Mühendislik Fakültesi / Faculty of Engineering
Permanent URI for this collectionhttps://hdl.handle.net/11727/1401
Browse
6 results
Search Results
Item A New Approach to Fuzzy Initial Value Problem(2014) Gasilov, N. A.; Fatullayev, A. G.; Amrahov, S. E.; Khastan, A.; https://orcid.org/0000-0002-9955-8439; AEN-1756-2022In this paper, we consider a high-order linear differential equation with fuzzy initial values. We present solution as a fuzzy set of real functions such that each real function satisfies the initial value problem by some membership degree. Also we propose a method based on properties of linear transformations to find the fuzzy solution. We find out the solution determined by our method coincides with one of the solutions obtained by the extension principle method. Some examples are presented to illustrate applicability of the proposed method.Item Solution Method for A Boundary Value Problem with Fuzzy Forcing Function(2015) Gasilov, N. A.; Amrahov, S. E.; Fatullayev, A.G.; Hashimoglu, I. F.; 0000-0002-9955-8439; 0000-0001-7747-5467; AEN-1756-2022; AAF-3339-2020In this paper, we present a new approach to a non-homogeneous fuzzy boundary value problem. We consider a linear differential equation with real coefficients but with a fuzzy forcing function and fuzzy boundary values. We assume that the forcing function is a triangular fuzzy function. Unlike previous studies, we look for a solution that is a fuzzy set of real functions (not a fuzzy-valued function). Each of these real functions satisfies the boundary value problem with some membership degree. We have developed a method that finds this solution, and demonstrated its effectiveness using a test example. To show that the approach can be extended to other types of fuzzy numbers, we extended it to the trapezoidal case. For a particular example, we used the product t-norm to demonstrate how a new solution type can be obtained. (C) 2015 Elsevier Inc. All rights reserved.Item 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-1756-2022We 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.Item On A Solution of the Fuzzy Dirichlet Problem for the Heat Equation(2016) Gasilov, N. A.; Amrahov, S. Emrah; Fatullayev, A. G.; AEN-1756-2022In real-world applications, the behavior of the system is determined by physics laws and described by differential equations. In particular, the heat transfer is determined by Fourier's law of heat conduction and described by partial differential equation of parabolic type. When we construct a mathematical model, we need several parameters, a lot of them are obtained from measurements, or observations. In general, no measurement is perfect and these parameters become uncertain. Fuzzy sets are a useful tool to model such uncertainties. Thus, mathematical models arise, where due to physical laws the dynamics is crisp (certain) but some parameters (such as the source term, initial and boundary values) are fuzzy. In this paper, we consider such a model. Namely, we investigate fuzzy Dirichlet problem for the heat equation with fuzzy source function and fuzzy initial-boundary conditions. Most of the researchers assume a solution of a fuzzy differential equation as a fuzzy-valued function. But this approach is accompanied with some known difficulties. We are motivated by the fact that the fuzzy-valued function is not the only tool to model the uncertainties, changing with time. We are looking for a solution in the form of a fuzzy set (bunch) of real functions. We assume that the source term and initial-boundary conditions are modeled by triangular fuzzy functions, which are a special kind of fuzzy bunches. To determine the solution, we split the given fuzzy Dirichlet problem to three subproblems. The first subproblem provides the crisp solution. The other two subproblems give the uncertainties due to initial-boundary conditions and due to source function. We show that these uncertainties are triangular fuzzy functions. On the basis of the obtained results, we establish the existence and uniqueness theorem for the solution, under commonly accepted conditions. We propose a solution method and explain it on numerical examples. (C) 2015 Elsevier Masson SAS. All rights reserved.Item 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-3339-2020In 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.Item Numerical solution of linear inhomogeneous fuzzy delay differential equations(2019) Fatullayev, A.G.; Gasilov, Nizami A.; Amrahov, Sahin Emrah; AAN-9386-2020We investigate inhomogeneous fuzzy delay differential equation (FDDE) in which initial function and source function are fuzzy. We assume these functions be in a special form, which we call triangular fuzzy function. We define solution as a fuzzy bunch of real functions such that each real function satisfies the equation with certain membership degree. We develop an algorithm to find the solution, and we provide the existence and uniqueness results for the considered FDDE. We also present an example to show the applicability of the proposed algorithm.