A fourth-order spline method for singular two-point boundary-value problems

摘要

This paper describes two methods for the solution of (weakly) singular two-point boundary-value problems: Consider the uniform mesh xi = ih, h = 1/N, i = 0(1)N. Define the linear functionals Li(y) = y(xi) and Mi(y) = (x−α(xαy′)′\xv;x=xi. In both these methods a piecewise ‘spline’ solution is obtained in the form s(x) = si(x), x\wE; [xi−1, xi], i = 1(1)N, where in each subinterval si(x) is in the linear span of a certain set of (non-polynomial) basis functions in the representation of the solution y(x) of the two-point boundary value problem and satisfies the interpolation conditions: Li−1(s) = Li−1(y), Li(y), Mi−1(s) = Mi−1(y), Mi(s) = Mi(y). By construction s and x−α(xαs′)′ \wE; C[0,1]. Conditions of continuity are derived to ensure that xαs′ \wE; C[0, 1]. It follows that the unknown parameters yi and Mi(y), i = 1(1)N − 1, must satisfy conditions of the form: The first method consists in replacing Mi(y) by fnof(xi, yi) and solving (*) to obtain the values yi; this method is generalization of the idea of Bickley [2] for the case of (weakly) singular two-point boundary-value problems and provides order h2 uniformly convergent approximations over [0, 1]. As a modification of the above method, in the second method we generate the solution yi at the nodal points by adapting the fourth-order method of Chawla [3] and then use the conditions of continuity (*) to obtain the corresponding smoothed approximations for Mi(y) needed for the construction of the spline solution. We show that the resulting new spline method provides order h4 uniformly convergent approximations over [0, 1]. The second-order and the fourth-order methods are illustrated computationally.