Abstract
In this study, by a parametric representation of interval numbers, two parametric representations for interval-valued functions are presented. Then, using these representations, the calculus of interval-valued functions and interval differential equations are investigated with two different approaches. In the first approach, the interval differential is transformed into a crisp problem. In the second approach, two solutions are obtained with the characterization of the solutions of two ordinary differential equation systems.
Highlights
The interval-valued analysis and interval differential equations are specific cases of setvalued analysis and set differential equations, respectively [ – ]
Interval analysis is introduced as an attempt to handle interval uncertainty, while interval differential equations are natural models for describing dynamic systems under uncertainty, and this approach is useful in many applications areas, such as physics and engineering [, ]
By using the parametric representation ( ) and its corresponding definitions of the p-derivative and integral, the interval differential equation is converted to a crisp problem
Summary
The interval-valued analysis and interval differential equations are specific cases of setvalued analysis and set differential equations, respectively [ – ]. By the parametric representations ( ) and ( ), some concepts such as limit, continuity, derivative, and integral for interval-valued functions will be defined in a parametric form. The concept of p-derivative for an interval-valued function will be defined, which depends upon the existence of the derivative of fc(t)(x) for every value t.
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