Sahin Emrah Amrahov

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.

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.

Strong Solutions of the Fuzzy Linear Systems

We consider a fuzzy linear system with crisp coefficient matrix and with an arbitrary fuzzy number in parametric form on the right-hand side. It is known that the well-known existence and uniqueness theorem of a strong fuzzy solution is equivalent to the following: The coefficient matrix is the product of a permutation matrix and a diagonal matrix. This means that this theorem can be applicable only for a special form of linear systems, namely, only when the system consists of equations, each of which has exactly one variable. We prove an existence and uniqueness theorem, which can be use on more general systems. The necessary and sufficient conditions of the theorem are dependent on both the coefficient matrix and the right-hand side. This theorem is a generalization of the well-known existence and uniqueness theorem for the strong solution.

Fuzzy Rule-Based Image Segmentation technique for rock thin section images

Image segmentation is a process of partitioning the images into meaningful regions that are ready to analyze. Segmentation of rock thin section images is not trivial task due to the unpredictable structures and features of minerals. In this paper, we propose Fuzzy Rule-Based Image Segmentation technique to segment rock thin section images. Proposed technique uses RGB images of rock thin sections as input and gives segmented into minerals images as output. In order to show an advantage of proposed technique the rock thin section images were also segmented by known Fuzzy C-Means technique. Both techniques were applied to many different rock thin section images. The obtained results of proposed Fuzzy Rule-Based Image Segmentation and Fuzzy C-Means techniques were compared. Implementation results showed that proposed image segmentation technique has better accuracy than known ones.

On solutions of initial-boundary value problem for fuzzy partial differential equations

In this paper, we investigate linear partial differential equations with fuzzy source function, and with fuzzy initial and boundary conditions. Usually, researchers consider solutions of fuzzy differential equations in the form of fuzzy-valued functions. On the contrary, in this study, we are looking for a solution in the form of fuzzy set (bunch) of real functions. To demonstrate the proposed approach we use Dirichlet problem for the heat equation. We assume the source function, and the initial and boundary conditions to be in a special form, which we name as triangular fuzzy function. We show that the uncertainties of the solution due to these parameters are triangular fuzzy functions too. The solution for the example, which we discuss in the paper, is expressed by an analytical formula. If we use numerical methods, we can find the solution in the suggested sense for each problem from the examined class.