Numerical Solution Of Ordinary Differential Equations Based On The B-spline

The ordinary differential equations have a long history, and they are widely used in different fields. Through the whole 20^th century, the research on numerical solutions of the ordinary differential equations made enormous progress. Particularly, with the fast development of computer technology and some famous mathematics soft wares, more and more new ideas and approaches which surpass classic methods could be realized. At the same time, the field of numerical methods research has been expanded.In this dissertation, we mainly use the B-splines functions in the problem of the numerical solutions of the ordinary differential equations. First we got a numerical method for initial value problem of ordinary differential equations with B-splines. Then we gave an inexplicit recursion formula of the equations. Also we made error analysis and stability analysis of this problem. The result showed that its local truncation error was 0(h⁵) but the global truncation error was 0(h⁴). Also it was stable. After that we use the B-splines functions and the Bernstein polynomials as basises respectively to approach the solutions of the boundary value problem of the periodic oscillation ordinary differential equations. We got the numerical solutions by solving some system of equations. In the end of the paper we gave some corresponding examples and drew the pictures of the absolute errors by using a Mathematica program. Then we made error analysis of the problem though examples. We discussed the different situations of the error in different basises. The result showed that the approximation effect was getting better and better by increasing the degrees of Bernstein polynomials or by the increasing segments of the B-splines functions.