Spline collocation method with four parameters for solving a system of fourth-order boundary-value problems

In this paper, a spline collocation method is applied to solve a system of fourth-order boundary-value problems associated with obstacle, unilateral and contact problems. The presented method is dependent on four collocation points to be satisfied by four parameters θ j ∈(0, 1], j=1(1) 4 in each subinterval. It turns out that the proposed method when applied to the concerned system is a fourth-order convergent method and gives numerical results which are better than those produced by other spline methods [E.A. Al-Said and M.A. Noor, Finite difference method for solving fourth-order obstacle problems, Int. J. Comput. Math. 81(6) (2004), pp. 741–748; F. Geng and Y. Lin, Numerical solution of a system of fourth order boundary value problems using variational iteration method, Appl. Math. Comput. 200 (2008), pp. 231–241; J. Rashidinia, R. Mohammadi, R. Jalilian, and M. Ghasemi, Convergence of cubic-spline approach to the solution of a system of boundary-value problems, Appl. Math. Comput. 192 (2007), pp. 319–331; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using non polynomial spline technique, Appl. Math. Comput. 185 (2007), pp. 128–135; S.S. Siddiqi and G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Appl. Math. Comput. 190(1) (2007), pp. 652–661; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using cubic spline, Appl. Math. Comput. 187(2) (2007), pp. 1219–1227; Siraj-ul-Islam, I.A. Tirmizi, F. Haq, and S.K. Taseer, Family of numerical methods based on non-polynomial splines for solution of contact problems, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 1448–1460]. Moreover, the absolute stability properties appear that the method is A-stable. Two numerical examples (one for each case of boundary conditions) are given to illustrate practical usefulness of the method developed.

[1]  S. M. Mahmoud,et al.  On a class of spline-collocation methods for solving second-order initial-value problems , 2009, Int. J. Comput. Math..

[2]  M. Kerimov,et al.  Modern numerical methods for ordinary differential equations , 1980 .

[3]  Fazhan Geng,et al.  Numerical solution of a system of fourth order boundary value problems using variational iteration method , 2008, Appl. Math. Comput..

[4]  Muhammad Aslam Noor,et al.  Finite difference method for solving fourth-order obstacle problems , 2004, Int. J. Comput. Math..

[5]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[6]  Siraj-ul-Islam,et al.  FAMILY OF NUMERICAL METHODS BASED ON NON-POLYNOMIAL SPLINES FOR SOLUTION OF CONTACT PROBLEMS , 2008 .

[7]  Shahid S. Siddiqi,et al.  Solution of the system of fourth-order boundary value problems using non-polynomial spline technique , 2007, Appl. Math. Comput..

[8]  Jalil Rashidinia,et al.  Convergence of cubic-spline approach to the solution of a system of boundary-value problems , 2007, Appl. Math. Comput..

[9]  Hassan Mohamed El-Hawary,et al.  On some 4-Point Spline Collocation Methods for Solving Ordinary Initial Value Problems , 2002, Int. J. Comput. Math..

[10]  Muhammad Aslam Noor,et al.  Quartic spline method for solving fourth order obstacle boundary value problems , 2002 .

[11]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[12]  Shahid S. Siddiqi,et al.  Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method , 2007, Appl. Math. Comput..

[13]  Shahid S. Siddiqi,et al.  Solution of the system of fourth order boundary value problems using cubic spline , 2007, Appl. Math. Comput..