Homotopy Method for Solving Finite Level Fuzzy Nonlinear Integral Equation
Alan Jalal Abdulqader
University Gadjah mada, Department of Mathematics and atural Science, Faculty MIPA, Yogyakarta, Indonesia
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To cite this article:
Alan Jalal Abdulqader. Homotopy Method for Solvind Finite Level Fuzzy Nonlinear Integral Equation. Applied and Computational Mathematics. Vol. 4, No. 4, 2015, pp. 245257. 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 nonempty fuzzy set can always be normalized by dividing
7The empty set respectively
8 for all x
9 for all x
10 is a fuzzy set whose membership function is defined by
11Given 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 perexplained 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].
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]:
1 Addition
=[
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
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=toone 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 onetoone then we have
(22)
proof:
Since H is one –toone , then
Theorem 4: [8]
Let be a fuzzy mapping , be two operators where is onetoone . 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 –toone 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 nonempty 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 nonempty 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 nonempty 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 nonzero 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 zeroorder deformation equation (26) becomes
(27)
and when since 0 and , the zeroorder 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  ExactHAM 
0  (0,0.4)  (0.00000026768,0.4)  2.56186267595E8 
0.1  (0.095310179804,0.4)  (0.95310206285,0.4)  2.53309949127E8 
0.2  (0.182321556793,0.4)  (0.182321579887,0.4)  2.19432191028E8 
0.3  (0.262364264467,0.4)  (0.262364280170,0.4)  1.45529119865E8 
0.4  (0.336472236621,0.4)  (0.336472248418,0.4)  1.06476171213E8 
0.5  (0.405465108108,0.4)  (0.405465120209,0.4)  1.84023705163E8 
0.6  (0.470003629245,0.4)  (0.470003629794,0.4)  6.84976264597E9 
0.7  (0.530628251062,0.4)  (0.587786677053,0.4)  6.0872384910E10 
0.8  (0.587786664902,0.4)  (0.587786677053,0.4)  3.5510718809E9 
0.9  (0.641853886172,0.4)  (0.641853890141,0.4)  2.8196508461E9 
1  (0.693147180559,0.4)  (0.693147181293,0.4)  4.163761557E10 
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  ExactHAM 
0  (0.0754266889,1)  (0.0754266889,1)  5.53723733531E15 
0.1  (0.3807520383,1)  (0.3807520383,1)  5.21804821573E15 
0.2  (0.6488067254,1)  (0.6488067254,1)  4.5519144009615 
0.3  (0.8533516897,1)  (0.8533516897,1)  3.2196467714115 
0.4  (0.9743646449,1)  (0.9743646449,1)  1.7763568394015 
0.5  (1.0000000000,1)  (1.0000000000,1)  0 
0.6  (0.9277483875,1)  (0.9277483875,1)  1.77635683940E15 
0.7  (0.7646822990,1)  (0.7646822990,1)  3.21964677141E15 
0.8  (0.5267637791,1)  (0.5267637791,1)  4.55191440096E15 
0.9  (0.2372819503,1)  (0.2372819503,1)  0.52735593669E15 
1  (0.0754266889,1)  (0.0754266889,1)  0.55372373353E15 
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.
References