Applied and Computational Mathematics
Volume 4, Issue 4, August 2015, Pages: 245-257

Homotopy Method for Solving Finite Level Fuzzy Nonlinear Integral Equation

University Gadjah mada, Department of Mathematics and atural Science, Faculty MIPA, Yogyakarta, Indonesia

Alan Jalal Abdulqader. Homotopy Method for Solvind Finite Level Fuzzy Nonlinear Integral Equation. Applied and Computational Mathematics. Vol. 4, No. 4, 2015, pp. 245-257. doi: 10.11648/j.acm.20150404.13

Abstract: In this paper, non – linear finite fuzzy Volterra integral equation of the second kind (NFVIEK2) is considered. The Homotopy analysis method will be used to solve it, and comparing with the exact solution and calculate the absolute error between them. Some numerical examples are prepared to show the efficiency and simplicity of the method.

Keywords: Fuzzy Number, Finite Level, Volterra Integral Equation of Second Kind, Homotopy Analysis Method, Fuzzy Integral

1. Introduction

In this chapter, we construct a new method to find a solution of the nonlinear fuzzy integral equation.

(1)

where. Park et al., consider the existence of solution 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 solution 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. So we need some background material about fuzzy metric space, fuzzy contraction mapping and related mathematical notions. These notions are fundamental, and absolutely essential in proving the existence and uniqueness of (1) .We will discuss some method in order to find the solutions of nonlinear fuzzy integral equation of second kind.

2. Basic Concepts

Let X be a space of object , let  be a fuzzy set in X then one can define the following concepts related to fuzzy subset  of X [1,6] :

1- The support of  in the universal X is crisp set , denoted by :

2- The core of a fuzzy set  is the set of all point , such that

3- The height of a fuzzy set  is the largest membership grade over X, i.e hgt(

4- Crossover point of a fuzzy set  is the point in X whose grade of membership in  is 0.5

5- Fuzzy singleton is a fuzzy set whose support is single point in X with

6- A fuzzy set  is called normalized if it’s height is 1; otherwise it is subnormal

Note:

A non-empty fuzzy set  can always be normalized by dividing

7-The empty set respectively

8- for all x

9-  for all x

10- is a fuzzy set whose membership function is defined by

11-Given two fuzzy sets, , their standard intersection, , are fuzzy sets and their membership function are defined for all

3. a – Cut Sets

Definition 1: ( The -cut set  of a fuzzy set A is made up of membership whose membership is not less than , [3,5,9]

The following properties are satisfied for all ]

i-

ii-

iii-

iv-

v-

Remarks 3:

1- The set of all level , that represent distinct  – cuts of a given fuzzy set [17]

2- The support of  is exactly the same as the strong  of  for

3- The core of  is exactly the same as the of  for .

4- The height of  may also be viewed as the supremum of  for which

5- The membership function of a fuzzy set  can be expressed in terms of the characteristic function of it is s according to the formula:

4. Convex Fuzzy Sets

We can generalize the definition of convexity to fuzzy sets. Assuming universal set  is defined in the set of real numbers . If all convex, then the fuzzy set with these  is convex [12, 20]

Definition 2:

A fuzzy set  on R is convex if and only if [13] :

Remarks 4:

Assume that  is convex for all  then if  and moreover  for any  by the convexity of . Consequently

Assume that  satisfies equation (1), we need to prove that For any  and for any  by equation (1)

i.e is convex.

Definition 3. (Extension of fuzzy set ) Let  be a fuzzy set defined on then we can obtain a fuzzy set  [14, 23]

Definition 4: (Extension Principle) We can generalize the per-explained extension of fuzzy set. Let  be Cartesian product of universal set  be r- fuzzy sets in the universal set. Cartesian product of fuzzy sets  yields a fuzzy set [14,24,19]

define as

Let function  be from space

Then fuzzy set  can be obtained by function  and fuzzy sets  as follows:

Here,  is the inverse image of ,  is the membership of

In following example, we will show that fuzzy distance between fuzzy sets can be defined by extension principle.

5. Intervals

"real number" implies a set containing whole real numbers and "positive numbers" implies a set holding numbers excluding negative numbers. "positive number less than equal to 10 (including 0)" suggests us a set having numbers from 0 to 10. That is [1,4,11,22]

A=

Or

Since the crisp boundary is involved, the outcome of membership function is one or zero. In general, when interval is defined on set of real number R this interval is said to be a subset of R. For instance, if interval is denoted as , we may regard this as one kind of sets. Expressing the interval as membership function is shown in the following .

If  this interval indicates a point. That is

Fig. 1. Interval .

Definition 5: (fuzzy number) If a fuzzy set is convex and normalized, and its membership function is defined in  and piecewise continuous, its is called as fuzzy number so fuzzy number (fuzzy set ) represents a real number interval whose boundaries is fuzzy Fig 2, [3,26,5,6].

Fig. 2. Sets denoting intervals and fuzzy number.

Fuzzy number should be normalized and convex. Here the condition of normalization implies that maximum membership value is 1

The convex condition is that the line by  is continuous and interval satisfies the following relation:

This condition may also be written as,

Fig. 3. -cut of fuzzy number.

5.1. Operation of  Interval

Operation on fuzzy numbers can be generalized from that of crisp interval. First of all, we referred to  interval of fuzzy number  as crisp set

So  is a crisp interval. If  interval  for fuzzy number B is given

operation between  and  can be described as follows [6,15]:

=[

2- Subtraction

=[

3- Multiplication

4- Division

=

5- Invers

6- Minimum

=[

7- Maximum

=[

5.2. Operation on Fuzzy Numbers

Let  be the set of all fuzzy numbers on real line R. Using extension principle. A binary operation * can be extended into (*) to combine two fuzzy numbers A and B. Moreover, if  are the membership functions of A and B assumed to be continuous functions on R [2,7,16]

(2)

Theorem 1: Let A, B and C be a fuzzy numbers. The following holds [9]:

1-

2-  -(-A)=A

3-  A\1=A

4-  A/B=A.1/B

5-

6-  (-r)A=-(rA)

7-  (-A)B=-(A.B)=A(-B)

8-  A/r=(1/r)A

9-

6. Other Types of Fuzzy Numbers

Carrying out computations with fuzzy quantities and in particular with fuzzy numbers, can be complicated. There are some special classes of fuzzy numbers for which computations of their sum, for example .is easy. One such class is that of triangular fuzzy number, another one is that of trapezoidal fuzzy number.

In this paper we discuss about new type for fuzzy number name finite level fuzzy number[11,17,21].

Remarks 5:

Lets talk about the operation of trapezoidal fuzzy number as in the triangular fuzzy number

1.   Addition and Subtraction between trapezoidal (triangular )fuzzy numbers become trapezoidal (triangular ) fuzzy number

2.   Multiplication, Division and inverse need not be trapezoidal (triangular) fuzzy numbers

3.   Max and Min operation of trapezoidal (triangular) fuzzy numbers is not always in the form of trapezoidal (triangular) fuzzy numbers

But in may cases, the operation results from multiplication or division are approximated trapezoidal shape. As I triangular fuzzy number, addition and subtraction are simply defined, and multiplication and division operations should be done by using membership function

i-

ii-

iii-

iv-

The multiplication and the addition of two triangular ( trapezoidal) fuzzy numbers is not a triangular (trapezoidal) fuzzy number , so it will not form a group structure. Now, we will construct a new of fuzzy numbers ( which we shall call it finite level fuzzy numbers), such that the addition and multiplication of two finite level fuzzy numbers will be also finite level fuzzy number. The construction of this new type of fuzzy numbers will as follows [25,14,20]:

Given n ,N be two positive integers

Let  be the set of all fuzzy numbers  defined on R , such that

The operations of this type of fuzzy numbers can be defined by

Let  such that

According to equation (2) we have

(3)

If we perform the * operation between A and B, we will get the following table

 * [Min{ 1

Now, from this table it is clear that the convex of A*B is

(4)

According to equation (3 and 4) in this case can be written as

(5)

where

and

So  is fuzzy number and

7. Fuzzy Equations

A fuzzy equation is an equation whose coefficients and / or variable are fuzzy sets of R. The concept of equation can be extended to deal with fuzzy quantities in several ways. Consider the simple equation , then the fuzzy equation

(6)

means that the fuzzy set  is the same as  Note that it is forbidden to shift terms from one side to another . For instance, the equation is not equal to the first may have solution, while the second surely dose not, since  is fuzzy and 0 is scalar.

We can solve the fuzzy equation (6) if we consider the fuzzy variables and the fuzzy coefficient as a fuzzy numbers of the form  In another word[15],

(7)

again using equation (7) to solve equation (6)

(8)

Finally the fuzzy equation

implies that

(9)

So the solution of the fuzzy equation (6) is a fuzzy number

(10)

Fuzzy function of crisp Variable

Two points of view can be developed depending on whether the image of  is a fuzzy set  on , or  is mapped to  through a fuzzy set of functions .

Definition 6:

A fuzzy mapping  is a mapping from , namely In other words, to each , corresponds a fuzzy set  defined on , whose membership function is  and [8]

A fuzzy set of mapping  can be constructed in the following way,

Define a function  such that , ( where  is the set of all functions

Definition 7: Given a fuzzy set of mappings  with , we can construct a fuzzy function  is a fuzzy set , as follows[21]:

Definition 8:

Given a fuzzy function set on  with  and a function  Then there exists a fuzzy function  such that[25]

Fig. 4. fuzzy function .

Definition 9: Given a fuzzy set of mappings  with and a functional

. Then we can construct a fuzzy functional  such that[27]

Therefore

Fig. 5. fuzzy functional.

Example 1:

Let G be the set of all integrable functions. The integration  can be considered as a functional where

Then the fuzzy integral  can be defined the equation above

Given a fuzzy mapping  such that

(11)

Definition 10:

Let T be a fuzzy set such that then T will be finite if In another word,  where

Definition 11: rewrite the definition 8 , if fuzzy mapping  is finite , then can be written as

Any fuzzy set of mapping F, constructed from  also will be finite , and

This implies that

Now, if given a finite set of mappings , then we have

Definition 12:

Given a finite mapping  , and a functional, then a fuzzy functional in this case , can be defined by[27]

(12)

Definition 13:

The integral of a finite fuzzy mapping  is given by

(13)

Definition 14: Starting from the fuzzy mapping  with , for any , we can define the  as follows [17,24]:

(14)

For a fuzzy set of mappings F with  and it can be constructed using (13) as

(15)

Theorem 2: [19]Let A be a fuzzy set such hat  then

(16)

Theorem 3: [11] let  be a fuzzy function.Due to above theoerm we always ha

(17)

8. a - Level Fuzzifying Function

Consider a fuzzy function, which shall be integrated over the crisp interval. The fuzzy function  is supposed to be fuzzy number; we shall further assume that - level curves[3,8,17]:

(18)

have exactly two continuous solutions:

and only one solution:

(19)

which is also continuous ; are defined such that

(20)

These functions will be called - level curves of

Definition 15:

Let a fuzzy function , such that for all  is a fuzzy number and  are curves as defined in equation (20), [22,27]

The fuzzy integral of  is then defined as the fuzzy set

,

where and + stands for the union opertors

Remark 5:

1-  A fuzzy mapping having a one curve will be called a normalized fuzzy mapping

2-  A continuous fuzzy mapping is a fuzzy mapping  is continuous for all

3-  The concept of fuzzy interval is convex, normalized fuzzy set of R whose membership function is continuous.

Fig. 6. level fuzzifying function.

Definition 16:.A fuzzy mapping  such that  , in other words, to each

.

A fuzzy set of mapping F can be constructed in the following way, Define a function  such that  , ( where is the set of all functions .

(21)

9. Fuzzy Operator

In the Eq(21) . we consider a fuzzy mapping  such that  with  The functional of  over X was defined as a fuzzy set

In this part , we shall deal with the operator of fuzzy function F, which will denoted

In this part , we shall deal with the operator of fuzzy function F, which will denoted

Definition 17 : Given a fuzzy function  with  and an operator

. Then we can construct a fuzzy operator  such that

Therefore ,

(21)

When  is non=-to-one operator then equation (21) will be

Lemma1:

Let F be a fuzzy mapping , Let T and H be two operator such that  , and H is one-to-one then we have

(22)

proof:

Since H is one –to-one , then

Theorem 4: [8]

Let  be a fuzzy mapping , be two operators  where  is one-to-one . Then there exist a fuzzy operators such that

Proof:

By Lemma , we have

Definition 18. Given a finite fuzzy set of mappings  , and an operator

. The fuzzy operator  of F can be defined by

(23)

If  is a one –to-one equation (90) will be

(24)

Remarks 6:[25]

Given a fuzzy mapping  . Then we have

i-

ii-

iii-

Theorem 5: let  be real fuzzy mapping from X to the set  such that

i-

ii-

Proof :

(i)

Where

(ii)

Definition 19:

Let  be the set of real number and  all fuzzy subsets defined on R . G.Zang defined the fuzzy number  as follows :

is normal , that is there exists  such that

Foe every  is closed interval , denoted by

Using Zaheh’s notation  is the fuzzy set on R defined by

Definition 20:

Let  we define the following operation as [1,7,20]:

1- if

2- if

3- if

4- ,

5- forevery

=

6-

7-

8-

Definition 21: Let

1-  If there exists  such that  for every , then  is said to have an upper bound

2-  If there exists  such that  for every  ,then  is said to have an lower bound

3-  is said to be bounded if  has both upper and lower bounds.

is said to be bounded if the set  is bounded

Definition 22: Let be a metric space , and let  be the set of all non-empty compact subset of X. The distance between A and B , for each  is defined by the Hausdorff metric [18,27]

Theorem 6. (H(x),D) is a metric space

Definition 23: A fuzzy set  is compact if all its level sets  is compact subset in the metric space (X,d)

Definition 24: Let  be the set of all non-empty compact fuzzy subset of X. the distance between  defined by

such that

where D is the Haousdorff metric defined in H(x)

Theorem 7:  is a metric space , if  is a metric space

Theorem 8:  is complete metric space ,if (X,d) is a complete metric space.

Now, when  and , since for each fuzzy number  we know that  is a closed interval ], then  is compact , and hence  is a non-empty compact subset in

Definition 23. The distance between fuzzy numbers is given by

}}

Theorem 5. is a metric space

Theorem 6. If

Proof:

}

definition 25. Let  . Then the sequence  is said to converge to  in fuzzy distance , denoted by

if for any given  there exists an integer  for  A sequence  is said to be a Cauchy sequence if for every  there exists an integer  such that

For  A fuzzy metric space  is called the complete metric space if every Cauchy sequence in  is converges .

Theorem 7. The sequence  is converge in the metric  if and only if is a Cauchy sequence .

Theorem 8.

Definition 26: A fuzzy mapping is called levelwise continuous at  if the mapping  is continuous at  with respect to the Hausdorff metric on  for all when , this definition can be generalized to

Definition 27: A fuzzy mapping  is called levelwise continuous at point  provided , for any fixed  and arbitrary  there exists  such that

whenever

for all

Definition 28:

Let is defined levelwise by the equation

Theorem 9. If  levelwise continuous and Supp( is bounded , then F is integrable

Proof: Directly from definition (27)

Theorem 10.Let  be integrable and  . Then

1-

2-

Theorem 10:

(Existence and uniqueness For a Solution Of fuzzy nonlinear integral Equation )

Assume the following conditions are satisfied

.

10. Solution of Fuzzy Nonlinear Integral Equations

Our treatment of fuzzy nonlinear volterra ntegral equation centerel mainly on illustrations of the known methods of finding exact, or numerical solution. In this paper we present new techniques for solving fuzzy nonlinear volterra integral equations by using Honotopy analysis method .

9.1. Homotopy Analysis Method

Consider

(24)

Where  is an operator , is known function and x is independent variable.

Let  ) denoted an initial guess of the exact solution ,  an auxiliary parameter,  an auxiliary function , and auxiliary linear operator ,with the property Then using  as an embedding parameter , we construct such a homotopy.

(25)

It should be emphasized that we have great freedom to choose the initial guess the auxiliary linear operator  , the non-zero auxiliary parameter  , and the auxiliary function .

Enforcing the homotopy (25) to zero, i.e

=0

we have the so – called zero- order deformation equation

(26)

When , the zero-order deformation equation (26) becomes

(27)

and when  since 0 and  , the zero-order deformation equation (26) is equivalent to

(28)

Thus according to (27) and (28), as the embedding parameter  increases from 0 to 1,  varies continuously. From the initial  the exact solution . Such a kind of continuous variation is called deformation in homotopy .

By Taylor’s theorem  can be expanded in power series of  as follows

(29)

Where

(30)

If the initial guess , the auxiliary linear parameter , the non zero auxiliary parameter h , and the auxiliary function  are property chosen so that the power series (29) of  converges at , we have under these assumptions the solution series .

(31)

Finaly we get

Where

Hence , the solution of equation (24)

we denoted the nth- order approximation to solve

9.2. Solve Fuzzy HAM

Consider the fuzzy nonlinear integral equation with fuzzy difference kernel

Where

Then

Now , make use equation (7), we get

which implies that for each

(32)

Now , we can apply the HAM to equation (32) , we get

Let

=0

When q=0

when q=1

Where

Finally we get

Where

we denoted the nth- order approximation to solve ,

Example 1

Consider the fuzzy nonlinear integral equation

ln(x+1)+2ln2(1−xln2+x)−2x− ,sin(πx)

with the exact solution to equation is ln(x + 1), and

by using equation (9), and (10)

Where

Then

Now , make use equation (7), we get

which implies that for each

by using HAM method to solve this formula , we get

Table 1. shows numerical result calculated according the exact solution and Homotopy analysis method for  , where h=0.5.

 Exact ln(x + 1), HAM Exact-HAM 0 (0,0.4) (0.00000026768,0.4) 2.56186267595E-8 0.1 (0.095310179804,0.4) (0.95310206285,0.4) 2.53309949127E-8 0.2 (0.182321556793,0.4) (0.182321579887,0.4) 2.19432191028E-8 0.3 (0.262364264467,0.4) (0.262364280170,0.4) 1.45529119865E-8 0.4 (0.336472236621,0.4) (0.336472248418,0.4) 1.06476171213E-8 0.5 (0.405465108108,0.4) (0.405465120209,0.4) 1.84023705163E-8 0.6 (0.470003629245,0.4) (0.470003629794,0.4) 6.84976264597E-9 0.7 (0.530628251062,0.4) (0.587786677053,0.4) 6.0872384910E-10 0.8 (0.587786664902,0.4) (0.587786677053,0.4) 3.5510718809E-9 0.9 (0.641853886172,0.4) (0.641853890141,0.4) 2.8196508461E-9 1 (0.693147180559,0.4) (0.693147181293,0.4) 4.163761557E-10

Table 1 solution to the example 1

at

by using the equation (9) and ( 10) , we get

Table 2. shows numerical result calculated according the exact solution and Homotopy analysis method for , where h=0.5.

 Exact . HAM Exact-HAM 0 (0.0754266889,1) (0.0754266889,1) 5.53723733531E-15 0.1 (0.3807520383,1) (0.3807520383,1) 5.21804821573E-15 0.2 (0.6488067254,1) (0.6488067254,1) 4.55191440096-15 0.3 (0.8533516897,1) (0.8533516897,1) 3.21964677141-15 0.4 (0.9743646449,1) (0.9743646449,1) 1.77635683940-15 0.5 (1.0000000000,1) (1.0000000000,1) 0 0.6 (0.9277483875,1) (0.9277483875,1) 1.77635683940E-15 0.7 (0.7646822990,1) (0.7646822990,1) 3.21964677141E-15 0.8 (0.5267637791,1) (0.5267637791,1) 4.55191440096E-15 0.9 (0.2372819503,1) (0.2372819503,1) 0.52735593669E-15 1 (-0.0754266889,1) (-0.0754266889,1) 0.55372373353E-15

Table 2 solution to the example 1

11. Conclusion

The proposed method is a powerful procedure for solving fuzzy nonlinear integral equations. The examples analyzed illustrate the ability and reliability of the method presented in this paper and reveals that this one is very simple and effective. The obtained solutions, in comparison with exact solutions admit a remarkable accuracy. Results indicate that the convergence rate is very fast, and lower approximations can achieve high accuracy.

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 Contents 1. 2. 3. 4. 5. 5.1. 5.2. 6. 7. 8. 9. 9.1. 9.2. 10. 11.
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