Interval versions of Milne’s multistep methods

Abstract

The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far.

Interval Runge-Kutta Methods with Variable Step Sizes

“…However, there is no restriction to apply the presented procedure to other kinds of interval methods of Runge-Kutta type (explicit and implicitsee, e.g., [16,17,23,24,29]). It should also be noted that although interval methods based on high-order Taylor se-ries are commonly considered as most universal, sometimes other interval methods (not only of Runge-Kutta type) give better enclosures of the exact solutions (see examples presented in [20][21][22]24] and Example 3). This is the main reason to consider also possibilities of applying such methods for solving various initial value problems.…”

“…In interval methods based on Runge-Kutta methods and in interval multistep methods a constant step size has been used. Although the methods based on high-order Taylor series seem to be most universal, in [20][21][22]24] we have shown that in some cases the interval methods of the second and third kind give better enclosures of solutions.…”

Interval methods of Adams-Bashforth type with variable step sizes

“…The methods based on high-order Taylor series use variable step sizes and seem to be the most universal. But it should be noted that in [25][26][27] and [30] we have shown that in some cases the interval methods of the second and third kinds give better enclosures of the exact solutions. Therefore, it is worth to take into account also such methods.…”