(untitled) (Journal of Computational Mathematics, 1991)
In this paper, a class of integration formulas is derived from the approximation so that the first derivative can be expressed within an interval [nh,(n+1)h] as dydt=-P(y-y_n)+f_n+Q_n(t). The class of formulas is exact if the differential equation has the shown form, where P is a diagonal matrix, whose elements -pj=y_jf_j(t_n,y_n),j=1,…,m are constant in the interval [nh,(n+1)h], and Q_n(t) is a polynomial in t. Each of the formulas derived in this paper includes only the first derivative f and -pj=y_jf_j(t_n,y_n). It is identical with a certain Runge-Kutta method. In particular, when Q_n(t) is a polynomial of degree two, one of our formulas is an extension of Treanor's method, and possesses better stability properties. Therefore the formulas derived in this paper can be regarded as a modified or an extended form of the classical Runge-Kutta methods. Preliminary numerical results indicate that our fourth order formula is superior to Treanor's in stability properties.