## 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, ∞).

##### URI

http://hdl.handle.net/20.500.11851/3565http://theicmme.org/docs/Abstracts_Book/ICMME-2016_ABSTRACTS_BOOK.pdf