A fourth order spline method for singular two-point boundary value problems (abstract only)

We describe a new spline method for the (weakly) singular two-point boundary value problem: (x α y)′ = f(x,y), 0 We construct our spline approximation s(x) for the solution y(x) of the two-point boundary value problem (1) such that while (x α s′)′ e C[0,1], with the uniform mesh x i = ih, i = 0(1)N, in each sub-interval [x i ,s i+1 ], our spline approximation s(x) linearly spans 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 interpolition conditions L i (s) = L i (y), L i+1 (s) = L i+1 (y) (2) M i (s) = M i (y), M i+1 (s) = M i+1 (y), where L i (y) = y(x i ) and M i (y) = (x α y′)′)| x=x i The resulting method provides order h 2 uniformly convergent approximations for the solution over [0, 1]. We then describe a modification of the above method. In the modified method, we generate the solution at the nodal points by using the recently proposed fourth order method of Chawla [M.M. Chawla, “A fourth order finite difference method based on uniform mesh for singular two-point boundary value problems”, J. Comput. Appl. Math., to appear] and then use the “conditions of continuity” to obtain the smoothed approximations for the linear functionals M i (y) needed for the construction of the spline solution. We show that the resulting method provides order h 4 uniformly convergent spline approximations for the solution y(x) over [0, 1]. The second and the fourth order of the methods described above are verified computationally.