Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems

Abstract

We show that each isolated solution, y(t), of the general nonlinear two-point boundary value problem (*): y’=f(t,y), a < t < b, g(y(a),y(b))=0 can be approximated by the (box) difference scheme (**):[u_j - u_(j-1)]/h_j = f(t_(j-½),[u_j + u_(j-1)]/2), 1 ≦ j ≦ J, g(U_0,U_J) = O. For h = max_(1 ≦j≦J)h_j sufficiently small, the difference equations (**) are shown to have a unique solution {U_j}^J_0} in some sphere about {y(t_j)}^J_0, and it can be computed by Newton’s method which converges quadratically. If y(t) is sufficiently smooth, then the error has an asymptotic expansion of the form u_j - y(t_j) = Σ^(m)_(v=1) h^(2v) e_v(t_j) + O(h^(2m+2), so that Richardson extrapolation is justified. The coefficient matrices of the linear systems to be solved in applying Newton’s method are of order n(J + l) when y(t) ∈ ℝ^n. For separated endpoint boundary conditions: g_1(y(a)) = 0, g_2(y(b)) = 0 with dim g_1 = p, dim g_2 = q and p + q = n, the coefficient matrices have the special block tridiagonal form A ≡ [B_j, A_j, C_j] in which the n x n matrices B_j(C_j) have their last q (first p) rows null. Block elimination and band elimination without destroying the zero pattern are shown to be valid. The numerical scheme is very efficient, as a worked out example illustrates.