Numerical and analytic method for solvingproposal New Type for fuzzy nonlinear volterra integral equation

In this paper, we proved the existence and uniqueness and convergence of the solution of new type for nonlinear fuzzy volterra integral equation . The homotopy analysis method are proposed to solve the new type fuzzy nonlinear Volterra integral equation . We convert a fuzzy volterra integral equation for new type of kernel for integral equation, to a system of crisp function nonlinear volterra integral equation . We use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy volterra integral equation . Some numerical examples is given and results reveal that homotopy analysis method is very effective and compared with the exact solution and calculate the absolute error between the exact and AHM .Finally using the MAPLE program to solve our problem .


Introduction
The solutions of integral equations have a major role in the field of science and engineering. Since few of these equations can be solved explicitly, it is often necessary to resort to numerical techniques which are appropriate combinations of numerical integration and interpolation [7,12]. There are several numerical methods for solving linear Volterra integral equation [11,23] and system of nonlinear Volterra integral equations [3]. Borzabadi and Fard in [5] obtained a numerical solution of nonlinear Fredholm integral equations of the second kind.The concept of fuzzy numbers and fuzzy arithmetic operations were first introduced by Zadeh [14],Dubois and Prade [21]. We refer the reader to [10] for more information on fuzzy numbers and fuzzy arithmetic.The topics of fuzzy integral [20] and fuzzy integral equations (FIE) which growing interest for some time, inparticular in relation to fuzzy control, have been rapidly developed in recent years. The fuzzy mapping functionwas introduced by Chang and Zadeh [14]. Later, Dubois and Prade [8] presented an elementary fuzzy calculusbased on the exten-sion principle also the concept of integration of fuzzy functions was first introduced byDubois and Prade [21]. Babolian et al. and Abbasbandy et al. in [10,11] obtained a numerical solution of linearFredholm fuzzy integral equations of the second kind, while finding an approximate solution for the fuzzynonlinear kinds.is more difficult and a numerical method in this case can be found in [4]In this paper, we present a novel and very simply numerical method ( Homotopy Analysis method ) for solving fuzzy nonlinear volterra integral equation . .2.Basic concepts Basic definitions of fuzzy number are given in [1,2,10,15,17,20] as follows: Definition 2.1. Fuzzy number. A fuzzy number is a map : R → a, b , which satisfying (1) u is upper semi-continuous function, (2) u(x) = 0 outside some interval a, d The set of all fuzzy numbers (as given by Definition 13 ) is denoted by E 1 and is a convex cone. An alternative definition for parameter from of a fuzzy number is given by Kaleva [14 ].
In the distance between The u and v , it is prove (E 1 , D) is a complete metric space .
Definition 2.5. The integral of a fuzzy function was define in [14] by using the Riemann integral concept . Let f: a, b E 1 . For Fuzzy function, for each partition p={t 0, …, t n } of a, b and for arbitrary ξ i ∈ t i−1 , t i , 1 ≤ i ≤ n, suppose If the fuzzy function f t is continuous in metric D,its definite the integral exists and also It should be noted that the fuzzy integral can be also defined using the Lebesguetype approach. However, if f t is continuous, both approaches yield the same value .More details about the properties of the fuzzy integral Proposition 2.1. A function F, G: I → E n be integrable and φ ∈ R. ∈ then Proposition 2.2. For any p, q, r, s ∈ E n and φ ∈ R, then the following hold iii-D p + r, q + s = D(p, q)

3.Novel formula fuzzy nonlinear volterra integral equation
The fuzzy nonlinear integral equation with integral kernel which is discussed in this work is the fuzzy nonlinear Volterra integral equation of the second kind (FNVIE-2) as follows: x a u(s, α))ds))dt (7) For each 0≤α≤1anda≤x≤b. WecanseethatEq. (6)converttoasystem of nonlinear Volterra integral equations in crisp case for each 0 ≤ α ≤ 1 and a ≤ t ≤ b. Now, we explain homotopy analysis methods as approximating solution of this system of nonlinear integral equations in crisp case. then, we find approximate solutions for u (x), a ≤ x ≤ b 0 ≤ c ≤ x

Homotopy analytic method "HAM"
Now we apply homotopy analytic method for solve the system (7) and obtain a recursion scheme for it. Prior to apply HAM for the system (7). We suppose that the kernel have four cases for kernal's . We see that eq(15) is convert to system of nonlinear crisp fuzzy volterra integral equations For solving system (9) by HAM, we construct the zeroth-order deformation equations: Where ∈ [ , ] is the embedding parameter, his nonzero auxiliary parameter, is an auxiliary linear operator ( ) is an auxiliary function , It should be noted that ∅ , ; = 0 ; ∅ , ; = 0 ; differential the zerothorder deformation equation (10) times with respect to the embedding parameter and dividing them by ! and ! finally setting = 0 ,we obtain the so called − ℎ order deformation equations From Eqs (14) and (15), we have that 0 where,  The following conditions are satisfy We take an initial guess 0 = ( ), an auxiliary operator = , an nonzero auxiliary parameter h=-1 and auxiliary function H(x)=1, this is substituted in Eq (19) to give the recurrence relation .
Then there exist a unique fuzzy solution of u(x) in Eq (19) and the successive iteration

numerical example
In this prrt we will discuss case of new formula by using homotopy analysis method.
And comparing the approximate method with he known exact solution and calculate the absolute error between the exact and approximate. Also give some finger for all cases.

Now we discus this example
We will discuss Now the case of formula 2 , And , , 1 ,