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

Authors: Jalil Rashidinia, Reza Jalilian

Abstract:

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.

Keywords: Quintic non-polynomial spline, Boundary formula, Convergence, Obstacle problems.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1055577

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.