American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: http://www.sciepub.com/journal/ajna Editor-in-chief: Emanuele Galligani
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American Journal of Numerical Analysis. 2015, 3(1), 18-24
DOI: 10.12691/ajna-3-1-3
Open AccessArticle

Quartic B – Spline Method for Solving a Singular Singularly Perturbed Third-Order Boundary Value Problems

Hradyesh Kumar Mishra1, and Sonali saini1

1Department of Mathematics Jaypee University of Engineering & Technology Guna-473226(M.P) India

Pub. Date: March 04, 2015

Cite this paper:
Hradyesh Kumar Mishra and Sonali saini. Quartic B – Spline Method for Solving a Singular Singularly Perturbed Third-Order Boundary Value Problems. American Journal of Numerical Analysis. 2015; 3(1):18-24. doi: 10.12691/ajna-3-1-3

Abstract

In this paper, we study the numerical solution of singular singularly perturbed third-order boundary value problems (BVPs) by using Quartic B-spline method. An efficient algorithm is presented here to solve the approximate solution of the given problem. To understand our method, we introduce the Quartic B-spline basis function in the form of at the different knots. After that we derive our method by using numerical difference formulas to construct the approximate values. Then we use the linear sequence of Quartic B-spline to get the numerical solution of the system of equations. These systems of equations are solved by using MATLAB. Three examples are illustrated to understand the present method.

Keywords:
Singular singularly perturbed two-point problem third-order BVPs Basis function Quartic B-spline

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References:

[1]  Ghazala Akram, Quartic Spline Solution Of A Third Order Singularly Perturbed Boundary Value Problem, ANZIAM J, 53, (2012), 44-58.
 
[2]  C.M. Bender, S.A. Orszag., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
 
[3]  C. De Boor, A Practical Guide To Splines, Springer - Verlag, New York, 1978.
 
[4]  Jincai Chang, Qianli Yang, Long Zhao, Comparison of B-Spline Method and Finite Difference Method to Solve BVP of Linear ODEs, Journal of Computers, 6(10), (2011), 2149-2155.
 
[5]  M. Cui, F. Geng, A computational method for solving third-order singularly perturbed boundary-value problems, Applied Mathematics and Computation, 198, (2008), 896-903.
 
[6]  Yogesh Gupta, Pankaj Srivastava, A Computational Method For Solving Two-Point Boundary Problems Of Order Four, International Journal of Computer Technology and Applications, 2 (5), (2011), 1426-1431.
 
[7]  Yogesh Gupta, Pankaj Kumar Srivastava, Manoj Kumar, Application of B-spline to Numerical Solution of a System of Singularly Perturbed Problems, Mathematica Aeterna, 1 (6), (2011), 405-415.
 
[8]  P.Jorgensen, D.E.Dutkay, “Methods from Multiscale Theory and Wavelets Applied to Non-linear Dynamics”, Operator Theory: Advances and Applications, 167 (2006), 87-126.
 
[9]  M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, International Journal of Computer Mathematics, 82, (2005), 67-76.
 
[10]  Arshad Khan, Islam Khan, Tariq Aziz, Sextic spline solution of singularly perturbed boundary value problem, Applied Mathematics and Computation, 181, (2006), 432-439.
 
[11]  M. Kumar, P. Singh and H.K. Mishra “An initial-value technique for singularly perturbed boundary value problems via cubic spline”, International Journal of Computer Methods, Engineering, Science and Mechanics, 8, (2007), 419-427.
 
[12]  Feng-Gong Lang, Xiao-Ping Xu, A new cubic B-spline method for linear fifth order boundary value problems, Journal of Applied Mathematics and Computing, 36 (1-2), (2011), 101-116.
 
[13]  J.J.H Miller, E. O’ Riordan, G.I. Shiskin, Fitted Numerical Methods for singular perturbation problem, World Scientific, Singapore, 1996.
 
[14]  H.K.Mishra, Atulya K.Nagar, “He-Laplace Method for Linear and Nonlinear Partial Differential Equations” Journal of Applied Mathematics, 2012, (2012)1-16
 
[15]  H.K. Mishra, Sonali Saini, Various Numerical Methods for Singularly Perturbed Boundary Value Problems, American Journal of Applied Mathematics and Statistics, 2 (3), (2014), 129-142.
 
[16]  R.K. Mohanty, Urvashi Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Applied Mathematics and Computation, 172 (1), (2006), 531-544.
 
[17]  R.E., O’ Malley, Jr, A singular singularly-perturbed linear boundary value problem, SIAM Journal of Mathematical Analysis, 10, (1979), 695-708.
 
[18]  J.U. Pandya And H.D. Doctor, Numerical Solution Of Third-Order Singularly Perturbed ODE Of Convection-Diffusion Type Using Spline Collocation, International Journal Of Mathematics And Scientific Computing, 2(2), (2012), 81-85.
 
[19]  J. Prakash, O.D. Makinde, “Radiative Heat Transfer to Blood Flow Through A Stenotic Artery in the Presence of Magnetic Field” Latin American Applied Research, 41, (2011), 273-277.
 
[20]  P.M. Prenter, Spline and Variational Methods, Wiley Classic Edition Publication in 1989.
 
[21]  J. Rashidnia, R. Mohammadi, M. Ghasemi, Cubic Spline solution of Singularly perturbed boundary value problems with significant first derivatives, Applied mathematics and Computation, 190(2), (2007), 1762-1766.
 
[22]  H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verleg, New York, 1996.
 
[23]  R.D. Russell, and L.F. Shampine, Numerical methods for singular boundary value problems, SIAM Journal of Numerical Analysis, 12, (1975), 13-36.
 
[24]  Ikram A. Tirmizi, Fazal-i-Haq and Siraj-ul-islam, Non-polynomial spline solution of singularity perturbed boundary value problem, Applied Mathematics and Computation, 196 (1), (2008), 6-16.