Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

: The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.


Introduction
Fuzzy integral equations are one of the important branches of fuzzy analysis theory and they are applied as an adequate apparatus in mathematical modeling in biology, chemistry, physics, engineering, etc. (see, for example, [1][2][3][4]). One of the first applications of fuzzy integration was given by Wu and Ma [5], who investigated the fuzzy Fredholm integral equation of the second kind.
In connection with the application, it is very important to such kind of problems. If methods for solving such problems with uncertainty are developed, then, many real life models in different fields with imprecise variable can be solved easily and accurately. Some fixed point theorems for complete fuzzy metric space are given in [6][7][8]. In recent years, many mathematicians have studied a solution to fuzzy integral equations by numerical methods [9][10][11][12][13][14].
Liao employed the basic idea of the homotopy in topology to propose a general analytic method for nonlinear problems, namely HAM (see the monograph [15], and the papers [32] , [33], [34]). This method is based on the concept of creating function series. If the series converges, its sum is the solution of this system of equations. Later, HAM has been successfully applied to solve many types of nonlinear problems such as multiple solutions of nonlinear boundary value problems ( [16]), Abel fuzzy integral equations ( [17]), partial differential equations ( [18]), two-dimensional linear Volterra fuzzy integral equations ( [19]), and fuzzy linear Volterra integral equations of the second kind ( [20]). The paper presents an application of HAM for solving the following nonlinear Volterra-Fredholm fuzzy integral equations with two variables (2D-NVFFIE) u(s, t) =g(s, t) ⊕ (FR) t c k 1 (t, τ) G 1 (u(s, τ))dτ⊕  (1) is called the Fredholm equation with respect to the position and Volterra with respect to the time. This equation is used in many problems of mathematical physics, theory of elasticity, contact problems and mixed problems of mechanics of continuous media (see [2,21]). In [22], the Adomian decomposition method is applied for solving this equation.
The structure of this paper is organized as follows: In Section 2, some basic notations used in fuzzy calculus are introduced. In Section 3, we present HAM. In Section 4, we apply HAM for the parametric form of 2D-NVFFIE. In Section 5, we prove the convergence of the proposed method and we give an error estimate. In Section 6, a numerical example is illustrating the application of the presented above procedure for approximately solving of the studied equation.

Preliminaries
In this section, we review the fundamental notations of fuzzy set theory to be used throughout this paper. (iii) there are the real numbers a and b with c ≤ a ≤ b ≤ d, such that u is increasing on [c, a], decreasing on [b, d] and u(x) = 1 for each x ∈ [a, b]; (iv) u(rx + (1 − r)y) ≥ min{u(x), u(y)} for any x, y ∈ R, r ∈ [0, 1] .
By E 1 we denote the set of all fuzzy numbers. Any real number a ∈ R can be interpreted as a fuzzy numberã = χ(a) and therefore R ⊂ E 1 . Denote For u, v ∈ E 1 , k ∈ R the addition and the scalar multiplication are defined by [u According to [24], we can summarize the following algebraic properties: (ii) u ⊕0 =0 ⊕ u = u for any u ∈ E 1 ; (iii) with respect to0, none u ∈ E 1 \ R, u =0 has opposite in (E 1 , ⊕); (iv) for any a, b ∈ R with a, b ≥ 0 or a, b ≤ 0 and any u ∈ E 1 we have (a + b) u = a u ⊕ b u; (v) for any a ∈ R and any u, v ∈ E 1 we have a (u ⊕ v) = a u ⊕ a v; (vi) for any a, b ∈ R and any u ∈ E 1 we have a (b u) = (ab) u and 1 u = u.
As a distance between fuzzy numbers we use the Hausdorff metric.

Definition 2 ([24]
). For arbitrary fuzzy numbers u = (u r − , u r Lemma 1 ([24]). The following properties of the above distance hold: For any fuzzy-number-valued function f : I ⊂ R → E 1 we define the functions f (., r), f (., r) : These functions are called the left and right r−level functions of f . We will use the notion of Henstock integral for fuzzy-number-valued functions defined as follows:: The fuzzy number I is named the fuzzy-Henstock integral of f and will be denoted by (FH) b a f (t)dt.
When the function δ : [a, b] → R + is constant, then we obtain the Riemann integrability for fuzzy-number-valued functions ( [23]). In this case, I ∈ E 1 is called the fuzzy-Riemann integral of f on [a, b], being denoted by (FR) b a f (t)dt. Consequently, the fuzzy-Riemann integrability is a particular case of the fuzzy-Henstock integrability, and therefore the properties of the integral (FH) will be valid for the integral (FR), too.

Definition 4 ([26]).
A fuzzy-number-valued function f : A → E 1 is called: On the set C(A, We see that (C(A, E 1 ), D * ) is a complete metric space.

The Homotopy Analysis Method
We will give a brief overview of the main used method-HAM. Homotopy analysis method transforms the considered equation into the corresponding deformation equation. Using this method, we solve the operator equation where N is the operator, u is the unknown function and Ω is any domain of the variable z.
Taking the Maclaurin series of function Φ(z; p) with respect to the parameter p, we obtain where u 0 (z) = Φ(z; 0) and If the above series converges for p = 1, we obtain the required 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 (m ≥ 1): whereũ m−1 (z) = {u 0 (z), u 1 (z), ..., u m−1 (z)}, and and Apply L −1 to both sides of (7) and obtain If we are not able to determine the sum of the series in (6), then for the approximate solution of the considered equation we accept the partial sum of this series Choosing in an appropriate way the convergence control parameter h, we can influence the convergence region of the created series and the rate of this convergence ( [15,29]). One of the methods to select the value of this parameter is the so-called "optimization method" ( [15]). In this method, we define the squared residual of the governing equation where s n (z) and u i (z) are defined by (11) and (10) respectively and they depend on h.
The optimum value of the convergence control parameter is obtained by determining the minimum of this squared residual. The effective region of the convergence control parameter is additionally defined by Choosing a different value of the convergence control parameter 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 the convergence control parameter is called the basic optimal HAM ( [15]).

Applying HAM to 2D-NVFFIE (1)
In this section we introduce the parametric form of the integral Equation (1) and then we will apply HAM for solving this equation.
From (9) for the operator R m , m ≥ 1 we obtain By the definitions of the appropriate operators we obtain where

Convergence of the HAM
In this section we prove convergence of HAM and find an error estimate. We introduce the following conditions: (iii) The inequality holds.
We will use the following result: 22]). Let the conditions (i)-(iii) be fulfilled. Then the integral Equation (1) has an unique solution.
Then the sum of the series (20) is the unique solution of Equation (14).
Then the value of the convergence control parameter h can be selected such that the series (20) converges in A.
Proof. Let E = (C(A, R), . ) be the Banach space of all continuous functions on A and s n (s, t, r) = n ∑ i=0 u i (s, t, r) for (s, t) ∈ A, r ∈ [0, , 1].
We will prove that {s n } is a Cauchy sequence in E. We get where Let m ≥ 1 and n > m. Then we obtain Consequently, from conditions (ii) and (iii) we obtain where β h = |1 + h| + |h|α. Let n = m + 1 then Using the triangle inequality and (22) we have s n − s m ≤ s m+1 − s m + s m+2 − s m+1 + · · · + s n − s n−1 We choose the value of the parameter h ∈ (−2, 0) such that Then β h = |1 + h| + |h|α ∈ (0, 1) and 1 − β h n−m < 1. Therefore, s n − s m < β h m 1−β h u 1 Since u 1 < ∞ then s n − s m → 0 as m → ∞,and the sequence {s n } is a Cauchy sequence in E. Therefore, the series Similarly, we have {s n } is a Cauchy sequence . Then the error of the approximate solution can be estimated as follows where β h = |1 + h| + |h|α and α is defined by (19).

Numerical Example
In this section, we will illustrate the obtained theoretical results on a numerical example. s 2 t 4 (3 − r))(3 − r)).
Then the exact solution of (1) in this partial case is given by u exact (s, t, r) = (st(1 + r), st(3 − r)).
These equalities lead to more simple calculations in the case of R comparatively with the "fuzy" case E 1 .

Conclusions
In this paper, HAM is applied for solving the two-dimensional nonlinear Volterra-Fredholm fuzzy integral equations, where the solution is found in the form of a series. It is shown that if this series is convergent, its sum is the solution of the considered equation. Sufficient conditions for the convergence of this series are given. Additionally, the error of the approximate solution, taken as the partial sum of generated series, is estimated. The presented example shows that the investigated method is effective in solving the equations of considered kind.