An Indicator Operator Algorithm for Solving a Second Order Intuitionistic Fuzzy Initial Value Problem

Abstract

L. Zadeh [1] was the first who introduced the concept of fuzzy settheory as an extension of the classical notion of the set theory. He remindedpeople that things are not always black or white; there may besome grey colours in life. Hence he simply assigned so called the membershipfunction to each element in a classical set and started the fuzzyset theory. However in some cases the membership concept in a fuzzyset is itself uncertain. This uncertainty may be because of the subjectivityof expert knowledge, complexity of data or imprecision of themodels. Since the fuzzy set theory considers only the degree of membership,it does not involve the degree of uncertainty for the membership.To handle such situations, the generalized concepts of fuzzy set theory are used [1]-[5]. One of these generalizations is intuitionistic fuzzy settheory which was given by Atanassov [2]. Atanassov introduced intuitionisticfuzzy set concept by extending the definition of fuzzy set afteranalyzing the shortcomings of it. He defined the intuitionistic fuzzy setconcept by introducing the the nonmembership function into the fuzzyset such that sum of both is less than one. And in his further researches,he showed the exclusive properties of intuitionistic fuzzy sets [6]-[13].In the past decades, the intuitionistic fuzzy set theory has penetratedinto different research areas, such as decision making [14]-[17], clusteringanalysis [18], medical diagnosis [19]-[20], pattern recognition [21]-[23] In recent years the topic of fuzzy differential equations has beenrapidly grown to model the real life situations where the observed datais insufficient [24]-[27]. Especially to describe the relation between velocityand acceleration, second order differential equations has greaterimportance in science. Therefore many approaches [27]-[29] were givento solve second order fuzzy differential equations. However there areonly few works [30]-[32] to observe the intuitionistic fuzzy differentialequations.In this work, we have examined the solution of the following second orderintuitionistic fuzzy initial value problems given in Eq. (1)-(2) usingintuitionistic Zadeh’s Extension Principle [13]. 𝑦 ′′(𝑥) + 𝑎1 𝑖 𝑦 ′ (𝑥) + 𝑎2 𝑖 𝑦(𝑥) = ∑𝑏 ̅𝑗 𝑖𝑔𝑗(𝑥) 𝑟 𝑗=1 𝑦(0) = 𝛾 ̅0 𝑖 ; 𝑦′(0) = 𝛾 ̅1 𝑖 Here 𝛾 ̅0 𝑖 ; 𝛾 ̅1 𝑖 and 𝑏̅ 𝑗 𝑖 (j=1, 2, 3,…, r ) are intuitionistic fuzzy numbers. 𝑔𝑗(𝑥) (j=1, 2, 3,…,r ) are continuous forcing functions on the interval [0, ∞).