Browsing by Author "Gasilov, Nizami A."

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A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters
(2019) Gasilov, Nizami A.; Kaya, Mujdat; AAN-9386-2020
In many real life applications, the behavior of the system is modeled by a boundary value problem (BVP) for a linear differential equation. If the uncertainties in the boundary values, the right-hand side function and the coefficient functions are to be taken into account, then in many cases an interval boundary value problem (IBVP) arises. In this study, for such an IBVP, we propose a different approach than the ones in common use. In the investigated IBVP, the boundary values are intervals. In addition, we model the right-hand side and coefficient functions as bunches of real functions. Then, we seek the solution of the problem as a bunch of functions. We interpret the IBVP as a set of classical BVPs. Such a classical BVP is constructed by taking a real number from each boundary interval, and a real function from each bunch. We define the bunch consisting of the solutions of all the classical BVPs to be the solution of the IBVP. In this context, we develop a numerical method to obtain the solution. We reduce the complexity of the method from O(n(5)) to O(n(2)) through our analysis. We demonstrate the effectiveness of the proposed approach and the numerical method by test examples.
A New Curve Fitting Based Rating Prediction Algorithm For Recommender Systems
(2022) Yilmaz, A. R.; Amrahov, Sahin Emrah; Gasilov, Nizami A.; Yigit-Sert, Sevgi
The most algorithms for Recommender Systems (RSs) are based on a Collaborative Filtering (CF) approach, in particular on the Probabilistic Matrix Factorization (PMF) method. It is known that the PMF method is quite successful for the rating prediction. In this study, we consider the problem of rating prediction in RSs. We propose a new algorithm which is also in the CF framework; however, it is completely different from the PMF-based algorithms. There are studies in the literature that can increase the accuracy of rating prediction by using additional information. However, we seek the answer to the question that if the input data does not contain additional information, how we can increase the accuracy of rating prediction. In the proposed algorithm, we construct a curve (a low-degree polynomial) for each user using the sparse input data and by this curve, we predict the unknown ratings of items. The proposed algorithm is easy to implement. The main advantage of the algorithm is that the running time is polynomial, namely it is theta(n2), for sparse matrices. Moreover, in the experiments we get slightly more accurate results compared to the known rating prediction algorithms.
Numerical solution of linear inhomogeneous fuzzy delay differential equations
(2019) Fatullayev, A.G.; Gasilov, Nizami A.; Amrahov, Sahin Emrah; AAN-9386-2020
We 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.
On differential equations with interval coefficients
(2019) Gasilov, Nizami A.; Amrahov, Sahin Emrah; AAN-9386-2020
In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept. Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case.
On Exact Solutions Of A Class Of Interval Boundary Value Problems
(2022) Gasilov, Nizami A.
In this article, we deal with the Boundary Value Problem (BVP) for linear ordinary differ-ential equations, the coefficients and the boundary values of which are constant intervals. To solve this kind of interval BVP, we implement an approach that differs from commonly used ones. With this approach, the interval BVP is interpreted as a family of classical (real) BVPs. The set (bunch) of solutions of all these real BVPs we define to be the solution of the interval BVP. Therefore, the novelty of the proposed approach is that the solution is treated as a set of real functions, not as an interval-valued function, as usual. It is well-known that the existence and uniqueness of the solution is a critical issue, especially in studying BVPs. We provide an existence and uniqueness result for interval BVPs under consideration. We also present a numerical method to compute the lower and upper bounds of the solution bunch. Moreover, we express the solution by an analytical formula under certain conditions. We provide numerical examples to illustrate the effectiveness of the introduced approach and the proposed method. We also demonstrate that the approach is applicable to non-linear interval BVPs.
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.
Solving A Nonhomogeneous Linear System of Interval Differential Equations
(2018) Gasilov, Nizami A.; Amrahov, Sahin Emrah; AAN-9386-2020
In most application problems, the exact values of the input parameters are unknown, but the intervals in which these values lie can be determined. In such problems, the dynamics of the system are described by an interval-valued differential equation. In this study, we present a new approach to nonhomogeneous systems of interval differential equations. We consider linear differential equations with real coefficients, but with interval initial values and forcing terms that are sets of real functions. For each forcing term, we assume these real functions to be linearly distributed between two given real functions. We seek solutions not as a vector of interval-valued functions, as usual, but as a set of real vector functions. We develop a method to find the solution and establish an existence and uniqueness theorem. We explain our approach and solution method through an illustrative example. Further, we demonstrate the advantages of the proposed approach over the differential inclusion approach and the generalized differentiability approach.
Solving A System Of Linear Differential Equations With Interval Coefficients
(2021) Gasilov, Nizami A.; AEN-1756-2022
In this study, we consider a system of homogeneous linear differential equations, the coefficients and initial values of which are constant intervals. We apply the approach that treats an interval problem as a set of real (classical) problems. In previous studies, a system of linear differential equations with real coefficients, but with interval forcing terms and interval initial values was investigated. It was shown that the value of the solution at each time instant forms a convex polygon in the coordinate plane. The motivating question of the present study is to investigate whether the same statement remains true, when the coefficients are intervals. Numerical experiments show that the answer is negative. Namely, at a fixed time, the region formed by the solution's value is not necessarily a polygon. Moreover, this region can be non-convex. The solution, defined in this study, is compared with the Hukuhara-differentiable solution, and its advantages are exhibited. First, under the proposed concept, the solution always exists and is unique. Second, this solution concept does not require a set-valued, or interval-valued derivative. Third, the concept is successful because it seeks a solution from a wider class of set-valued functions.