Optimal order spline methods for nonlinear differential and integro-differential equations

Abstract

In this work, we analyze new robust spline approximation methods for mth order boundary value problems described by nonlinear ordinary differential and integro-differential equations with m linear boundary conditions. Our main aim is to introduce a cost-effective alternative to the highly successful orthogonal collocation 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 ∈ C m (in contrast to C m − 1 in the orthogonal collocation case), so that u h ( m) ∈ 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 2 r − 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 significant 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-differential equation, compute solutions of a nonlinear ordinary differential equation, and solve a boundary value problem with a thin boundary layer occurring in a nonlinear convection problem.