Optimal order spline methods for nonlinear differential and integro-differential equations Original Research Article

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 uh of order m + r, but with one higher degree of smoothness than for the standard orthogonal collocation approximation: we require uh ∈ Cm (in contrast to Cm − 1 in the orthogonal collocation case), so that uh(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 2r − 1. There is one respect in which the present method has an advantage over orthogonal collocation. Firstly, because of the increased continuity of uh, 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.