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

Abstract

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 ′′