A spline method for solving fourth order singularly perturbed boundary value problem

Talat Sultana

doi:10.26637/mjm301/006

Abstract

In this paper, singularly perturbed boundary value problem of fourth order ordinary differential equation with a small positive parameter multiplying with the highest derivative of the form
$$
\begin{gathered}
\varepsilon u^{(4)}(x)+p(x) u^{\prime \prime}(x)+q(x) u(x)=r(x), 0 \leq x \leq 1, \\
u(0)=\gamma_0, u(1)=\gamma_1, u^{\prime \prime}(0)=\eta_0, u^{\prime \prime}(1)=\eta_1, 0 \leq \varepsilon \leq 1
\end{gathered}
$$
is considered. We have developed a numerical technique for the above problem using parametric and polynomial septic spline method. The method is shown to have second and fourth order convergent depending on the choice of parameters involved in the method. Truncation error and boundary equations are obtained. The method is tested on an example and the results are found to be in agreement with the theoretical analysis.

Keywords:

Parametricsepticsplines, Polynomialsepticsplines, Boundaryvalueproblems, Boundaryequations

Mathematics Subject Classification:

65D07, 65L10, 65L11

Authors Imprint References Funding Related Articles Downloads

Talat Sultana Department of Mathematics, Janki Devi Memorial College, University of Delhi, New Delhi-60, India.

Pages: 51-61
Date Published: 01-01-2015
Vol. 3 No. 01 (2015): Malaya Journal of Matematik (MJM)

A. H. Nayfeh, Introduction to Perfurbation Methouls, John Wiley and Sons, New York, (1981).

B. Semper, Locking in finite element approximation of long thin extensible beams, LMA ]. Numer. Anal., $14(1994), 97-109$ DOI: https://doi.org/10.1093/imanum/14.1.97

E. C. Garland, Graded mesh difference schemes for singularly pertubed two point boundary value problems, Math. Contput., 51(1988), 631-657. DOI: https://doi.org/10.1090/S0025-5718-1988-0935072-1

G. Akram and S. S. Siddiqi, End conditions for interpolatory septic splines, Inf. J. Comput. Math., $12(82)(2005), 1525-1540$ DOI: https://doi.org/10.1080/00207160412331291099

G. B. Loghmani and M. Ahmadinia, Numerical solution of singularly perturbed boundary value problems based on optimal control strategy, Acta, Appl. Math., 112(2010), 69-78. DOI: https://doi.org/10.1007/s10440-009-9553-y

G. Sun and M. Stynes, Finite element methods for singularly perturbed higher order elliptic two point boundary value problems I: Reaction diffusion type, IMA J. Nuner. And., 15(1995), 117-139.

G. Sun and M. Stynes, Finite element methods for singularly perturbed higher order elliptic two point boundary value problems II: Reaction diffusion type, IMA J. Nuner. Anal, 15(1995), 117-139, DOI: https://doi.org/10.1093/imanum/15.1.117

H. G. Roos and M. Stynes, A uniformly convergent discretization method for a fourth order singular perturbation problem, Bonner Math. Schriffen, 228 (1991), 30-40.

H. G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Egantions, Springer, New York, (1996). DOI: https://doi.org/10.1007/978-3-662-03206-0

J. Jayakumar and N. Ramanujam, A computational method for solving singular perturbation problems. Appl. Math. Comput, 55(1993), 31-48. DOI: https://doi.org/10.1016/0096-3003(93)90004-X

J. Jayakumar and $mathrm{N}$. Ramanujam, A numerical method for singular perturbation problems arising in chemical reactor theory, Conput. Math. Applic, $5(27)(1994), 83-99$. DOI: https://doi.org/10.1016/0898-1221(94)90078-7

K. Niederdrenk and H. Yserentant, The uniform stability of singularly perturbed discrete and continuous boundary value problems, Naner. Math., 41(1983), 223-253. DOI: https://doi.org/10.1007/BF01390214

M. Feckan, Singularly perturbed higher order boundary value problems, I. Differential Eagations, $3(1)(1994), 79-102$ DOI: https://doi.org/10.1006/jdeq.1994.1076

M. K. Jain, Numerical Solufion of Diffenential Equations, Wiley eastern Limited, New Delhi, India, (1979).

M. K. Kadalbajoo and V. Gupta, A brief survey on numerical methods for solving singularly perturbed problems, Appl. Math. Conput., 217(2010), 3641-3716. DOI: https://doi.org/10.1016/j.amc.2010.09.059

M. K. Kadalbajoo and Y. N. Reddy, Numerical treatment of singularly perturbed two point boundary value problems, Appl. Math. Comput,, 21(1987), 93-110. DOI: https://doi.org/10.1016/0096-3003(87)90020-8

M. K. Kadalbajoo and Y. N. Reddy, Approxinate nethods for fhe nanericul solution of singular pertuerhation problens, Appl. Math. Comput, 21 (1987) 85-199. DOI: https://doi.org/10.1016/0096-3003(87)90001-4

P. Henrici, Discnte Variable Methods in Ondinary Diffenential Equations, Wiley, New York, (1962).

R. K. Bawa, Spline based computational technique for linear singularly perturbed boundary value problems, Appl. Math. Conput., 167(2005), 225-236. DOI: https://doi.org/10.1016/j.amc.2004.06.112

S. Natesan and $mathrm{N}$. Ramanujam, A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers, App. Math. Conput., 93(1998), 259-275. DOI: https://doi.org/10.1016/S0096-3003(97)10056-X

T. Aziz and A. Khan, A spline method for second order singularly perturbed boundary value problems, 1. Conput. Appl. Math.s 147(2002), 445-452. DOI: https://doi.org/10.1016/S0377-0427(02)00479-X

V. Shanthi and $mathrm{N}$. Ramanujam, A numerical method for boundary value problems for singularly perturbed fourth order ordinary differential equations, Appl. Math. Comput., 129(2002), 269-294. DOI: https://doi.org/10.1016/S0096-3003(01)00040-6

V. Shanthi and N. Ramanujam, Asymptotic numerical methods for singularly perturbed fourth order ordinary differential equations of reaction diffussion type, Int. I. Compuf. Mnth. Applic., 46(2003), 463-478. DOI: https://doi.org/10.1016/S0898-1221(03)90039-3

V. Shanthi and N. Ramanujam, A boundary value technique for boundary value problems for singularly perturbed fourth order ordinary differential equations, Comput. Math. Applics. 47(2004), 1673-1688. DOI: https://doi.org/10.1016/j.camwa.2004.06.015

V. Shanthi and N. Ramanujam, Asymptotic numerical method for boundary value problems for singularly perturbed fourth order ordinary differential equations with a weak interior layer, Appl. Math. Comput., 172(2006), 252-266. DOI: https://doi.org/10.1016/j.amc.2005.01.140