A new numerical approach for solving high-order non-linear ordinary differential equations
There have been many numerical solution approaches to ordinary dierential equations in the literature. However, very few are eective in solving non-linear ordinary dierential equations (ODEs), particularly when they are of order higher than one. With modern symbolic calculation packages, such as Maple and Mathematica, being readily available to researchers, we shall present a new numerical method in this paper. Based on the repeated use of a symbolic calculation package and a second-order nite-dierence scheme, our method is particularly suitable for solving high-order non-linear dierential 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 cannot be written explicitly in terms of all the other lower-order terms, iterations are only required before the actual time marching begins.Copyright ? 2003 John Wiley & Sons, Ltd.