Numerical Methods for Boundary Value Problems with Application to Bifurcation Problems

The method of orthogonal collocation for solving boundary value problems in ordinary diierential equation is described, and its relationship to classical nite diierence methods is illustrated. Numerical examples demonstrate the capabilities of software based on orthogonal collocation for analyzing nonlinear diierential equations. A new discretiza-tion method for the solution of elliptic partial diierential equations is described. It is indicated how this method can also be viewed as a collocation method. Numerical results illustrate its accuracy. (BVP) for the ordinary diierential equation (ODE) u 00 (t) + a(t) u 0 (t) + b(t) u(t) = f(t); t 2 0; 1]; u(0) = u(1) = 0: Assume that the coeecient functions and the inhomogeneous term are suuciently smooth and that the corresponding homogeneous problem only admits the zero solution. Our main interest is nonlinear ODE BVPs, but to keep the presentation simple, we describe numerical methods only for linear equations. Introduce a mesh 0 = t 0 < t 1 < t 2 < < t N = 1, with h j t j ? t j?1 and h max j t j. The method of orthogonal collocation with piecewise polynomials consists of nding p 2 C 1 0; 1] such that p is a polynomial of degree m + 1 in each mesh interval t j?1 ; t j ], (j = 1; 2; ; N), and such that p satisses the collocation equations p 00 (z ji) + a(z ji) p 0 and the boundary conditions p(0) = p(1) = 0 : Here the m collocation points z ji i = 1; ; m lie in the jth mesh interval t j?1 ; t j ] and are distinct, but otherwise arbitrary. It is easily checked that the number of equations is formally enough to determine p. A rigorous existence and convergence proof shows that the global accuracy of the approximate solution is given by max 0;1] j p(t) ? u(t) j= O(h m+1): However, if in each subinterval t j?1 ; t j ] the m collocation points are chosen as the (scaled and translated) roots of the mth degree Legendre orthogonal polynomial, then we have superconvergence 1] at the main meshpoints t j : max j j p(t j) ? u(t j) j= O(h 2m): 1