Afet Golayoglu Fatullayev

Numerical solution of the inverse problem of determining an unknown source term in a heat equation

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. A numerical examples are presented.

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.

An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation

An inverse problem for the determination of an unknown spacewise-dependent coefficient in a parabolic equation is considered. The problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. The iterative fixed point projection method is applied to solve the reformulated problem. The comparison analysis of proposed method with a least square method and some numerical examples are presented.

Numerical procedures for determining unknown source parameter in parabolic equations

Numerical procedures for the solution of an inverse problem of determining unknown source parameter in a parabolic equation are considered. Two different numerical procedures are studied and their comparison analysis is presented. One of these procedures obtained by introducing transformation of unknown function, while the other based on trace type functional formulation of the problem.

Numerical solution of the inverse problem of determining an unknown source term in a two-dimensional heat equation

A numerical procedure for an inverse problem of determination of unknown source term in two-dimensional parabolic equation is presented. The approach of the proposed method is to approximate unknown function by a piecewise linear function whose coefficients are determined from the solution of minimisation problem based on the overspecified data. Some numerical examples are presented.

Numerical procedure for the determination of an unknown coefficients in parabolic equations

A numerical procedure for an inverse problem of determination of unknown coefficients in a class of parabolic differential equations is presented. The approach of the proposed method is to approximate unknown coefficients by a piecewise linear function whose coefficients are determined from the solution of minimization problem based on the overspecified data. Some numerical examples are presented.

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.

Numerical procedure for the simultaneous determination of unknown coefficients in a parabolic equation

A numerical procedure for an inverse problem of simultaneously determining unknown coefficients in a linear parabolic equation subject to the specifications of the solution at internal points along with the usual initial boundary conditions is considered. By using some transformation the problem is reformulated to nonlocal parabolic problem. Some numerical examples using the proposed numerical procedure are presented.

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.