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Volume 7, Issue 3
A Parameter-Uniform Finite Difference Method for Singularly Perturbed Linear Dynamical Systems

S. Valarmathi & J. J. H. Miller

Int. J. Numer. Anal. Mod., 7 (2010), pp. 535-548.

Published online: 2010-07

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  • Abstract

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise-uniform mesh is constructed, which is used, in conjunction with a classical finite difference discretization, to form a new numerical method for solving this problem. It is proved that the numerical approximations obtained from this method are essentially first order convergent uniformly in all of the parameters. Numerical results are presented in support of the theory.

  • Keywords

Linear dynamical system, multiscale, initial value problem, singularly perturbed, finite difference method, parameter-uniform convergence.

  • AMS Subject Headings

65L05, 65L12, 65L20, 65L70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-7-535, author = {}, title = {A Parameter-Uniform Finite Difference Method for Singularly Perturbed Linear Dynamical Systems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2010}, volume = {7}, number = {3}, pages = {535--548}, abstract = {

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise-uniform mesh is constructed, which is used, in conjunction with a classical finite difference discretization, to form a new numerical method for solving this problem. It is proved that the numerical approximations obtained from this method are essentially first order convergent uniformly in all of the parameters. Numerical results are presented in support of the theory.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/736.html} }
TY - JOUR T1 - A Parameter-Uniform Finite Difference Method for Singularly Perturbed Linear Dynamical Systems JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 535 EP - 548 PY - 2010 DA - 2010/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/736.html KW - Linear dynamical system, multiscale, initial value problem, singularly perturbed, finite difference method, parameter-uniform convergence. AB -

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise-uniform mesh is constructed, which is used, in conjunction with a classical finite difference discretization, to form a new numerical method for solving this problem. It is proved that the numerical approximations obtained from this method are essentially first order convergent uniformly in all of the parameters. Numerical results are presented in support of the theory.

S. Valarmathi & J. J. H. Miller. (1970). A Parameter-Uniform Finite Difference Method for Singularly Perturbed Linear Dynamical Systems. International Journal of Numerical Analysis and Modeling. 7 (3). 535-548. doi:
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