Optimal Order Spline Methods for Nonlinear Diierential and Integro-diierential Equations

In this work, we analyse new robust spline approximation methods for mth order boundary value problems described by nonlinear ordinary diierential and integro-diierential equations with m linear boundary conditions. Our main aim is to introduce a cost-eeective alternative to the highly successful orthogonal col-location method, and to prove stability and convergence properties similar to the orthogonal collocation method. Our method is a discrete Petrov-Galerkin method, in which we seek a spline approximation u h of order m + r, but with one higher degree of smoothness than for the standard orthogonal collocation approximation: we require u h 2 C m (in contrast to C m?1 in the orthogonal collocation case), so that u (m) h 2 C: We show that optimal order convergence and superconvergence at break points hold without any mesh restriction, provided only that a certain underlying quadrature rule has degree of precision at least 2r ?1: There is one respect in which the present method has an advantage over orthogonal collocation. Firstly, because of the increased continuity of u h ; the number of unknown variables is reduced (almost halved in certain practical cases), which is a signiicant reduction for complex nonlinear problems. Orthogonal collocation method itself can be obtained by changing a few parameters in our method. To test our theoretical results and demonstrate the generality of the method, we consider an application to a catalytic combustion model problem involving nonlinear integro-diierential equation, compute solutions of a nonlinear ordinary diierential equation, and solve a boundary value problem with a thin boundary layer occuring in a nonlinear convection problem. Running title: Optimal order spline methods