A convergence analysis of orthogonal spline collocation for solving two-point boundary value problems without the boundary subintervals

We consider a new Hermite cubic orthogonal spline collocation (OSC) scheme to solve a two-point boundary value problem (TPBVP) with boundary subintervals excluded from the given interval. Such TPBVPs arise, for example, in the alternating direction implicit OSC solution of parabolic problems on arbitrary domains. The scheme involves transfer of the given Dirichlet boundary values to the end points of the interior interval. The convergence analysis shows that the scheme is of optimal fourth order accuracy in the maximum norm. Numerical results confirm the theoretical results.