Convergence of cubic-spline approach to the solution of a system of boundary-value problems J. Rashidinia, R. Mohammadi * , R. Jalilian, M. Ghasemi School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran Abstract We use cubic spline to derive some consistency relations which are then used to develop a numerical method for the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Boundary formula of order Oðh 8 Þ are formulated. The most common approach for convergence analysis are using monotonicity of the coefficient matrix. But here we study a new approach and give the convergence of prescribed method, so that the matrix associated with the system of linear equations that arises, is not required to be monotone. Numerical examples are given to show the appli- cability and efficiency of our method. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Cubic spline; System of boundary-value problems; Obstacle problems; Convergence analysis; Monotone matrix 1. Introduction It is known that contact, unilateral, obstacle, and equilibrium problems arising in different branches of pure and applied sciences, can be studied using the notion of variational inequalities. In recent years, this has emerged as an interesting and important branch of applicable mathematics. The idea and approach is being applied to various fields of mathematical and engineering sciences (e.g. elasticity, mechanics, transportation, fluid flow through porous media, optimal control, structural analysis, and economics, etc.) [1–7] and references therein. The area of contact problems in elasticity forms an important foundation for the applications of var- iational inequalities. It has been shown by Kikuchi and Oden [6] that the problem of equilibrium of elastic bodies in contact with a rigid frictionless foundation can be studied in the framework of variational inequal- ities. If the obstacle function is known, then we can apply the technique of Lewy and Stampacchia [7] to characterize the variational inequality by a system of boundary-value problems without constraints. Noor and Al-Said [1] have developed some finite-difference and quintic spline methods for solving fourth-order system of differential equations associated with obstacle and unilateral problems. The possibility of using 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.03.008 * Corresponding author. E-mail address: r_mohammadi@mathdep.iust.ac.ir (R. Mohammadi). Applied Mathematics and Computation 192 (2007) 319–331 www.elsevier.com/locate/amc