Generalized Hukuhara differentiability of interval-valued functions and interval differential equations

Stefanini, Luciano; Bede, Barnabás

doi:10.1016/j.na.2008.12.005

Cited by 516 publications

(273 citation statements)

References 45 publications

(64 reference statements)

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“…Bede and Stefanini [14] introduced the definition and theorem of the generalized Hukuhara as follows: …”

Section: F T Y T Y T Rmentioning

confidence: 99%

Fuzzy version of a developed fourth order Runge Kutta method for solving differential equations with fuzzy initial values

Ramli

Din²,

Salleh³

2016

AIP Conference Proceedings

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Abstract. In this paper, numerical algorithm is applied for solving fuzzy differential equations based on generalized Hukuhara differentiability. In order to enhance the order of accuracy of the solutions, a developed fourth order Runge-Kutta method is apply for solving fuzzy differential equations. The purpose of this study is to explore the explicit methods which we believe that most of the explicit methods can be improved and modified to cater and solve fuzzy differential equations. This paper is divided into six sections. In the first section, some basic definitions and theorem are reviewed. The numerical examples are given to illustrate the efficiency of the method and the comparison with the existing method is discussed.

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“…From ( (2) is not a linear space since an interval does not have inverse element and therefore subtraction does not have adequate properties (see [3,13]). More recently, Stefanini and Bede in [13] have introduced the following difference between two intervals.…”

Section: The Space Of Intervals and Interval-valued Functionsmentioning

confidence: 99%

“…More recently, Stefanini and Bede in [13] have introduced the following difference between two intervals.…”

Section: The Space Of Intervals and Interval-valued Functionsmentioning

confidence: 99%

A note on optimality conditions to interval optimization problems

Chalco-Cano

Osuna-Gómez

Hernández‐Jiménez

et al. 2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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In this article we examine to necessary and sufficient optimality conditions for interval optimization problems. We introduce a new concept of stationary point for an interval-valued function based on the gH-derivative. We show the importance the this concept from a practical and computational point of view. We introduce a new concept of invexity for gH-differentiable interval-valued function which generalizes previous concepts and we prove that it is a sufficient optimality condition. Finally, we show that the concepts of differentiability, convexity and invexity for interval-valued functions based on the differentiability, convexity and invexity of its endpoint functions are not adequate tools for interval optimization problems.

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“…Based on the results established in [5], when both f (x) and f + (x) are differentiable w.r.t. x for all 's, then thecuts of the gH-derivative of f are As f (x) and f + (x) de ne the -cuts of the fuzzy number f (x) for each x, clearly they are monotonic and almost everywhere differentiable w.r.t.…”

Section: Lu-parametric Representationmentioning

confidence: 99%

Solution of Fuzzy Differential Equations with generalized differentiability using LU-parametric representation

Bede¹,

Stefanini

2011

Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)

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Abstract-The paper uses the LU-parametric representation of fuzzy numbers and fuzzy-valued functions, to obtain valid approximations of fuzzy generalized derivative and to solve fuzzy differential equations. The main result is that a fuzzy differential initial-value problem can be translated into a system of in nitely many ordinary differential equations and, by the LU-parametric representation, the in nitely many equations can be approximated ef ciently by a nite set of four ODEs. Some examples are included.

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“…From then on, the theory of interval analysis and its application developed greatly with the joint effort of many researchers. Especially, in order to build the theory of optimization of interval value [5,7,8], theory of differential equation of interval value [1,2,6] and the differential theory of fuzzy value [3], several kinds of differentiability of interval-valued functions were invented, and the related theories were built.…”

Section: Introductionmentioning

confidence: 99%

Study on differentiability problems of interval-valued functions

Bao¹,

Li²,

Bai³

2017

J. Nonlinear Sci. Appl.

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In this paper, we give the concepts of H-directional differentiability and D-directional differentiability of interval-valued functions. Then we discuss the properties of H-directional differentiable interval-valued functions and D-directional differentiable interval-valued functions. The necessary and sufficient conditions for the H-directional differentiability are given together with the sufficient conditions and the necessary and sufficient conditions for D-directional differentiability of interval-valued functions. Then we discuss the relationship between the two directional differentiability and prove these directional differentiability can be equivalent under a certain conditions.

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Generalized Hukuhara differentiability of interval-valued functions and interval differential equations

Cited by 516 publications

References 45 publications

Fuzzy version of a developed fourth order Runge Kutta method for solving differential equations with fuzzy initial values

A note on optimality conditions to interval optimization problems

Solution of Fuzzy Differential Equations with generalized differentiability using LU-parametric representation

Study on differentiability problems of interval-valued functions

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