Browsing by Author "Fatullayev, Afet Golayoglu"

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Relationship Between Bede-Gal Differentiable Set-Valued Functions and Their Associated Support Functions
(2016) Amrahov, Sahin Emrah; Khastan, Alireza; Gasilov, Nizami; Fatullayev, Afet Golayoglu; AEN-1756-2022
In this study, we adapt the concept of the Bede-Gal derivative, which was initially suggested for fuzzy number-valued functions, to set-valued functions. We use an example to demonstrate that this concept overcomes some of the shortcomings of the Hukuhara derivative. We prove some properties of Bede-Gal differentiable set-valued functions. We also study the relationship between a Bede-Gal differentiable set-valued function and its value's support function, which we call the associated support function. We provide examples of set-valued functions that are not Bede-Gal differentiable whereas their associated support functions are differentiable. We also present some applications of the Bede-Gal derivative to solving set-valued differential equations. (C) 2015 Elsevier B.V. All rights reserved.
Solution Method For A Non-Homogeneous Fuzzy Linear System Of Differential Equations
(2018) Gasilov, Nizami A.; Fatullayev, Afet Golayoglu; Amrahov, Sahin Emrah; AAN-9386-2020; 0000-0001-7747-5467; AAF-3339-2020
In this paper, we propose a new solution method to non-homogeneous fuzzy linear system of differential equations. The coefficients of the considered system are crisp while forcing functions and initial values are fuzzy. We assume each forcing function be in a special form, which we call as triangular fuzzy function and which represents a fuzzy bunch (set) of real functions. We construct a solution as a fuzzy set of real vector-functions, not as a vector of fuzzy-valued functions, as usual. We interpret the given fuzzy initial value problem (fuzzy IVP) as a set of crisp (classical) IVPs. Such a crisp IVP is obtained if we take a forcing function from each of fuzzy bunches and an initial value from each of fuzzy intervals. The solution of the crisp IVP is a vector-function. We define it to be an element of the fuzzy solution set and assign a membership degree which is the lowest value among membership degrees of taken forcing functions and initial values in the corresponding fuzzy sets. We explain our approach and solution method with the help of several illustrative examples. We show the advantage of our method over the differential inclusions method and its applicability to real-world problems. (C) 2018 Elsevier B.V. All rights reserved.
Solution of Linear Differential Equations with Fuzzy Boundary Values
(2014) Gasilov, Nizami; Amrahov, Sahin Emrah; Fatullayev, Afet Golayoglu; https://orcid.org/0000-0002-9955-8439; AEN-1756-2022
We investigate linear differential equations with boundary values expressed by fuzzy numbers. In contrast to most approaches, which search for a fuzzy-valued function as the solution, we search for a fuzzy set of real functions as the solution. We define a real function as an element of the solution set if it satisfies the differential equation and its boundary values are in intervals determined by the corresponding fuzzy numbers. The membership degree of the real function is defined as the lowest value among membership degrees of its boundary values in the corresponding fuzzy sets. To find the fuzzy solution, we use a method based on the properties of linear transformations. We show that the fuzzy problem has a unique solution if the corresponding crisp problem has a unique solution. We prove that if the boundary values are triangular fuzzy numbers, then the value of the solution at a given time is also a triangular fuzzy number. The defined solution is the same as one of the solutions obtained by Zadeh's extension principle. For a second-order differential equation with constant coefficients, the solution is expressed in analytical form. Examples are given to describe the proposed approach and to compare it to a method that uses the generalized Hukuhara derivative, which demonstrates the advantages of our method. Crown Copyright (C) 2013 Published by Elsevier B.V. All rights reserved.