Spline in Compression Methods for Singularly Perturbed 1D Parabolic Equations with Singular Coefficients

Affiliation(s)

Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, New Delhi, India.

Department of Mathematics, Deenbandhu Chhotu Ram University of Science & Technology, Murthal, India.

Department of Mathematics, Bharati Vidyapeeth’s College of Engineering,Guru Gobind Singh Indraprastha University, New Delhi, India.

Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, New Delhi, India.

Department of Mathematics, Deenbandhu Chhotu Ram University of Science & Technology, Murthal, India.

Department of Mathematics, Bharati Vidyapeeth’s College of Engineering,Guru Gobind Singh Indraprastha University, New Delhi, India.

ABSTRACT

In this article, we discuss three difference schemes; for the numerical solution of singularity perturbed 1-D parabolic equations with singular coefficients using spline in compression. The proposed methods are of accurate and applicable to problems in both cases singular and non-singular. Stability theory of a proposed method has been discussed and numerical examples have been given in support of the theoretical results.

In this article, we discuss three difference schemes; for the numerical solution of singularity perturbed 1-D parabolic equations with singular coefficients using spline in compression. The proposed methods are of accurate and applicable to problems in both cases singular and non-singular. Stability theory of a proposed method has been discussed and numerical examples have been given in support of the theoretical results.

Cite this paper

R. Mohanty, V. Dahiya and N. Khosla, "Spline in Compression Methods for Singularly Perturbed 1D Parabolic Equations with Singular Coefficients,"*Open Journal of Discrete Mathematics*, Vol. 2 No. 2, 2012, pp. 70-77. doi: 10.4236/ojdm.2012.22013.

R. Mohanty, V. Dahiya and N. Khosla, "Spline in Compression Methods for Singularly Perturbed 1D Parabolic Equations with Singular Coefficients,"

References

[1] P. Henrici, “Discrete Variable Methods in Ordinary Differential Equations,” John Wiley, New York, 1962.

[2] J. H. Ahlberg, E. N. Nilson and J. H. Walsh, “The Theory of Splines and Their Applications,” Academic Press, New York, 1967.

[3] T. N. E. Greville, “Theory and Applications of Spline Functions,” Academic Press, New York, 1969.

[4] R. E. O’Malley, “Introduction to Singular Perturbations,” Academic Press, New York, 1974.

[5] L. R. Abrahamsson, H. B. Keller and H. O. Kreiss, “Difference Approximations for Singular Perturbations of System of Ordinary Differential Equations,” Numerische Mathematik, Vol. 22, No. 5, 1974, pp. 367-391. doi:10.1007/BF01436920

[6] P. M. Prenter, “Splines and Variational Methods,” John Wiley, New York, 1975.

[7] C. de Boor, “A Practical Guide to Splines, Applied Mathematical Science Series 27,” Springer-Verlag, New York, 1978.

[8] P. W. Hemker and J. J. H. Miller, “Numerical Analysis of Singular Perturbation Problems,” Academic Press, New York, 1979.

[9] A. Berger, J. M. Solomon and M. Ciment, “An Analysis of a Uniformly Accurate Difference Method for a Singular Perturbation Problem,” Mathematics of Computation, Vol. 37, No. 155, 1981, pp. 79-94. doi:10.1090/S0025-5718-1981-0616361-0

[10] B. Kreiss and H. O. Kreiss, “Numerical Methods for Singular Perturbation Problems,” SIAM Journal on Numerical Analysis, Vol.18, No. 2, 1981, pp. 262-276. doi:10.1137/0718019

[11] A. Segal, “Aspects of Numerical Methods for Elliptic Singular Perturbation Problems,” SIAM Journal on Scientific and Statistical Computing, Vol. 3, No. 3, 1982, pp. 327-349. doi:10.1137/0903020

[12] M. K. Jain and T. Aziz, “Numerical Solution of Stiff and Convection-Diffusion Equations using Adaptive Spline Function Approximation,” Applied Mathematical Modelling, Vol.7, No. 1, 1983, pp. 57-62. doi:10.1016/0307-904X(83)90163-4

[13] J. J. H. Miller, E. O’Riordan and G. I. Shishkin, “On Piecewise Uniform Meshes for Upwind and Central Difference Operators for Solving Singularly Perturbed Problems,” IMA Journal of Numerical Analysis, Vol. 15, No. 1, 1995, pp. 89-99. doi:10.1093/imanum/15.1.89

[14] M. K. Kadalbajoo and K. C. Patidar, “Numerical Solution of Singularly Perturbed Two Point Boundary Value Problems by Spline in Compression,” International Jour- nal of Computer Mathematics, Vol. 77, No. 2, 2001, pp. 263-284. doi:10.1080/00207160108805064

[15] R. K. Mohanty, N. Jha and D. J. Evans, “Spline in Compression Methods for the Numerical Solution of Singularly Perturbed Two Point Singular Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 81, No. 5, 2004, pp. 615-627. doi:10.1080/00207160410001684307

[16] I. Khan and T. Aziz, “Tension Spline Method for Second Order Singularly Perturbed Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 82, No. 12, 2005, pp. 1547-1553. doi:10.1080/00207160410001684280

[17] A. Khan, I. Khan and T. Aziz, “A Survey on Parametric Spline Function Approximation,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 983-1003. doi:10.1016/j.amc.2005.01.112

[18] R. K. Mohanty, R. Kumar and V. Dahiya, “Spline in Tension Methods for Singularly Perturbed One Space Dimensional Parabolic Equations with Singular Coefficients,” Neural Parallel & Scientific Computations, Vol. 20, No. 1, 2012, pp. 81-92.

[19] Y. Saad, “Iterative Methods for Sparse Linear Systems,” SIAM Publication, Philadelphia, 2003.

[20] L. A. Hageman and D. M. Young, “Applied Iterative Methods,” Dover Publication, New York, 2004.

[1] P. Henrici, “Discrete Variable Methods in Ordinary Differential Equations,” John Wiley, New York, 1962.

[2] J. H. Ahlberg, E. N. Nilson and J. H. Walsh, “The Theory of Splines and Their Applications,” Academic Press, New York, 1967.

[3] T. N. E. Greville, “Theory and Applications of Spline Functions,” Academic Press, New York, 1969.

[4] R. E. O’Malley, “Introduction to Singular Perturbations,” Academic Press, New York, 1974.

[5] L. R. Abrahamsson, H. B. Keller and H. O. Kreiss, “Difference Approximations for Singular Perturbations of System of Ordinary Differential Equations,” Numerische Mathematik, Vol. 22, No. 5, 1974, pp. 367-391. doi:10.1007/BF01436920

[6] P. M. Prenter, “Splines and Variational Methods,” John Wiley, New York, 1975.

[7] C. de Boor, “A Practical Guide to Splines, Applied Mathematical Science Series 27,” Springer-Verlag, New York, 1978.

[8] P. W. Hemker and J. J. H. Miller, “Numerical Analysis of Singular Perturbation Problems,” Academic Press, New York, 1979.

[9] A. Berger, J. M. Solomon and M. Ciment, “An Analysis of a Uniformly Accurate Difference Method for a Singular Perturbation Problem,” Mathematics of Computation, Vol. 37, No. 155, 1981, pp. 79-94. doi:10.1090/S0025-5718-1981-0616361-0

[10] B. Kreiss and H. O. Kreiss, “Numerical Methods for Singular Perturbation Problems,” SIAM Journal on Numerical Analysis, Vol.18, No. 2, 1981, pp. 262-276. doi:10.1137/0718019

[11] A. Segal, “Aspects of Numerical Methods for Elliptic Singular Perturbation Problems,” SIAM Journal on Scientific and Statistical Computing, Vol. 3, No. 3, 1982, pp. 327-349. doi:10.1137/0903020

[12] M. K. Jain and T. Aziz, “Numerical Solution of Stiff and Convection-Diffusion Equations using Adaptive Spline Function Approximation,” Applied Mathematical Modelling, Vol.7, No. 1, 1983, pp. 57-62. doi:10.1016/0307-904X(83)90163-4

[13] J. J. H. Miller, E. O’Riordan and G. I. Shishkin, “On Piecewise Uniform Meshes for Upwind and Central Difference Operators for Solving Singularly Perturbed Problems,” IMA Journal of Numerical Analysis, Vol. 15, No. 1, 1995, pp. 89-99. doi:10.1093/imanum/15.1.89

[14] M. K. Kadalbajoo and K. C. Patidar, “Numerical Solution of Singularly Perturbed Two Point Boundary Value Problems by Spline in Compression,” International Jour- nal of Computer Mathematics, Vol. 77, No. 2, 2001, pp. 263-284. doi:10.1080/00207160108805064

[15] R. K. Mohanty, N. Jha and D. J. Evans, “Spline in Compression Methods for the Numerical Solution of Singularly Perturbed Two Point Singular Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 81, No. 5, 2004, pp. 615-627. doi:10.1080/00207160410001684307

[16] I. Khan and T. Aziz, “Tension Spline Method for Second Order Singularly Perturbed Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 82, No. 12, 2005, pp. 1547-1553. doi:10.1080/00207160410001684280

[17] A. Khan, I. Khan and T. Aziz, “A Survey on Parametric Spline Function Approximation,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 983-1003. doi:10.1016/j.amc.2005.01.112

[18] R. K. Mohanty, R. Kumar and V. Dahiya, “Spline in Tension Methods for Singularly Perturbed One Space Dimensional Parabolic Equations with Singular Coefficients,” Neural Parallel & Scientific Computations, Vol. 20, No. 1, 2012, pp. 81-92.

[19] Y. Saad, “Iterative Methods for Sparse Linear Systems,” SIAM Publication, Philadelphia, 2003.

[20] L. A. Hageman and D. M. Young, “Applied Iterative Methods,” Dover Publication, New York, 2004.