Solving two - dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

: The purpose of the paper is to ﬁ nd an approximate solution of the two - dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method ( HAM ) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we ﬁ nd approx - imate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

and then we apply HAM to it. In Section 5, we obtain sufficient conditions for convergence of the proposed method and error estimate. In Section 6, we give an example. In Section 7, the conclusion is drawn.

Basic concepts
First, we present some notions and results about fuzzy numbers and fuzzy-number-valued functions, and fuzzy integrals. In [20], the convexity of the fuzzy numbers ( + ( − ) ) ≥ { ( ) ( )} u rx r y u x u y 1 m i n , , for any ∈ x y R , , ∈ [ ] r 0, 1 is shown.
We denote with E 1 the set of all fuzzy numbers and = ( ∞) are functions, such that u is increasing and u is decreasing.
Let ∈ u v E , 1 , ∈ k R. The addition and the scalar multiplication are defined by With respect to ⊕ in E 1 the neutral element is denoted by = { } χ 0 0 . In [21], the algebraic properties of fuzzy numbers are given. We use the Hausdorff metric to define a distance between fuzzy numbers. For any fuzzy-number-valued function , , 1 we define the left and right r-level functions, respectively, ( ) ( ) → f r f r A R .,., , .,., : for each ( ) ∈ s t A , and ∈ [ ] r 0, 1 .

Definition 3. [22] A fuzzy-number-valued function
, then we say that f is continuous on A.
Define the metric ( ) = ( ( ) ( )) * max , , , on the set ; is continuous . 1 1 We see that ( ( ) ) * C A E D , , 1 is a complete metric space. In [23], the notion of Henstock integral for fuzzy-number-valued functions is defined as follows. such that for any δ-fine and σ-fine divisions we have where ∑ denotes the fuzzy summation.
Then, I is called the two-dimensional Henstock integral of f and is denoted by If the above δ and σ are constant functions, then one recaptures the concept of Riemann integral. In this case, ∈ I E 1 will be called two-dimensional integral of f on A and will be denoted by  Also, if f is continuous, then ( ) f r .,., and ( ) f r .,., are continuous for any ∈ [ ] r 0, 1 , and consequently, they are Henstock integrable.

Basic idea of HAM
According to [6], we will give a brief overview of the main used method. HAM transforms the considered equation into the corresponding deformation equation. Using this method we solve the operator equation where N is the nonlinear operator, u is the unknown function and Ω is the domain of z. Define the homotopy operator in the following way: where ∈ [ ] p 0, 1 is an embedding parameter, ≠ h 0 is the convergence control parameter [24,25] and u 0 is the initial approximation of the solution of equation (1). The linear operator L is the auxiliary with property ( ) = L 0 0. This operator can be arbitrarily selected. The practice is to choose L so that the equations, obtained in the next stages of the procedure, would be as simple to solve as possible.
From the equation ( ( )) = z p Φ ; 0, we get the so-called zero-order deformation equation: If series (4) converges for = p 1, then we get the sought solution In order to determine the function u m we differentiate m-times, with respect to parameter p, the left and right-hand side of formula (3), then the obtained result is divided by ! m and substituted with = p 0, which gives the so-called mth-order deformation equation (7), we get and By appropriate selection of the convergence control parameter h, we can influence the convergence region of series and the rate of this convergence [15,26]. One of the methods for selecting the value of convergence control parameter is the so-called h-curve. To obtain this curve we need to investigate the behavior of a certain quantity of the exact solution as a function of parameter h [27]. Another method is the so-called "optimization method" proposed in the study by Liao [15]. In this method, we define the squared residual of the governing equation: The optimum value of the convergence control parameter is obtained by finding the minimum of this squared residual, whereas the effective region of the convergence control parameter h is defined as To speed up the calculations, Liao [15] suggested to replace the integral in formula 11 by its approximate value obtained by applying the quadrature rules. By choosing a different value of the control parameter h than the optimal one, but still belonging to the effective region, we also obtain the convergent series, only the rate of convergence is lower. A version of the method with the above described selection of optimal value of the convergence control parameter is called the basic optimal HAM [15].

Using of HAM
We consider 2D-NFVIE of the following form: , , 1 are continuous fuzzy-number valued functions, is continuous function and → G E E : 1 1 is continuous function on E 1 .
According to [2], we introduce the parametric form of the integral equation (13).
be the parametric form of the functions ( ) u s t , and ( ) g s t , . So, the parametric form of equation (13) is as follows: , we have H u x y r u x y r G β u x y r β u x y r F u x y r u x y r G β u x y r β u x y r , , , , Then,

s t x y G u x y r k s t x y F u x y r u x y r k s t x y k s t x y H u x y r u x y r k s t x y k s t x y G u x y r k s t x y H u x y r u x y r k s t x y k s t x y F u x y r u x y r k s t x y
Then the parametric form of equation (13) is as follows: where χ m is defined by (8).
For the operator R m , ≥ m 1 from (9) we obtain

Existence and convergence
In this section, we prove convergence of the presented method and obtain the error estimation between the exact and the approximate solution.

Lemma 3. [29]
Let the functions ∈ ( × ) We introduce the following conditions: , according to the continuity of k and Theorem 1. Let conditions (i)-(iii) be fulfilled. Then the integral equation (13) has a unique solution. and > ε 0. Since u is continuous it follows that for > 0 .,. : . Applying Lemma 3, it follows that the function → F A E : is continuous on A for any ∈ u X. Since ∈ g X, we conclude that the operator ( ) T u is continuous on A for any ∈ u X. Now, we prove that → T X X : is a contraction. Let arbitrary ∈ u v X , . From conditions (ii) and (iii) we have  Under the condition < α 1 the operator T is contraction; therefore, by the Banach fixed-point theorem for contraction, there exists a unique solution to problem (13) and this completes the proof. is convergent, and the sum of this series is the solution of equation (14).

D A u s t A v s t D FR FR k s t x y G u x y x y FR FR k s t x y G v x y x y k s t x y D G u x y G v x y x y M L D u x y v x y x y M L D u v s t A
Proof. Let the series in (19) be convergent . Then from the necessary condition for convergence of the series, it follows     , ∈ [ ] r 0, 1 . We will prove that the sequence { } S n is a Cauchy sequence in B. Then from condition (iii) we have    Hence, from conditions (ii) and (iii) we obtain  Using the triangle inequality we have, One can easily note that It implies that if condition < α 1 is fulfilled, then we are able to choose the value of parameter h such that inequality (20) will be satisfied (for this aim it is enough to take any ∈ [− ) h 1, 0 ), which means that

Numerical results
In this section, we give a numerical example to illustrate the obtained theoretical results. . The results are expressed in Table 1.

Conclusion
In this paper, HAM is used for solving the nonlinear fuzzy Volterra integral equations in two variables of the second kind. The solution of the discussed equation is obtained in the form of a series, the elements of which are iteratively determined. It is shown that if this series is convergent, its sum is the solution of the considered equation. The sufficient conditions for convergence of this series are received and then its sum is solution of the considered equation. The error of approximate solution, taken as the partial sum of generated series, is estimated. Example illustrating the use of the investigated method are presented as well.
Funding information: The research was supported by the Bulgarian National Science Fund under Project KP-06-N32/7.

Conflict of interest:
The author states no conflict of interest.