On the stability of a classical second order method for solving a class of two-point boundary value problems (Q1316161)

The authors consider the stability of a classical second order method for the following two-point boundary value problem: \(y^{(4)} + f(x,y) = 0\), \(0 \leq x \leq 1\), \(y(0) = A_ 1\), \(y(1) = B_ 1\), \(y''(0) = A_ 2\), \(y''(1) = B_ 2\); \(A_ 1\), \(A_ 2\), \(B_ 1\) and \(B_ 2\) being constants. It is shown that, for sufficiently small \(h\), this method is stable for all \(\partial f/\partial y\) satisfying \(-\pi^ 4 < \partial f/\partial y < 4K^ 4\), where \(K\) is the smallest positive root of \(\text{tan }K = \text{tanh }K\).