New Interval Calculus with Application to Interval Differential Equations
Wei Liu, Muhammad Aamir Ali, Yanrong An
TL;DR
The paper tackles the instability and nonlinearity of traditional interval arithmetic by constructing a linear, even Hilbert, space $\mathbb{R}_{\mathcal{I}}$ with new arithmetic and a compatible distance, norm, and inner product. It then develops a hybrid calculus for interval-valued functions, introducing a derivative that blends classical and multiplicative components and a Riemann-type integral with robust convergence properties. The main contributions include a unified derivative/integral framework, an existence theorem for interval differential equations via Schauder's fixed point, and convergence results that improve computational robustness over gH-based methods. These developments offer a practical, rigorous toolkit for uncertainty quantification and interval analysis in applications requiring interval differential equations and related computations.
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.