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 AM  Vol.3 No.9 , September 2012
Uniform Difference Scheme on the Singularly Perturbed System
Abstract: This paper is concerned with the numerical solution for singular perturbation system of two coupled second ordinary differential equations with initial and boundary conditions, respectively. Fitted finite difference scheme on a uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed.
Cite this paper: I. Amiraliyeva, "Uniform Difference Scheme on the Singularly Perturbed System," Applied Mathematics, Vol. 3 No. 9, 2012, pp. 1029-1035. doi: 10.4236/am.2012.39152.
References

[1]   R. E. O’Malley, “Singular Perturbations Methods for Ordinary Differential Equations,” Springer Verlag, New York, 1991. doi:10.1007/978-1-4612-0977-5

[2]   A. H. Nayfeh, “Introduction to Perturbation Techniques,” Wiley, New York, 1993.

[3]   A. M. Ilin, “A Difference Scheme for a Differential Equation with a Small Parameter Affecting the Highest Derivative,” Matematicheskie Zametki, Vol. 6, 1969, pp. 237-248 (Russian).

[4]   E. R. Doolan, J. J. H. Miller and W. H. A. Schilders, “Uniform Numerical Methods for Problems with Initial and Boundary Layers,” Boole Press, Dublin, 1980.

[5]   P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan and G. I. Shishkin, “Robust Computational Techniques for Boundary Layers,” Chapman-Hall/CRC, New York, 2000.

[6]   H. G. Roos, M. Stynes and L. Tobiska, “Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection Diffusion and Flow Problems,” Springer-Verlag, Berlin, Heidelberg, 2008.

[7]   T. Linss, “Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Lecture Notes in Mathematics,” Springer, Heidelberg, 1985.

[8]   G. M. Amiraliyev and H. Duru, “A Uniformly Convergent Finite Difference Method for a Initial Value Problem,” Applied Mathematics and Mechanics, Vol. 20, No. 4, 1999, pp. 363-370. doi:10.1007/BF02458564

[9]   G. M. Amiraliyev, “The Convergence of a Finite Difference Method on Layer-Adapted Mesh for a Singularly Perturbed System,” Applied Mathematics and Computation, Vol. 162, No. 3, 2005, pp. 1023-1034. doi:10.1016/j.amc.2004.01.015

[10]   S. Natesan and B. S. Deb, “A Robust Computational Method for Singularly Perturbed Coupled System of Reaction—Diffusion Boundary-Value Problems,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 353-364. doi:10.1016/j.amc.2006.09.120

[11]   S. Hemavathi, T. Bhuvaneswari, S. Valarmathi and J. J. H. Miller, “A Parameter Uniform Numerical Method for a System of Singularly Perturbed Ordinary Differential Equations,” Applied Mathematics and Computation, Vol. 191, No. 1, 2007, pp. 1-11. doi:10.1016/j.amc.2006.05.218

[12]   T. Linss and M. Stynes, “Numerical Solution of Systems of Singularly Perturbed Differential Equations,” Computational Methods in Applied Mathematics, Vol. 9, No. 2, 2009, pp. 165-191.

[13]   T. Linss, “Analysis of a System of Singularly Perturbed Convection-Diffusion Equations with Strong Coupling,” SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 2009, pp. 1847-1862. doi:10.1137/070683970

[14]   Z. D. Cen, A. M. Xu and A. Le, “A Second-order Hybrid Finite Difference Scheme for a System of Singularly Perturbed Initial Value Problems,” Journal of Computational and Applied Mathematics, Vol. 234, No. 12, 2010, pp. 3445-3457. doi:10.1016/j.cam.2010.05.006

 
 
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