Numerical Methods for Sixth-Order Boundary Value Problems

A second-order convergent finite difference method is discussed for the numerical solution of the special nonlinear sixth-order boundary-value problem w(vi) (x) — f(x, w), a < x < b, w(a) = A0, w″(a) = A2, w(iv)(a) — A4, w(b) — B0, w″(b) — B2, w(iv)(b) — B4. Adaptation to the special linear problem with differential equation w(vi) (x) = q(x)w(x) + r(x) is considered briefly. Fourth- and sixth-order convergence is obtained by using the numerical method on two and three grids, respectively, and taking a linear combination of the individual results relating to the extra grid(s). Special formulas are developed for application to grid points adjacent to the boundaries x — a and x — b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at other points of each grid.