Uniform Difference Scheme on the Singularly Perturbed System

Author(s)
Ilhame G. Amiraliyeva

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.

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.

I. Amiraliyeva, "Uniform Difference Scheme on the Singularly Perturbed System,"

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

[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