New Interval Calculus with Application to Interval Differential Equations

TL;DR

This paper introduces a new interval calculus framework with novel arithmetic operations that form a Hilbert space, simplifying the analysis and solution of interval differential equations while maintaining classical calculus properties.

Contribution

The paper develops a new interval arithmetic and calculus framework that forms a Hilbert space, streamlining the solution process for interval differential equations and improving robustness over existing methods.

Findings

New interval arithmetic operations form a Hilbert space.

Unified calculus framework simplifies solutions of interval differential equations.

Robustness and computational efficiency are improved compared to gH-derivative methods.

Abstract

This paper presents a systematic study of the calculus of interval-valued functions and its application to interval differential equations. To this end, first, we introduce new interval arithmetic operations. Under new operations, the space of interval numbers becomes a strict linear space, and indeed a Hilbert space, whereas the traditional interval arithmetic yields only a semilinear space with a defective algebraic structure. Secondly, by basing derivative and integral of interval-valued functions on the proposed operations, we retain every essential property of classical calculus while seamlessly incorporating ideas from the multiplicative calculus. The resulting unified hybrid framework eliminates the tedious case-by-case inspection of switching points required by the gH-derivative, leading to a markedly streamlined computational procedure. Finally, we establish an existence theorem for solutions of interval differential equations within the new calculus and corroborate its validity and practicality through representative examples. In contrast to the gH-derivative approach, the number of potential solutions does not explode doubly with additional switching points, ensuring robustness in both theory and computation.