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^αy)′ = f(x,y), 0 < x < 1, αε(0,1) (1) y(0) = A, y(1) = B.

We construct our spline approximation s(x) for the solution y(x) of the two-point boundary value problem (1) such that while (x^αs′)′ ε 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^αy′)′)|_x=xi

The resulting method provides order h² 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.