Fitted mesh methods based on non-polynomial splines for singularly perturbed boundary value problems with mixed shifts

T. Prathap; R. Nageshwar Rao; T. Prathap; R. Nageshwar Rao

doi:10.3934/math.20241285

AIMS Mathematics

2024, Volume 9, Issue 10: 26403-26434. doi: 10.3934/math.20241285

Previous Article Next Article

Research article Special Issues

Fitted mesh methods based on non-polynomial splines for singularly perturbed boundary value problems with mixed shifts

T. Prathap ,
R. Nageshwar Rao ^,

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, Tamil Nadu, India

Received: 25 April 2024 Revised: 01 August 2024 Accepted: 07 August 2024 Published: 11 September 2024
MSC : 65L10, 65L11, 65L12, 65L20, 65L50, 65L70

In this paper, numerical schemes based on non-polynomial splines, namely, spline in compression, tension, and adaptive spline, are constructed for singularly perturbed boundary value problems with mixed shifts. A convergence analysis is carried out on the proposed methods. A comparitive study of the results is performed on test problems and presented in the form of tables. Graphs are drawn to illustrate the behavior of the solution to the problems.
- singular perturbations,
- delay,
- differential equations,
- numerical methods,
- fitted mesh,
- splines
Citation: T. Prathap, R. Nageshwar Rao. Fitted mesh methods based on non-polynomial splines for singularly perturbed boundary value problems with mixed shifts[J]. AIMS Mathematics, 2024, 9(10): 26403-26434. doi: 10.3934/math.20241285

Related Papers:

Abstract

In this paper, numerical schemes based on non-polynomial splines, namely, spline in compression, tension, and adaptive spline, are constructed for singularly perturbed boundary value problems with mixed shifts. A convergence analysis is carried out on the proposed methods. A comparitive study of the results is performed on test problems and presented in the form of tables. Graphs are drawn to illustrate the behavior of the solution to the problems.

References

[1]	R. Bellman, K. L. Cooke, Differential-difference equations, Academic Press, New York, 1963.
[2]	A. Bellen, M. Zennaro, Numerical methods for delay differential equations, Oxford Science Publications, New York, 2003.
[3]	C. G. Lange, R. M. Miura, Singular perturbation analysis of boundary value problems for differential-difference equations. Ⅴ. Small shifts with layer behavior, SIAM J. Appl. Math., 54 (1994), 249–272. https://doi.org/10.1137/S0036139992228120 doi: 10.1137/S0036139992228120
[4]	C. G. Lange, R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations. Ⅵ. Small shifts with rapid oscillations, SIAM J. Appl. Math., 54 (1994), 273–283. https://doi.org/10.1137/S0036139992228119 doi: 10.1137/S0036139992228119
[5]	M. K. Kadalbajoo, K. K. Sharma, Numerical treatment of a mathematical model arising from a model of neuronal variability, J. Math. Anal. Appl., 307 (2005), 606–627. https://doi.org/10.1016/j.jmaa.2005.02.014 doi: 10.1016/j.jmaa.2005.02.014
[6]	M. K. Kadalbajoo, K. C. Patidar, K. K. Sharma, $\epsilon$-Uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs, Appl. Math. Comput., 182 (2006), 119–139. https://doi.org/10.1016/j.amc.2006.03.043 doi: 10.1016/j.amc.2006.03.043
[7]	M. K. Kadalbajoo, V. P. Ramesh, Hybrid method for numerical solution of singularly perturbed delay differential equations, Appl. Math. Comput., 187 (2007), 797–814. https://doi.org/10.1016/j.amc.2006.08.159 doi: 10.1016/j.amc.2006.08.159
[8]	L. Sirisha, K. Phaneendra, Y. N. Reddy, Mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition, Ain Shams Eng. J., 9 (2018), 647–654. https://doi.org/10.1016/j.asej.2016.03.009 doi: 10.1016/j.asej.2016.03.009
[9]	M. M. Woldaregay, G. F. Duressa, Higher-order uniformly convergent numerical scheme for singularly perturbed differential difference equations with mixed small shifts, Int. J. Differ. Equat., 2020. https://doi.org/10.1155/2020/6661592 doi: 10.1155/2020/6661592
[10]	M. M. Woldaregay, G. F. Duressa, Robust mid-point upwind scheme for singularly perturbed delay differential equations, Comput. Appl. Math., 40 (2021). https://doi.org/10.1007/s40314-021-01569-5 doi: 10.1007/s40314-021-01569-5
[11]	R. Ranjan, H. S. Prasad, A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts, J. Appl. Math. Comput., 65 (2021), 403–427. https://doi.org/10.1007/s12190-020-01397-6 doi: 10.1007/s12190-020-01397-6
[12]	D. Kumar, M. K. Kadalbajoo, A parameter uniform method for singularly perturbed differential-difference equations with small shifts, J. Numer. Math., 21 (2013), 1–22. https://doi.org/10.1515/jnum-2013-0001 doi: 10.1515/jnum-2013-0001
[13]	M. K. Jain, Spline function approximation in discrete mechanics, Int. J. Nonlin. Mech., 14 (1979), 341–345. https://doi.org/10.1016/0020-7462(79)90007-6 doi: 10.1016/0020-7462(79)90007-6
[14]	M. K. Kadalbajoo, R. K. Bawa, Variable-mesh difference scheme for singularly-perturbed boundary-value problems using splines, J. Optimiz. Theory Appl., 90 (1996), 405–416. https://doi.org/10.1007/BF02190005 doi: 10.1007/BF02190005
[15]	M. K. Kadalbajoo, K. C. Patidar, Numerical solution of singularly perturbed two-point boundary value problems by spline in tension, Appl. Math. Comput., 13 (2002), 299–320. https://doi.org/10.1016/S0096-3003(01)00146-1 doi: 10.1016/S0096-3003(01)00146-1
[16]	M. K. Kadalbajoo, K. C. Patidar, Spline techniques for the numerical solution of singular perturbation problems, J. Optimiz. Theory Appl., 112 (2002), 575–594. https://doi.org/10.1023/A:1017968116819 doi: 10.1023/A:1017968116819
[17]	T. Aziz, A. Khan, A spline method for second-order singularly perturbed boundary-value problems, J. Comput. Appl. Math., 147 (2002), 445–452. https://doi.org/10.1016/S0377-0427(02)00479-X doi: 10.1016/S0377-0427(02)00479-X
[18]	R. K. Mohanty, N. Jha, A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems, Appl. Math. Comput., 168 (2005), 704–716. https://doi.org/10.1016/j.amc.2004.09.049 doi: 10.1016/j.amc.2004.09.049
[19]	R. K. Mohanty, U. Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Appl. Math. Comput., 172 (2006), 531–544. https://doi.org/10.1016/j.amc.2005.02.023 doi: 10.1016/j.amc.2005.02.023
[20]	P. P. Chakravarthy, S. D. Kumar, R. N. Rao, D. P. Ghate, A fitted numerical scheme for second order singularly perturbed delay differential equations via cubic spline in compression, Adv. Differ. Equat., 300 (2015). https://doi.org/10.1186/s13662-015-0637-x doi: 10.1186/s13662-015-0637-x
[21]	P. P. Chakravarthy, S. D. Kumar, R. N. Rao, Numerical solution of second order singularly perturbed delay differential equations via cubic spline in tension, Int. J. Appl. Comput. Math., 3 (2017), 1703–1717. https://doi.org/10.1007/s40819-016-0204-5 doi: 10.1007/s40819-016-0204-5
[22]	A. R. Kanth, M. K. P. Murali, A numerical technique for solving nonlinear singularly perturbed delay differential equations, Math. Model. Anal., 23 (2018), 64–78. https://doi.org/10.3846/mma.2018.005 doi: 10.3846/mma.2018.005
[23]	R. B. Kellogg, A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput., 32 (1978), 1025–1039. https://doi.org/10.1090/S0025-5718-1978-0483484-9 doi: 10.1090/S0025-5718-1978-0483484-9
[24]	R. S. Varga, Matrix iterative analysis, Prentice-Hall Inc, Englewood Cliffs, New Jersey, 1962.
[25]	D. M. Young, Iterative solution of large linear systems, Academic Press, New York, 1971.
[26]	J. M. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., 11 (1975), 3–5. https://doi.org/10.1016/0024-3795(75)90112-3 doi: 10.1016/0024-3795(75)90112-3

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)