A new numerical approach for solving higher order nonlinear ordinary differential equations

There have been many numerical solution approaches to ordinary differential equations in the literature. However, very few are effective in solving non-linear ordinary differential equations (ODEs), particularly when they are of order higher than one. With the modern symbolic calculation packages, such as Maple and Mathematica, being readily available to researchers, we shall present a new numerical method in this thesis. Based on the repeated use of a symbolic calculation package and a second order finite difference scheme, our method is particularly suitable for solving higher order nonlinear differential equations arising from initial value problems. One important feature of our approach is that if the highest order derivative in an ODE can be written explicitly in terms of all the other terms of lower orders, our method requires no iterations at all. On the other hand, if the highest order derivative in an ODE can not be written explicitly in terms of all the other lower-order terms, iterations are only required before the actual time marching begins.