Abstract
In a number of our previous papers, we have proposed interval versions of multistep methods (explicit and implicit), including interval predictor-corrector methods, in which the step size was constant. In this paper, we present interval versions of Adams-Bashforth methods with a possibility to change step sizes. This possibility can be used to obtain interval enclosures of the exact solution with a width given beforehand.
Highlights
Using computers and approximate methods to solve problems described in the form of ordinary differential equations, we usually provide all calculations in floatingpoint arithmetic
Applying interval methods to approximate the solution of the initial value problem in floating-point interval arithmetic we can obtain enclosures of the solution in the form of intervals which contain both these errors and the truncation error
We can distinguish three main kinds of interval methods for solving the initial value problem: methods based on high-order Taylor series, explicit and implicit methods of Runge-Kutta type ([9, 10, 22, 28, 30, 37]), and explicit and implicit multistep methods ([9, 17,18,19, 22, 25, 26]), including interval predictor-corrector methods [27]
Summary
Using computers and approximate methods to solve problems described in the form of ordinary differential equations, we usually provide all calculations in floatingpoint arithmetic. Applying interval methods to approximate the solution of the initial value problem in floating-point interval arithmetic (see, e.g., [11]) we can obtain enclosures of the solution in the form of intervals which contain both these errors and the truncation error. In these methods by appropriate selection of step size, one can minimize the truncation error, i.e., obtain an approximate solution at some points (mesh points of a grid) with a tolerance given beforehand.
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