 # Hukuhara Derivative: Definition

Share on

Derivatives > Hukuhara derivative

Generalized Hukuhara differentiability is the most general type of differentiability for interval-valued functions . Hukuhara introduced the idea of the Hukahara derivative in his 1967 paper , which became a starting point for the study of set differential equations and fuzzy differential equations .

## Hukuhara Derivative: Definition

Hukuhara first described the Hukahara derivative of f at x as the fuzzy number (an imprecise number) f′(x), defined as:

A function f: (a, b) → ℝF is Hukahara differentiable if, for sufficiently small h > 0, the H-differences
f (x + h) ⊖ f(x) and f(x) ⊖ f(x – h) exist and if there is also an element f′(x) ∈ ℝF such that : Hukuhara defined the “H-difference” to overcome the problem of finding a suitable difference between two intervals for interval-valued functions. If A = B + C, then the H-difference is A and B (denoted by A – HB) equals C. For arbitrary pairs of intervals, the H-difference may or may not exist .

## Two Versions of the Hukuhara Derivative

The “classical” Hukuhara derivative from Hukuhara’s paper has a shortcoming: solutions for interval differential equations (IDEs), which are interval-valued mappings, have values with non non-decreasing length. This means that the uncertainty shown by the solutions grows over time .

Stefanini and Bede’s  research on interval differential equations introduced a new version of the Hukuhara derivative for interval-valued mappings. Their approach has solutions where values have non-increasing length.

## References

 Cano, Y. et al. (2012). Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Retrieved November 23, 2021 from: https://www.sciencedirect.com/science/article/abs/pii/S016501141200509X
 M. Hukuhara, Intégration des applications mesurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967) 205–223.
 Cano, Y. et al. Calculus for interval-valued functions using generalized Hukuhara
derivative and applications. Fuzzy Sets and Systems 219 (2013) 49–67.
 Bede, B. (2012). Mathematics of Fuzzy Sets and Fuzzy Logic. Springer Berlin Heidelberg.
 Malinowski, M. Interval differential equations with a second type Hukuhara derivative. Retrieved November 23, 2021 from: https://core.ac.uk/download/pdf/82239432.pdf
 L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71 (2009)
1311–1328.

CITE THIS AS:
Stephanie Glen. "Hukuhara Derivative: Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/hukuhara-derivative/
---------------------------------------------------------------------------  Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!