Methods of solving linear fuzzy integral equations

number: 
712
إنجليزية
Degree: 
Imprint: 
Mathematics and Computer Applications
Author: 
Samah Waleed Najieb
Supervisor: 
Dr.Adil M. Ahmed
Dr. Akram M. Al-Abood
year: 
2002
Abstract:

Integral equation are used as a standard tool in every field of science for building models which describe various phenomena. Indeed many physical phenomena lend themselves to description by integral equations. In this work, some background material about fuzzy number, fuzzy equation, fuzzy function, and related mathematical notions are defined. Since the set of fuzzy numbers with the operations of addition and multiplication does not have a group structure, so we construct a new special type of fuzzy numbers (which we shall call it finite level fuzzy number) such that the addition and multiplication of two finite level fuzzy numbers will be finite level fuzzy numbers. Solutions of fuzzy equation is also considered by assuming that the fuzzy coefficients and the fuzzy variables are finite level fuzzy numbers. The integral and differential of fuzzy function was defined in the literature, but we generalize these concepts (fuzzy integral and fuzzy differential) by constructing the functional of fuzzy function. Using Zadeh's extension principle, the operator of fuzzy function is defined. Fuzzy Laplace transform is considered as a fuzzy operator, and the problem of the linearity is addressed. Also, we show that the convolution theorem can be extended in fuzzy theory. Park et al., consider the existence of solutions of fuzzy integral equations in Banach spaces. But unfortunately, we could not see the proof of the existence theorem. For this reason, we prove the existence theorem for the solutions of fuzzy integral equations by extending the existence theorems for ordinary integral equations, and we think that our approach different from the approach of those authors. Finally, we construct three methods ( fuzzy Laplace transform method, method of successive approximation, quadrature method) to find the exact and numerical solutions of linear fuzzy integral equations of Volterra type, these methods are demonstrated with examples.