一个新的求解常微分方程的四阶三点一步法

One New Fourth Order Three Point Single Step Rule for Solving Ordinary Differential Equations

提高求解常微分方程初值问题一步法的阶数有两种途径,一种是通过增加使用导数的阶数来提高方法的阶数;另一种是通过增加使用函数值的个数来提高方法的阶数。但当f(x,y)比较复杂时,第一种途径会使计算变得很困难;而第二种途径对一个p阶的方法在每一步计算时要使用p个点的函数值,使得计算工作量较大。本文给出的新的四阶一步法,克服了传统的四阶一步法的缺点,既不使用高阶导数,同时在每一步计算时使用的函数值的个数又少于传统方法的阶数。

There are two different approaches to increase the order number of the single step method for solving ordinary differential equation initial value problems. The first is with the help of higher - order derivatives, while the second is by calculating the function values at intermediate points. The disadvantages of each of these approaches are as follows; If f(x ,y) is very complicated, then one might encounter difficulties with computation using the first approach, while the second approach requires the computation of p function values during each computation step, if one changes the method into a p - order single step method. In order to resolve these disadvantages, this paper presents a new fourth order single step method. The new method does not requre the partial derivatives of f(x,y). At the same time, the number of function values used in each step in the computation is less than the order number of the method.