A high order algorithm for ordinary boundary value problems

The method of analytic continuation has been used to obtain numerical solutions of nonlinear initial value problems. Here we formulate the problem in terms of characteristic functions that form a Partition of Unity. This method allows us to extend an approximate local solution to a global solution that can be expressed in closed form as a polynomial whose coefficients are piecewise constant functions. An error bound is calculated and used to prove uniform convergence to the exact solution as the number of partition points approaches infinity. Using a shooting method we also find approximate closed form solutions of nonlinear boundary value problems with arbitrarily high order of accuracy. The solution obtained is useful when a closed form expression is required. References E. T. Copson, Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, 1970. G. Birkhoff and G. Rota, Ordinary Differential Equations, Wiley, 1978. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, 1962. F. Viera, On approximate closed form solutions of linear ordinary differential equations, ANZIAM J. 47 (EMAC2005) pp.C245--C260, 2006. http://anziamj.austms.org.au/V47EMAC2005/Viera H. B. Keller, Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, Waltham, Mass, 1968. R. L. Burden and J. D. Faires, Numerical Analysis, PWS Publishing Company, Boston, 1993