Non-Polynomial Spline Solution of Fourth-Order Obstacle Boundary-Value Problems

In this paper we use quintic non-polynomial spline functions to develop numerical methods for approximation to the solution of a system of fourth-order boundaryvalue problems associated with obstacle, unilateral and contact problems. The convergence analysis of the methods has been discussed and shown that the given approximations are better than collocation and finite difference methods. Numerical examples are presented to illustrate the applications of these methods, and to compare the computed results with other known methods.

Authors:

• Jalil Rashidinia
• Reza Jalilian

Keywords:

• Quintic non-polynomial spline
• Boundary formula
• Convergence
• Obstacle problems.

References:

[1] AL-SAID, E.A. and NOOR, M.A., Quartic Spline Method for
Solving Fourth-Order Obstacle Boundary Value Problems, Journal
of Computational and Applied Mathematics Vol.143,pp.107-
116,2002.
[2] AL-SAID,E.A. NOOR,M.A. Computational Methods for Fourth-
Order Obstacle Boundary Value Problems, Comm. Appl. Nonlinear.
Anal. Vol.2,pp.73-83,1995.
[3] AL-SAID, E.A. NOOR,M.A. and RASSIAS,T.M., Cubic Splines
Method for Solving Fourth-Order Obstacle Problems, Appl. Math.
Comput.,Vol.174,
pp.180-187,2006.
[4] AL-SAID,E.A., NOOR,M.A., KAYA,D., Al-KHALED,K., Finite
difference method for solving fourth-order obstacle problems, Int.
J. Comput. Math. Vol.81,pp.741-748,2004.
[5] BAIOCCHI,C.C. and CALEO,A., Variational and quasivariational
Inequalities, John Wiley and Sons, New York, 1984.
[6] CHAWLA,M.M. and SUBRAMANIAN, High accuracy quintic
spline solution of fourth-order two-Point boundary value problems,
Int. J. Computer. Math. Vol.31,pp.87-94,1989.
[7] HENRICI,P., Discrete variable method in ordinary differential
equations, John Wiley, New York, 1961.
[8] JAIN,M.K., Numerical solution of differential equations, Second
Editions,Wiley Eastern Limited, 1984.
[9] JAIN,M.K., IYANGER,S.R.K. and SOLDHANHA,J.S.V., Numerical
solution of a fourth-order ordinary differential equation,
J. Engg. Math. Vol.11,pp.373-380,1977.
[10] KIKUCHI,N., and ODEN,J.T., Contact problem in elasticity,
SIAM, Publishing Co. Philadelphia,1988.
[11] KHALIFA,A.K. and NOOR,M.A., Quintic spline solutions
of a class of contact problems, Math. Comput. Modlelling
Vol.13,pp.51-58,1990.
[12] LEWY,H. and STAMPACCHIA,G., On the regularity of the
solution of the variational inequalities, Comm. pure and Appl.
Math. Vol.22,pp.153-188,1969.
[13] NOOR,M.A. and Al-SAID,E.A., Fourth order obstacle problems.
In: Th.M.Rassias and H.M.Srivastava(Eds), Analytic and
geometric inequalities and applications, kluwer Academic Publishers,
Dordrecht, Netherlands,pp.277-300,1999.
[14] NOOR,M.A. and AL-SAID,E.A., Numerical solution of
fourth order variational inequalities, Inter. J. Comput. Math.
Vol.75,pp.107-116,2000.
[15] PAPAMICHEL,N. and WORSEY,E.A., A cubic spline method
for the solution of linear fourth-order two-point boundary value
problem, J. Comput. Appl. Math. Vol.7,pp.187-189,1981.
[16] RASHIDINIA, J., Applications of splines to the numerical
solution of differential equations, Ph.D. thesis, Aligarh Muslim
University, Aligarh, India, 1994.
[17] SIRAJ-UL-ISLAM, TIRMIZI,S.I.A. and SAADAT ASHRAF, A
class of method based on non-polynomial spline function for the
solution of a special fourth-order boundary-value problems with
engineering applications, Appl. Math. Comput. Vol.174,pp.1169-
1180,2006.
[18] USMANI,R.A. and WARSI,S.A., Smooth spline solutions for
boundary value problems in plate deflection thoery, Comput.
Maths. with Appls. Vol.6, pp.205-211,1980.
[19] USMANI,R.A., Discrete variable method for a boundary
value problem with engineering applications, Math. Comput.
Vol.32,1087-1096,1978.
[20] Van DAELE,M., VANDEN BERGHE,G. and De MEYER,H., A
smooth approximation for the solution of a fourth-order boundary
value problem based on non-polynomial splines, J. Comput. Appl.
Math. Vol.pp.51,383-394,1994.