On Explicit Interval Methods of Adams-Bashforth Type

Abstract

Abstract. In our previous paper [1] we have considered implicit interval multistep methods of AdamsMoulton type for solving the initial value problem. On the basis of these methods and the explicit ones introduced by Sokin [2] we wanted to construct predictor-corrector (explicit-implicit) interval methods. However, it turned out that the formulas given by Šokin are incorrect even in the simplest case. Therefore, in this paper we direct our attention to the explicit interval methods of Adams-Bashforth type and modify the formulas of Šokin. For the modified explicit interval methods it is proved, like for the implicit interval methods considered in [1], that the exact solution of the problem belongs to interval-solutions obtained by these methods. Moreover, it is shown an estimation of the widths of such interval-solutions.

On interval predictor-corrector methods

“…In [19,26,28], we have proved that for the methods (30) and (37) (or (32) and (38), respectively) we have y (t k ) ∈ Y k , where y (t) is the exact solution of the initial value problem (1), and we have also estimated the widths w (Y k ) of the intervals Y k obtained by these methods.…”

“…As for the method (19), for interval equivalents of (25), it is important to write two coefficients ν * n+1 and ν * * n instead of one ν n+1 = ν * n+1 + ν * * n+1 (see Section 4 for details).…”

“…I. Shokin [41]. Unfortunately, as we have shown in [19] and [28], his formula fails even in the simplest case, i.e., when n = 1. Correcting a defective error term in Shokin's formula, we obtain the correct formula for interval methods of Adams-Bashforth type of the following form:…”

“…I. Shokin proposed an explicit interval method of Adams-Bashforth type. Unfortunately, it can be shown that his formula fails in the simplest case, but it can be easily corrected [19,28]. Other explicit interval multistep methods have been considered in [26][27][28] and implicit ones in [18,20,[26][27][28].…”

Interval methods of Adams-Bashforth type with variable step sizes

“…The CPU times have been equal to 36.019 s for the method (16), 1.152 s for (17), 0.344 s Fig. 1 Step size changes in problem (45) for the methods (16)- (19) for (18), and 0.415 s for the method (19). The changes of step sizes are shown in Fig.…”