Solving the First Kind Fuzzy Integral Equations Using a Hybrid Regularization Method and Bernstein Polynomials

A hybrid of regularization methodand Bernstein polynomials is used to solve the first kind fuzzy integral equations. In this paper, first the regularization method applied to convert the first kind fuzzy integral equation into the second kind fuzzy integral equation. Then by approximating Bernstein polynomials, the obtained second kind fuzzy integral equation is solved. In this method, one parameter was created in the second kind equation. When this parameter tends to zero, the solutions of integral equation of the second kind tend to solutions of the integral equation of the first kind. The obtained solutions are comparable to the solutions of the other similar methods. Performance of the mentioned method is illustrated by considering some example.


Introduction
Integral equations are employed in numerical Computations and in modeling of engineering science and other sciences (Jerri, 1999;Tikhonov, 1963;Wazwaz, 2011) as well.In manynonsingular integral equations of the first kind, especially Abel equations, the exact solution will not be achieved.In solving (Hanna, Roumeliotis & Kucera, 2005) Fredholm integral equations of the first kind the local regularization method is employed.Integral equations of the first kind with single kernel were solved by applying the Cauchy equation in (Okecha & Onwukwe, 2012).Chebyshev polynomials and Jacobi polynomials are considered in this paper.Chris et al. (Karimi & Jozi 2015), have solved equations of the first kind using least squares.Another type of integral equations is called Fuzzy integral equations and is a very useful tool in several applications such as engineering areas.Fredholm fuzzy integral equations are solved by numerical method (Abbasbandy.et al (Abbas, Babolian & Alavi, 2007)).Adomian decomposition method is employed to solve the Fredholm fuzzy integral equations of the second by Babolian, et al. (Babolian, Sadeghi & Goghary, 2005).Another approach to solve the integral equations of the first kind is to transform them to the integral equations of second kind by regularization method.In (Wazwaz, 2011) the first kind integral equations have been solved by hybrid regularization and Homotopy method.Another type of the first kind equations is the Abel integral equation which is presented in Fuzzy form.Jafarian et al. (Jafarian & Measoomy, 2014), solved Abel fuzzy integral equations by multi-layers neural network and "back-propagation learning".Here, first the regularization method was used for solving the non-fuzzy integral equation and integral equation of the first kind.Then considering transformation of the first kind equations to the second kind equations by regularization method, we need to solve the obtained second kind equation by choosing Bernstein polynomials.Then integral equation of the second kind will be solved by Bernstein polynomials Approximation.Finally, by hybridregularization and Bernstein methods the Abel fuzzy first kind integral equations will be solved.Note that the hybrid method increases accuracy of approximations.

The Regularization Method
The regularization methodwas established independently by Tikhonov in (Tikhonov, 1963) and Philips in (Philips, 1962) separately.In this method, integral equation of the first kind converts to integral equation of the second kind.By converting, the Fredholm first kind linear integral equation is defined as Converts to the approximation Fredholm integral equation Where, α in (3) and ( 4) is the small real valued positive parameter which is called regularization parameter.It is clear that (3) and ( 4) are the second kind integral equations.They are presented as follows respectively Lemma: Fredholm first kind linear integral equation ( 1) substituted with Fredholm second kind linear integral equation ( 3) and α is a small real valued positive parameter.Such that ( ) lim ( ) (Wazwaz, 2011).Tikhonov (Tikhonov, 1963) and Philips (Philips, 1962) proved that equations ( 5) or ( 6), when 0 α → , is convergent into value of ( ) u x and of (1), (2).Such that ( ) lim ( )

=
It is important to note that the fredholm integral equations off the first kind is an ill-posed problem.The solution for an ill-posed problem may not exist, and if it does exists; it may not be unique.

[ ]
p is the polynomial with order n presents as follows: , ( ) ( ) , , , , Bernstein applied this approximation to give a proof to Weirestrass theorem (Bernstein, 1912).
This theorem follows that, for any ∈ [0,1] and for any > 0 , there exists such that the in equality ‖ − ‖ < . Holds.
In chapter 1 of (Revlin, 1969), it has been proved that satisfy For more details see (Revlin, 1969); Assume that maximum . is on [ ] , 0 1 In this case, the error bound is shown as fallows This bound has presented in chapter 10 of (Devore, 1993) that shows for [ ] . On the other handVoronovskaya asymptotic error (Voronovskaya, 1932) is given by For numerical solution (12), we replace ( ) f t with Bernstein polynomial approximate (7).So we get that ( ) To find the unknowns ( ) i f n , , i 0 n =  , the equation ( 13) converts to linear system of A x y = by choosing j x , , , , j 0 1 n =  .In this linear form ( ) , , , After having the system A x y = solved, the values of ( ) And by replacing these values in (7) the approximate solution of integral equation ( 12) will be achieved.Since, in regularization method, integral equation of the first kind converts to integral equation of the second kind, in system (12) integral equation of the second kind is presented.Now recall that, in this paper we tend to solve integral equation of the first kind.The error bound is given in Theorem (2).Theorem 3.2 Consider the integral equation of the second kind (12).Assume that k(x, t)is continuous on the square [ ] , 2 0 1 and the solution to equation for some 2 Where B f x are the solution of assumed method.

Fuzzy Integral Equations of the First Kind
In this section, properties of symmetrical fuzzy numbers are discussed.( ) Let 1 E denote the sets of all fuzzy numbers based on described definition (Klir, Clair & Yuan, 1997).An alternative definition which yields the same 1 E is given by Kaleva (Kaleva, 1981;Ma, Friedman & Kandel, 2003).
With the above definitions it is clear that ( ) ( ) ( ) and also a fuzzy number Remark 2. For every desired fuzzy number, , ∈ , we define below parameter for a interval between two fuzzy numbers u and V: It is shown (Puri & Ralescu, 1986) that ( ) Abel integral equations are the most applied integral equations of the first kind (Wazwaz, 1997).

Abel Fuzzy Integral Equations
Let us define Abel integral ( ) ( ) ; ( ) Where α is a known constant and0 Where 0 r 1 ≤ ≤ andα is known constants that ( ) ( ) is the solutions to (17) which should be determined.
By assumption 1 2 α = in Eq. ( 17), the standard form of Abel non-linear fuzzy integral equation will be introduced as follows , , Where the function ( )  In order to solve the fuzzy non-linear integral equation of the first kind ( 18), first we need to convert it to the fuzzy non-linear integral equation of the second kind using regularization method, and then Bernstein approximations are employed to define the solution.As it can be seen, fuzzy non-linear integral equation of the first kind(3) converts to the fuzzy non-linear integral equation of the second kind (6) and the solutions of u α for system (6) when 0 α → is the exact solutions of (2).
Fuzzy form of the second kind using regularization method ( 18) is shown as Abel fuzzy integral equation ( 19) is solved by considering Bernstein polynomials then by decreasing value of α, we expect the obtained approximations of ( ) And Equations( 20) and ( 21) were solved by Bernstein polynomial approximations then we find c u α and d u α .
We note that ( , ) Finally, after finding the value of u α and u α , approximate solutions (18) are derived.
For small value of α, we repeat above process and every time new u α and u α derived and every time α becomes smaller.As expected: Now, we present some examples of non-fuzzy and fuzzy integral equations so that we can study the efficiency of the hybrid method.As was noted in the article, fuzzy integral equations could be transformed to two non-fuzzy integral equations.
First, to show the effect of hybrid method in solving the integral equations of first kind, here we will illustrate some examples of solving non-fuzzy integral equations of the first kind using ybrid method; then we discuss hybridregularization method and Bernstein polynomials in solving non-fuzzy equations of the first kind and then the main examples that are fuzzy Abel integral equations will be solved using hybrid method.Then transforming second kind equation ( 23) to linear equations system, results in solving the second kind equation ( 23).
For more description, we divide interval [ ] , 0 1 to 10 parts.The unknowns of linear equations are going to be ( ) then after solving the equations the unknowns will be calculated.
In this section for approximating a definite integral, Simpson method was applied and Gaussian elimination method is employed to solve the linear equations.
By starting from 1 α = the linear equations derived from system ( 23) is being solved and then we need to decrease the value of α in order to achieve exact solution?
Convergence steps, diagrams of exact solutions and the diagram obtained from applying regularization method for , ., ., . 1 0 1 0 001 0 000001 α = are illustrated as follows: Due to this example solved by using least squares method in (Karimi & Jozi 2015) so the results obtained from regularization method compared with system (Karimi & Jozi 2015) in the following table.It can be seen that the results obtained from regularization method are better than two methods ( ) (Karimi & Jozi, 2015).
Example 5.2 We consider the non-fuzzy integral equation of the first kind with (Maleknejad, Hashemizadeh & Ezzati, 2011) ( ) sin Where the exact solution is ( ) cos sin f x x x = − 0 x 1 ≤ ≤ , if we use regularization method, the equation ( 26) transforms to following second kind equation: ( ) sin ( ) Generally, in fuzzy form, by converting fuzzy equations to non-fuzzy equations, two equations such as (25) will be obtained.Therefore, equation of the second kind (25) would be solved using Bernstein polynomials.As previous examples, at first we solve the equation ( 25) for 1 α = and with a small value of α, we find exact solution and achieve to the solution of ( 24) with an efficient accuracy.In above table, this error is 1.664326 × 10 -21.The solution derived from hybrid of regularization and Bernstein polynomials approximations seem to be much more accurate than applying the solid Bernstein polynomials.So in this example the solution is better and more accurate; and also the implementation time is lasts about 2 minutes and makes cause no problem in calculating.Now we consider the examples of Abel fuzzy integral equations.Operation of hybridal method will be compared with examples in (Jafarian & Measoomy, 2014).Jafarian et al, in (Jafarian & Measoomy, 2014) solved the Abel fuzzy integral equations by multilayer neural networks.
Example 5.3 consider the following fuzzy Abel integral equation (Jafarian & Measoomy, 2014) With the exact solution of Eq. ( 26) is: ( ) For comparing the results of hybrid method with Jafarian et al, method (Jafarian & Measoomy, 2014) we illustrate the solutions in following table:  Considering that in many points, the results from a hybrid method of regularization and Bernstein polynomials are better than the results of neural networks.By considering these figures we can say that in the regularization method and with a small value of α we are able to achieve to more accurate solution in first kind equation.If the error is a little bit, we cannot study the figures because the errors less than 0.001 are not seen in the figures.So, for getting accurate comparing between the obtained results, consider the following table which includes the results derived from two methods.We assume α in hybrid method As it can be seen, in some points hybrid method is satisfied and in the other points FNN method is preferred.Of course we can say that the two methods in example 5 own the same efficiency.

Conclusions and Recommendations
In this paper the fuzzy integral equations of the first kind was solved by a hybrid method including regularization method and Bernstein polynomials approximation.While similar methods for solving integral equations of the first kind, especially Abel equations, have weak accuracy, so the obtained results for all examples of the first kind are acceptable and comparable to similar methods.In hybrid method the regularization results in converting the first kind integral equation to second kind integral equation and so we need a method for solving the second kind integral equation.We can choose this method over many other methods because the numbers of methods that solve the integral equations of the second kind are more than the numbers of methods to solve the first kind integral equations.Bernstein polynomials method is the acceptable method for solving the second kind equation and makes a solution for first kind equation that can be compared with other available methods.
The proposed hybrid method not only can solve the first kind fuzzy integral equations but also can solve the first kind non-fuzzy equations with a better accuracy.Especially this hybrid method cause acceptable solution to Abel fuzzy integral equations.
To change the hybrid method and study the ability of changing, we can replace the method of solving second kind equations.For instance, for approximating, we can use Chebyshev, hermit and Orthogonal polynomials instead of Bernstein polynomials.
The important advantage of this method over the other similar methods is the less run-time.For example in using the artificial neural networks the learning process increase the run-time but in the hybrid method run-time is not a problem at all.( ) ( ) , ., .
i i s r 0 1i 0 1i = Definition 4.2 A desired fuzzy number in a parametric form of functions is stated as ( ) ( ), ( ) u r u r , 0 r 1 ≤ ≤ and three following conditions are included: Is on a non-descending function, left continuous and bounded [ ] Is on a non-ascending function, left continuous and bounded [ ]
Is a predetermined data function and ( ) u x is the unknown function which should be calculated.The expression ( ) x t α − − is calledthe kernel of Abel integral equation, or simply Abel kernel, that is singular as → .If ( ) f x is acrisp function (non-fuzzy) then the solutions to equation (16) are crisp too.But if ( ) f x is fuzzy function then the equation may have only fuzzy solutions.Parametric form of Abel integral equation defines by ( ) is a function with real and known values and u x r appears only in integral.

Example 5 . 1
Consider the Fredholm integral equation of the first kind(Karimi & Jozi 2015)

Figure 1 .
Figure 1.Plotting the convergence by two regularization methods for Figure 2. The error function graph for example 5.2 for

.Figure 3 .
Figure 3.The exact and approximated solutions for example 5.3 for

Figure 4 .
Figure 4.The exact and approximated solutions for example 5.4

Table 3 .
The absolute error of numerical results for example 5.3