first order homogeneous ordinary differential equation with initial value as triangular...

* Corresponding Author. Email address: [email protected]

First order homogeneous ordinary differential equation with initial

value as triangular intuitionistic fuzzy number

Sankar Prasad Mondal1*, Tapan Kumar Roy1

(1) Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal,

India

Copyright 2014 © Sankar Prasad Mondal and Tapan Kumar Roy. This is an open access article distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract

In this paper first order homogeneous ordinary differential equation is described in intuitionistic fuzzy

environment. Here initial condition of the said differential equation is considered as triangular

intuitionistic fuzzy number. It is illustrated by numerical examples. Finally two elementary applications on

oil production and weight loss problems are described in intuitionistic fuzzy environment.

Keywords: Fuzzy set, Intuitionistic fuzzy number, Fuzzy differential equation. Triangular intuitionistic fuzzy

number.

1 Introduction

Zadeh [1] and Dubois and Parade [2] were the first who introduced the conception based on fuzzy

number and fuzzy arithmetic. Generalizations of fuzzy sets theory [1] is considered to be one of

Intuitionistic fuzzy set (IFS). Out of several higher-order fuzzy sets, IFS was first introduced by Atanassov

[3] have been found to be suitable to deal with unexplored areas.The fuzzy set considers only the degree of

belongingness and non belongingness. Fuzzy set theory does not incorporate the degree of hesitation

(i.e.,degree of non-determinacy defined as, sum of membership function and non-membership

function.To handle such situations, Atanassov [4] 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 [4].

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Journal of Uncertainty in Mathematics Science 2014 (2014) 1-17

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Volume 2014, Year 2014 Article ID jums-00003, 17 Pages

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International Scientific Publications and Consulting Services

Basic arithmetic operations of TIFNs is defined by Deng-Feng Li in [5] using membership and non

membership values. Basic arithmetic operations of TIFNs such as addition,subtraction and multiplication

are defined by Mahapatra and Roy in [6], by considering the six tuple number itself and division by

A.Nagoorgani & K.Ponnalagu [7].

Now-a-days, IFSs are being studied extensively and being used in different fields of Science and

Technology. Amongst the all research works mainly on IFS we can include Atanassov [4,8-11], Atanassov

and Gargov [12], Szmidt and Kacprzyk [13], Buhaescu [14], Ban [15], Deschrijver and Kerre [16],

Stoyanova [17], Cornelis et al. [18], Buhaesku [19], Gerstenkorn and Manko [20], Stoyanova and

Atanassov [21], Stoyanova [22], Mahapatra and Roy [23], Hajeeh [24], Persona et al. [25], Prabha et al.

[26], Nikolaidis and Mourelatos [27], Kumar et al.[28] and Wang [29], Shaw and Roy [30], Adak et

al.[31], A.Varghese and S.Kuriakose [32].

It is seen that in recent years the topic of Fuzzy Differential Equations (FDEs) has been rapidly grown. In

the year 1987, the term “fuzzy differential equation” was introduced by Kandel and Byatt [33]. To study

FDE there have been many conceptions for the definition of fuzzy derivative. Chang and Zadeh [34] was

someone who first introduced the concept of fuzzy derivative, later on it was followed up by Dobois and

Prade [35] who used the extension principle in their approach. Other methods have been discussed by Puri

and Ralescu [36], Goetschel and Voxman [37], Seikkala [38] and Friedman et al. [39,40], Y. Cano, H.

Flores [41], E. Hüllermeier [42], H.Y. Lan, J.J. Nieto [43], J.J. Nieto, R. López, D.N. Georgiou [44]. First

order linear fuzzy differential equations or systems are researched under various interpretations in several

papers (see [45,46,47,48,49]).There are only few papers such as [50,51,52,53] in which intuitionistic fuzzy

number are applied in differential equation.

Fuzzy differential equations play an important role in the field of biology, engineering, physics as well as

among other field of science. For example, in population models [54], civil engineering [55],

bioinformatics and computational biology [56], quantum optics and gravity [57], modeling hydraulic [58],

HIV model [59], decay model [60], predator-prey model [61], population dynamics model [62], Friction

model [63], Growth model [64], Bacteria culture model [65], bank account and drug concentration

problem [66], barometric pressure problem [67]. First order linear fuzzy differential equations are sort of

equations which have many applications among all the Fuzzy Differential Equation.

2 Preliminary concepts

Definition 2.1. Fuzzy Set: A fuzzy set in a universe of discourse X is defined as the following set of pairs

*( ( )) )+ Here : X [0,1] is a mapping called the membership value of x X in a fuzzy

set .

Definition 2.2. Height: The height ( ), of a fuzzy set ( ( ) ), is the largest membership

grade obtained by any element in that set i.e. ( )= ( )

Definition 2.3. Convex Fuzzy sets: A fuzzy set {( ( ))} is called convex fuzzy set if all

for every , - are convex sets i.e. for every element and and ( )

, -. Otherwise the fuzzy set is called non-convex fuzzy set.

Definition 2.4. Intuitionistic Fuzzy Set: Let a set be fixed. An IFS in is an object having the form

{ ( ) ( ) }, where the ( ) , - and ( ) , - define the degree

of membership and degree of non-membership respectively, of the element to the set , which is a

subset of , for every element of , ( ) ( ) .




Definition 2.5. ( )- Level Interval or ( )-cuts: A set of ( )-cut, generated by an IFS ,Where

, - are fixed number such that is defined as

2. ( ) ( )/ ( ) ( ) , -3

We define ( )- level Interval or ( )-cut, denoted by , as the crisp set of elements which

belongs to at least to the degree and which belong to at most to the degree .

Definition 2.6. Intuitionistic Fuzzy Number: An IFN is defined as follows

i) an intuitionistic fuzzy subject of real line

ii) normal,i.e., there is any such that ( ) (so ( ) )

iii) a convex set for the membership function ( ), i.e.,

( ( ) ) ( ( ) ( )) , -

iv) a concave set for the non-membership function ( ), i.e.,

( ( ) ) ( ( ) ( )) , -.

Definition 2.7. Triangular Intuitionistic Fuzzy number: A TIFN is a subset of IFN in R with following

membership function and non membership function as follows:

( )

{

and

( )

{

Where

and TIFN is denoted by (

).

Note 1: Here ( ) increases with constant rate for , - and decreases with constant rate for

, - but ( ) decreases with constant rate for , - and increases with constant rate for

, -.

Definition 2.8. Trpezoidal Intuitionistic Fuzzy number: A TrIFN is a subset of IFN in R with following

membership function and non membership function as follows:

( )

{

and




( )

{

Where

and TrIFN is denoted by (

).

Note -1: Here ( ) increases with constant rate for , - and decreases with constant rate for

, - but ( ) decreases with constant rate for , - and increases with constant rate for

, -.

Definition 2.9. Arithmetic operations of Intuitionistic Fuzzy Number based on ( )-cuts Methods: If

is a IFN, then ( )-level interval or ( )-cuts is given by

{, ( ) ( )- , -

, ( )

( )- , - with .

Here (i) ( )

( )

, - ( ) ( )

and (ii)

( )

( )

, -

( ) ( ).

It is expressed as *, ( ) ( )- , ( )

( )-+ , -.

For instance, if (

) then ( )-level interval or ( )-cuts is given by

*, ( ) ( )- , ( ) (

)-+ , -

Where , -.

Property 2.1. (a) If (

) and (with ) then is a TIFN

(

).

(b) If (

) and (with ) then is a TIFN

(

).

Property 2.2. If (

) and (

) are two TIFN, then

is also TIFN (

).

Property 2.3. If (

) and (


is also TIFN (

).

Property 2.4. If (

) and (

) are two TIFN, then their

product is also TIFN (

).

Property 2.5. If (

) and (


is also

TIFN

.

/.

Definition 2.10. If , ( ) ( ) ( )

( )- be the solution of the Intutionistic fuzzy

differential equation then the solution is written as Intutionistic fuzzy number as bellow

( ( ) ( ) ( ) ( )

( ) ( ))




Definition 2.11. Strong and Weak solution of Intuitionistic Fuzzy ordinary differential equation:

Consider the 1st order linear homogeneous intuitionistic fuzzy ordinary differential equation

with ( ) . Here k or (and) be triangular intuitionistic fuzzy number(s).

Let the solution of the above FODE be ( ) and its ( )-cut be

( ) [, ( ) ( ) ( )

( )-]

The solution is a strong solution if

(i) ( )

( )

, - ( ) ( )

and

(ii)

( )

( )

, -

( ) ( ).

Otherwise the solution is week solution.

Definition 2.12. [68] Let ( ) and ( ). We say that is strongly generalized differential

at (Bede-Gal differential) if there exists an element ( ) , such that

(i) for all sufficiently small, ( ) ( ), ( ) ( ) and the limits(in the

metric )

( ) ( )

( ) ( )

( )

Or

(ii) for all sufficiently small, ( ) ( ), ( ) ( ) and the limits(in the

metric )

( ) ( )

( ) ( )

( )

Or

(iii) for all sufficiently small, ( ) ( ), ( ) ( ) and the limits(in the

metric )

( ) ( )

( ) ( )

( )

Or

(iv) for all sufficiently small, ( ) ( ), ( )

( ) and the limits(in the

metric )

( ) ( )

( ) ( )

( )

( and at denominators mean

and

, respectively).

Definition 2.13. [69] Let be a function and denote ( ) ( ( ) ( )), for each , -.

Then

1) If is (i)-differentiable, then ( ) and ( ) are differentiable function and ( )

( ( ) ( )).

2) If is (ii)-differentiable, then ( ) and ( ) are differentiable function and ( )

( ( ) ( )).




Definition 2.14. Let , - . The integral of in , -, ( denoted by ∫ ( ) , -

or, ∫ ( )

) is

defined levelwise as the set if integrals of the (real) measurable selections for , - , for each , -. We

say that is integrable over , - if ∫ ( ) , -

and we have

0∫ ( )

1 0∫ ( )

∫

( )

1 for each , -.

3 Differential Equation with initial value as Triangular Intuitionistic Fuzzy number

Consider the differential equation

with ( ) (

) (3.1)

Case I:

Taking ( )-cut of equation (3.1) we get

(, ( ) ( )- ,

( ) ( )-) (, ( ) ( )- ,

( ) ( )-) (3.2)

With initial condition

( ) (, ( ) ( )- , ( )

( )- , -)

i.e.,

( ) ( )

( ) ( )

( )

( )

( )

( )


( ) ( )

( ) ( )

( )

( )

( )

( )

The solution is given by

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( )

( )

( ) ( )

Case II:

Let

Taking ( )-cut of equation (3.1) we get

(, ( ) ( )- ,

( ) ( )-) (, ( ) ( )- ,

( ) ( )-) (3.3)





( ) (, ( ) ( )- , ( )

( )- , -)

i.e.,

( ) ( )

( ) ( )

( )

( )

( )

( )


( ) ( )

( ) ( )

( )

( )

( )

( )

The solution is given by

( ) . ( ) ( )

/ ( ) .

( ) ( )

/ ( )

( ) . ( ) ( )

/ ( ) .

( ) ( )

/ ( )

( ) .

( )

( )

/ ( ) .

( )

( )

/ ( )

( ) .

( )

( )

/ ( ) .

( )

( )

/ ( )

4 Example

Example 4.1. Consider the FODE

with ( ) ( )

Solution: The solution is given by

( ) ( )

( ) ( )

( ) ( )

( ) ( )

Here ( )

,

( )

,

( )

,

( )

Hence the solution is a strong solution.




Figure 1: For

The solution is a triangular Intuitionistic fuzzy number which is written as

( )

The membership function and non-membership function as follows

( )

{

and

( )

{

Example 4.2. Consider the FODE

with ( ) ( )

Solution: The solution is given by

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )




Figure 2: For

The solution is a triangular Intutionistic fuzzy number which is written as

( )

The membership function and non-membership functions are as follows

( )

{

( )

( )

and

( )

{

5 Applications

Applications 5.1. Oil production Model: The rate of increase of world oil production y in million metric

tons per year was assumed to be proportional to y itself. Then what is the amount of oil after five years if

initially ( ) million metric ton oil is there.(the constant of

proportionality is ).

Solution:

with and ( ) million The solution

is

( ) ( )

( ) ( )

( ) ( )

( ) ( )




Figure 3: For years


( )


( )

{

and

( )

{

Applications 5.2. Weight-loss: Over-weight people on diet and treatment gradually reduce their weight y.

The time rate of decrease of weight y is assumed to be proportional to to the weight y itself. If initially the

wait is ( ) lb. What is the wait after 30 days?(The Constant of proportionality is

)

Solution:

where ( ) ( ) and

The solution is

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )




Figure 4: For t=30 days


(

)


( )

{

(

)

(

)

and

( )

{

6 Conclusion

In this paper we have solved first order homogeneous ordinary differential equation in intuitionistic

fuzzy environment. Here we have discussed initial value as intuitionistic fuzzy number. The intutionistic

fuzzy number is taken as triangular intuitionistic fuzzy number. Oil Production and Weight Loss problems

are discussed in intuitionistic fuzzy environment. For further work the same process can be solved by

using generalised triangular intutionistic fuzzy number, generalized trapizoidal intuitionistic fuzzy number,

generalized L-R type intuitionistic fuzzy number. Also we can follow the same for first order non

homogeneous ordinary differential equation. This process can be followed for any economical, bio-

mathematical and engineering sciences problem in fuzzy environment.




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