Browsing by Author "Gasilov, N. A."

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A New Approach to a Fuzzy Time-Optimal Control Problem
(2014) Amrahov, S. Emrah; Gasilov, N. A.; Fatullayev, A. G.; AAN-9386-2020
In this paper, we present a new approach to a time-optimal control problem with uncertainties. The dynamics of the controlled object, expressed by a linear system of differential equations, is assumed to be crisp, while the initial and final phase states are fuzzy sets. We interpret the problem as a set of crisp problems. We introduce a new notion of fuzzy optimal time and transform its calculation to two classical time-optimal control problems with initial and final sets. We examine the proposed approach on an example which is a problem of fuzzy control of mathematical pendulum.
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-1756-2022
In 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.
The Numerical Code TOKSCEN for Modelling Plasma Evolution in Tokamaks
(2015) Sadykov, A. D.; Sychugov, D. Yu.; Shapovalov, G. V.; Chektybaev, B. Zh.; Skakov, M. K.; Gasilov, N. A.; 0000-0002-9955-8439; AEN-1756-2022
The code TOKSCEN (TOKamak SCENario) for the modelling of plasma evolution is described in this paper. The modelling is based on numerical solution of the Grad-Shafranov equation of plasma equilibrium and circuit equations for eddy currents at each time step. The circuit equations for eddy currents are solved in matrix form using the technique of matrix inversion. The plasma current distribution should be given. The code enables the calculation of an increment of vertical instability at each time step of the plasma evolution. The algorithms of the code were used for the modelling of processes in KTM and other tokamaks.
On A Solution of the Fuzzy Dirichlet Problem for the Heat Equation
(2016) Gasilov, N. A.; Amrahov, S. Emrah; Fatullayev, A. G.; AEN-1756-2022
In 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.
On The Existence And Uniqueness Of A Solution To The Boundary Value Problem For Linear Ordinary Differential Equations
(ACTA MATHEMATICA UNIVERSITATIS COMENIANAE, 2024) Gasilov, N. A.
In this study, we investigate the Boundary Value Problem (BVP) for second order non-homogeneous linear differential equation with Dirichlet conditions. We derive a novel sufficient condition for the existence and uniqueness of a solution. The condition is formulated in terms of input parameters (coefficient functions and the length l of the interval, where the BVP is considered), not in secondary terms as Lipschitz coefficients. We compare the obtained sufficient condition with those for non-linear BVPs and demonstrate that it covers a significantly wider class of BVPs.
On the Numerical Solution of Linear Differential Equations with Interval Coefficients
(2017) Gasilov, N. A.; Amrahov, S. E.; https://orcid.org/0000-0001-7747-5467; AEN-1756-2022; AAF-3339-2020
In this study, we consider Initial Value Problem (IVP) for a linear differential equation with interval coefficients. The initial values of the problem are taken as intervals, too. We interpret the interval IVP as a set of classical IVPs, and we investigate the bunch (set) of their solutions. We define this bunch to be the solution of the interval IVP. We develop a numerical method to find the upper and lower bounds of the solution bunch. We apply the method to an example and illustrate the solution.
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-3339-2020
In 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.