Nizami Gasilov

Solution of linear differential equations with fuzzy boundary values

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 Zadehs 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.

A new approach to fuzzy initial value problem

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.

Morphology and causes of the Weddell Sea anomaly

The zone of anomalous diurnal variations in foF2, which is characterized by an excess of nighttime foF2 values over daytime ones, has been distinguished in the Southern Hemisphere based on the Intercosmos-19 satellite data. In English literature, this zone is usually defined as the Weddell Sea anomaly (WSA). The anomaly occupies the longitudes of 180°–360° E in the Western Hemisphere and the latitudes of 40°–80° S, and the effect is maximal (up to ∼5 MHz) at longitudes of 255°–315° E and latitudes of 60°–70° S (50°–55° ILAT). The anomaly is observed at all levels of solar activity. The anomaly formation causes have been considered based on calculations and qualitative analysis. For this purpose, the longitudinal variations in the ionospheric and thermospheric parameters in the Southern Hemisphere have been analyzed in detail for near-noon and near-midnight conditions. The analysis shows that the daytime foF2 values are much smaller in the Western Hemisphere than in the Eastern one, and, on the contrary, the nighttime values are much larger, as a result of which the foF2 diurnal variations are anomalous. Such a character of the longitudinal effect mainly depends on the vertical plasma drift under the action of the neutral wind and ionization by solar radiation. Other causes have also been considered: the composition and temperature of the atmosphere, plasma flows from the plasmasphere, electric fields, particle precipitation, and the relationship to the equatorial anomaly and the main ionospheric trough.

Solution method for a boundary value problem with fuzzy forcing function

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.

Linear differential equations with fuzzy boundary values

In this study, we consider a linear differential equation with fuzzy boundary values. We express the solution of the problem in terms of a fuzzy set of crisp real functions. Each real function from the solution set satisfies differential equation, and its boundary values belong to intervals, determined by the corresponding fuzzy numbers. The least possibility among possibilities of boundary values in corresponding fuzzy sets is defined as the possibility of the real function in the fuzzy solution.

Causes of NmF2 longitudinal variations at mid- and subauroral latitudes under summer nighttime conditions

The main factors controlling NmF2 longitudinal variations at mid- and subauroral latitudes have been studied. The data of the Intercosmos-19 topside sounding, obtained at high solar activity for summer nighttime conditions, have been used in the analysis. The contributions of the solar ionization, neutral wind, and temperature and composition of the thermosphere to NmF2 longitudinal variations have been estimated based on ionospheric models. It has been indicated that NmF2 variations in the unsunlit midlatitude ionosphere mainly depends on the residual electron density and its decay under the action of recombination. At subauroral latitudes under summer nighttime conditions, the ionosphere is partially sunlit, and ionization by solar radiation mainly contributes to NmF2 longitudinal variations, whereas the effect of the neutral wind is slightly less significant. These results also indicate how the contribution of different factors to NmF2 longitudinal variations changes at different latitudes.

A Geometric Approach to Solve Fuzzy Linear Systems

In this paper, linear systems with a crisp real coefficient matrix and with a vector of fuzzy triangular numbers on the right-hand side are studied. A new method, which is based on the geometric representations of linear transformations, is proposed to find solutions. The method uses the fact that a vector of fuzzy triangular numbers forms a rectangular prism in n-dimensional space and that the image of a parallelepiped is also a parallelepiped under a linear transformation. The suggested method clarifies why in general case different approaches do not generate solutions as fuzzy numbers. It is geometrically proved that if the coefficient matrix is a generalized permutation matrix, then the solution of a fuzzy linear system (FLS) is a vector of fuzzy numbers irrespective of the vector on the right-hand side. The most important difference between this and previous papers on FLS is that the solution is sought as a fuzzy set of vectors (with real components) rather than a vector of fuzzy numbers. Each vector in the solution set solves the given FLS with a certain possibility. The suggested method can also be applied in the case when the right-hand side is a vector of fuzzy numbers in parametric form. However, in this case, -cuts of the solution can not be determined by geometric similarity and additional computations are needed.

Simultaneous determination of unknown coefficients in a parabolic equation

A numerical method for an inverse problem of simultaneously determining unknown coefficients in a parabolic equation subject to the specifications of the solution at boundary points and given integral of the solution on space variable along with the usual initial and boundary conditions is proposed. The approach based on trace-type functional formulation of the problem is used. To avoid instability in this approach the Tikhonov regularization method is applied. Some numerical examples using the proposed algorithm are presented.

Relationship between Bede-Gal differentiable set-valued functions and their associated support functions

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 values 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.

Comparing numerical methods for inverse coefficient problem in parabolic equation

An inverse problem of identification of a coefficient depending on a solution in the one-dimensional parabolic equation is considered. Two different algorithms for the solution of this problem are studied and their comparison analysis is presented.