation, exponentially fitted method, differential-difference equations.

  • AMS Subject Headings

65L11

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-23, author = {}, title = {A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {1}, pages = {23--40}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1316nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5864.html} }
TY - JOUR T1 - A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 23 EP - 40 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1316nm UR - https://global-sci.org/intro/article_detail/nmtma/5864.html KW - Singular perturbations, parabolic partial differential equation, exponentially fitted method, differential-difference equations. AB -

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.

R. Nageshwar Rao & P. Pramod Chakravarthy. (2020). A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory. Numerical Mathematics: Theory, Methods and Applications. 7 (1). 23-40. doi:10.4208/nmtma.2014.1316nm
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