Intutionistic Fuzzy Difference Equation

Intutionistic Fuzzy Difference Equation

Sankar Prasad Mondal (National Institute of Technology Agartala, India), Dileep Kumar Vishwakarma (National Institute of Technology Agartala, India) and Apu Kumar Saha (National Institute of Technology Agartala, India)
DOI: 10.4018/978-1-5225-0914-1.ch005
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Abstract

In this chapter we solve linear difference equation with intuitionistic fuzzy initial condition. All possible cases are defined and solved them to obtain the exact solutions. The intuitionistic fuzzy numbers are also taken as trapezoidal intuitionistic fuzzy number. The problems are illustrated by two different numerical examples.
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Introduction

Fuzzy Sets and Intuitionistic Fuzzy Sets

Lotfi A. Zadeh (1965) published the paper on his new theory of Fuzzy Sets and Systems in the year 1965. Since the 1980s, this mathematical theory of “uncertain ammount” has been applied with great success in many different areas. Chang and Zadeh (1972) introduced the concept of fuzzy numbers in 1972. A lot of mathematicians have been studying the concept (one-dimension or n-dimension) fuzzy numbers, see for example Diamond and Kloeden (1994), Dubois and Prade (1978), Goetschel and Voxman (1986), Kaleva (1987). With the development of theories and applications of fuzzy numbers, this concept becomes more and more important.

Generalizations of fuzzy sets theory are considered to be one of Intuitionistic fuzzy set (IFS) theory. Out of several higher-order fuzzy sets, IFS was first introduced by Atanassov (1983) and have been found to be suitable to deal with unexplored areas. The fuzzy set considers only the degree of belongingness. Fuzzy set theory does not incorporate the degree of hesitation (i.e., degree of non-determinacy defined as, 1- sum of membership function and non-membership function. To handle such situations, Atanassov (1983) explored the concept of fuzzy set theory by intuitionistic fuzzy set (IFS) theory. The degree of acceptance in Fuzzy Sets is only considered, otherwise IFS is characterized by a membership function and a non-membership function so that the sum of both values is less than one (see Atanassov (1986)). Now-a-days, IFSs are being studied extensively and being used in different fields of Science and Technology.

Necessity for Difference Equation and Fuzzy Difference

In modeling of real natural phenomena, difference equations play an important role in many areas of discipline, exemplary in economics, biomathematics, sciences and engineering. Many experts in such areas widely use difference equations in order to make some problems under study more comprehensible.

Difference Equation

Recently the study of the behavior of difference equation and difference equation system is a topic of a great interest. A difference equation is an equation specifying the change in a variable between two periods. By using the difference equation we can study the concerning factors, which cause the change in the value of the given functions in different time periods. It is known that difference equation appears naturally as discrete system which has many applications in computer science, control engineering, ecology, population dynamics, queuing problems, statistical problems, stochastic time sires, geometry, psychology, sociology physics economics engineering etc. The theory of difference equations developed greatly during last three decays. The theory of difference equation occupies an important position in different fields. No doubt the theory of difference equation is play important role in mathematics as well as its applications.

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