System Of Fourth-order Boundary Value Problems Research Articles

Element-Free Galerkin Method Based on Block-Pulse Wavelets Integration for Solving Fourth-Order Obstacle Problem

We introduce improved element-free Galerkin method based on block pulse wavelet integration for numerical approximations to the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. Moving least squares (MLS) approach is used to construct shape functions with optimized weight functions and basis. Numerical results for test problems are presented in this article to elaborate the pertinent features for the proposed technique. Comparison with existing techniques shows that our proposed method based on integration technique provides better approximation at reduced computational cost.

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.

Galerkin's finite element formulation of the system of fourth-order boundary-value problems

In this article, a Galerkin's finite element approach based on weighted-residual is presented to find approximate solutions of a system of fourth-order boundary-value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece-wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non-polynomial spline and cubic non-polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011

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

Abstract — In this paper we use quintic non-polynomialspline functions to develop numerical methods for approxi-mation to the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral and contactproblems. The convergence analysis of the methods has beendiscussed and shown that the given approximations are betterthan collocation and finite difference methods. Numericalexamples are presented to illustrate the applications of thesemethods, and to compare the computed results with otherknown methods. Keywords —Quintic non-polynomial spline, Boundary for-mula, Convergence, Obstacle problems. I. I NTRODUCTION In this paper, we apply non-polynomial splinefunctions to develop numerical methods for ob-taining smooth approximations to the solution of asystem of fourth-order boundary-value problem ofthe form:u (4) =  f(x), a ≤ x ≤ c,g(x)u(x)+f(x)+r, c ≤ x ≤ d,f(x), d ≤ x ≤ b,(1)subjected to the boundary and continuity condi-tionsu(a) = u(b) = α 1 , u ′′

Convergence of cubic-spline approach to the solution of a system of boundary-value problems

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 O ( h 8 ) 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 applicability and efficiency of our method.

Family of numerical methods based on non-polynomial splines for solution of contact problems

In this paper a new technique based on quartic non-polynomial spline functions connecting spline functions values at mid knots and their corresponding values of the fourth-order derivatives is developed. This approach leads to a family of numerical methods for computing approximations to the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is shown that the present family of methods gives better approximations. Existing second and fourth-order finite-difference and spline functions based methods developed at mid knots become special cases of the new approach. Numerical examples are given to illustrate applicability and efficiency of the new methods.

Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method

A cubic non-polynomial spline technique is developed for the numerical solutions of a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. The end conditions consistent with the BVP are derived corresponding to the boundary conditions in terms of not only second derivatives but first derivatives as well. The present method has less computational cost and gives better approximations than those produced by other collocation, finite difference and spline methods. The method developed is compared with those developed by Khan et al. [A. Khan, M.A. Noor, T. Aziz, Parametric quintic-spline approach to the solution of a system of fourth-order boundary-value problems, Journal of Optimization Theory and Applications 122(2) (2004) 309–322], Al-Said and Noor [E.A. Al-Said, M.A. Noor, Computational methods for fourth-order obstacle boundary-value problems, Communications in Applied Nonlinear Analysis, 2 (1995) 73–83] and Siddiqi and Ghazala [S.S. Siddiqi, G. Akram, Solution of the system of fourth-order boundary-value problems using non polynomial spline technique, Applied Mathematics and Computation 185 (2007) 128–135] through different examples.

A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications

We use a quintic non-polynomial spline functions to develop a numerical method for computing approximations to the solution of a system of fourth-order boundary-value problems associated with plate deflection theory. We show that the present family of methods gives better approximations and generalize all the existing finite difference and spline functions based methods up to order six. Convergence of the methods is shown through standard convergence analysis. Numerical exampls are given to illustrate the applicability and efficiency of the new method.

Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary-Value Problems

In this paper, we use parametric quintic splines to derive some consistency relations which are then used to develop a numerical method for computing 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. Numerical evidence is presented to show the applicability and superiority of the new method over other collocation, finite difference, and spline methods.