Quartic spline method for solving fourth order obstacle boundary value problems (Q1612333)

The authors consider the following fourth-order obstacle boundary value problem \[ u^{(iv)}= \begin{cases} f(x),\quad &a\leq x\leq c,\\ g(x)u(x)+ f(x)+ r,\quad & c\leq x\leq d,\\ f(x),\quad & d\leq x\leq b\end{cases} \] with the boundary and continuity conditions \[ u(a)= u(b)=\alpha 1,\quad u''(a)= u''(b)=\alpha 2,\quad u(c)= u(d)=\beta 1,\quad u''(c)= u''(d)=\beta 2. \] For this problem uniform quartic polynomial splines are used to develop a new method, which is of order two and gives approximations which are better than those produced by other collocation and finite difference methods. Numerical examples and comparisons with other methods are given.