American Journal of Applied Mathematics
Volume 4, Issue 3, June 2016, Pages: 142-157

Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method

Sameer Qasim Hasan, Alan Jalal Abdulqader

College of Education, Department of Mathematics, Al-Mustansiriya University, Baghdad, Iraq

(S. Q. Hasan)

Sameer Qasim Hasan, Alan Jalal Abdulqader. Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method. American Journal of Applied Mathematics. Vol. 4, No. 3, 2016, pp. 142-157. doi: 10.11648/j.ajam.20160403.15

Received: May 4, 2016; Accepted: May 14, 2016; Published: May 28, 2016

Abstract: In this paper, we presented the convergence of the solution for the nonlinear fuzzy volterra integral equation with high computational and complexity to find the solution in analytical method, so we describable this solution by using Homotopy analysis method, by using Banach fixed point theory for existence and uniqueness. That with explained numerical examples. Finally using the MAPLE program to solve our problem.

Keywords: Fuzzy Number, Volterra nonlinear Integral Equation, Operator of Fuzzy Number, Fuzzy Integral, Homotopy Analysis Method

Contents

1. Introduction

The fuzzy integral equations are one of the important part of the fuzzy analysis theory that play major role in numerical analysis. The concept of fuzzy numbers and arithmetic operations on it was introduced by Zadeh [1] which was further enriched by Mizumoto and Tanaka [24]. Dubois and Prade [21] made a significant contribution by introducing the concept of LR fuzzy numbers and presented a computational formula for operations on fuzzy numbers. Also they [4, 7, 8] was introduced the concept of integration of fuzzy functions. Later, Goetschel and Voxman [5] preferred a Riemann integral type approach, Kaleva [25] chose to define the integral of fuzzy function, using the Lebesgue-type concept for integration. One of the first applications of fuzzy integration was given by Wu and Ma who investigated the fuzzy Fredholm integral equation of the second kind. Recently, some mathematician have studied solution of fuzzy integral equation by numerical method [1, 9, 13]. In present, we try to employ homotopy analysis method for solving new formula fuzzy nonlinear Volterra integral equation. Furthermore, also we study the existence of a unique solution and convergence of the HAM methods for new formula 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 , which satisfying

(1) u is upper semi- continuous function,

(2) u(x) = 0 outside some interval

(3) There are real numbers b, c such

i) u(x) is a monotonic increasing function on

ii) u(x) is a monotonic decreasing function on

iii) u(x) = 1 for all

The set of all fuzzy numbers (as given by Definition (2.1)) is denoted by  and is a convex cone. An alternative definition for parameter from of a fuzzy number is given by Kaleva [14].

Definition 2.2. A fuzzy number  in parametric form is a pair (of function , , which satisfies the following requirement:

i)  is a bounded left continuous non- decreasing function over [0, 1]

ii)  is a bounded left continuous non- increasing function over [0, 1]

iii) ,

Definition 2.3. For arbitrary fuzzy ),,),  and scalar , we define addition, subtraction, scalar product by  and multiplication are respectively as following:

1- (= ()+ ()), () = (()+()),

2- (= ()- ()), () = (()-()),

3-

(1)

4- multiplication:

(2)

Defined 2.4. For arbitrary Fuzzy numbers

(3)

In the distance between The , it is prove ( 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 ® . For Fuzzy function, for each partition ={t0, …, tn} of  and for arbitrary ,  suppose

(4)

.

The define integral of  over  is

(5)

If the fuzzy function  is continuous in metric its definite the integral exists and also

(6)

It should be noted that the fuzzy integral can be also defined using the Lebesgue – type approach. However, if  is continuous, both approaches yield the same value.More details about the properties of the fuzzy integral

3. Proposal Formula for 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:

(7)

where  ≥ 0,is a fuzzy function of x: a ≤ x ≤ b,  is analytic functions ,  are nonlinear function on [a, b]. For solving in parametric form of Eq. (7), consider () and (), 0 ≤  ≤ 1 and t, s [a, b] are parametric form of and , respectively. then, parametric form of Eq. (7) is as follows

(8)

let

(9)

For each 0≤≤1 and a≤x≤b. We can see that Eq. (7) convert to a system of nonlinear Volterra integral equations in crisp case for each 0 ≤ ≤ 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 (x), a ≤ x ≤ b

3.1. Generalize Homotopy Analytic Method "HAM"

Now we apply homotopy analytic method for solve the system (8) and obtain a recursion scheme for it. Prior to apply HAM for the system (8). We suppose that the kernel have four cases for kernal’s.

Where

We see that eq(7) is convert to system of nonlinear crisp fuzzy volterra integral equations

(9)

For solving system (9) by HAM, we construct the zeroth-order deformation equations:

(10)

Where  is the embedding parameter, his nonzero auxiliary parameter,  is an auxiliary linear operator  is an auxiliary function,  are initial guesses of  respectively and are unknown functions. Using the above zeroth – order deformation equations, with assumption

, we have

(11)

Obviously, when  and since , we obtain

And

(12)

Respectively. Thus as  increases from , the function

deforms from the initial guesses  to the solution of

Expanding  in Taylors series with respect  gives

(13)

Where

It should be noted that   differential the zeroth – order deformation equation (11) times with respect to the embedding parameter  and dividing them by  and  finally setting ,we obtain the so called  order deformation equations

(14)

Where

and

(15)

From Eqs (14) and (15), we have that

Similary

(16)

where,

(17)

Thus, if we choose a proper values of , the series (9) is convergent at . Hence the solution of system (10) in series from (homotopy solution series) is obtain as

(18)

We denoted the  order approximate solution with

(19)

3.2. Existence and Convergence Analysis

Consider  is a bounded

We also suppose the nonlinear operators  are satisfied in Lipschitz condition

Let

Now we will prove the existence and uniqueness and convergence of the solution, of the method by using Homotopy analysis method.

Theorem 3.2.1:

Let , then equation (7) has a unique solution.

Proof: Let  be two different solutions of equation (7) then

|

Then we let

We get

let

From which we get . Since , then  Implies

Theorem 3.2.2. The series solution  of equation (7) using HAM convergence when

Proof: The series of partition sum let  be arbitrary partial sums with . We are going to prove that  is a Cauchy sequence in this banach space

by using the theorem 3.2.1 we get

Let n=m+1, then

We have,

Since , we have  then

But |as then . We conclude that  is Cauchy sequence

Similarly, we have  is a Cauchy sequence, then we have

Finally we will have

Theorem 3.2.3: fuzzy nonlinear Volterra integral equation is convergent to the exact solution when using Homotopy analysis method.

Proof: for

So

Then

3.3. Numerical Example

In this part we will discuss all of cases for our 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

Example 1:

Now we discuss this example by using HAM

where

the intimal condition is

the exact solution

Now we will use the homotopy analysis method to solve the example 1 we can see the first term of this method

where

(1+h)*(-(1+h) h (x r-1/32 x^9 r^5)+h^6 (-1/45097156608 r^25 x^49+31/5872025600 r^21 x^41-41/73400320 r^17 x^33+3/81920 r^13 x^25-3/2048 r^9 x^17+1/32 x^9 r^5))-h*(1)/200 (1/2033753534126478065664 (-1/203375353412647806566400 (1+h) h^13 r^51-31/13240582904469258240000 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^12 r^46-1540541/197726038040074256384000000 h^18 r^51+31/587202560000 h^6 r^21 (-1/22548578304 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^25-339/6012954214400 h^12 r^30)-1/4509715660800 h^6 r^25 (1/22548578304 h^7 r^26 (1+h)+31/2936012800 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^21+1119/375809638400 h^12 r^26)) h^12 r^50-31/132405829044692582400 (1/219645381685659631091712 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^12 r^50+1661509/17083529686662415751577600000 h^18 r^55-1/4870492913664 h^6 r^25 (-1/22548578304 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^25-339/6012954214400 h^12 r^30)) h^12 r^46-15701517640283/165692679056158781069638940217915666371444736000000000 h^30 r^101+31/557305131435787132506149486592000 h^18 r^71 (-1/22548578304 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^25-339/6012954214400 h^12 r^30)-1/13574042241325137563395494695141376 h^18 r^75 (1/22548578304 h^7 r^26 (1+h)+31/2936012800 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^21+1119/375809638400 h^12 r^26))

-h*(x*r-(1/32)*x^9*r^5)-(1+h)*h*(x*r-(1/32)*x^9*r^5)-h*(-h^5*(1/45097156608)*r^25*x^49+(31/5872025600)*r^21*x^41-(41/73400320)*r^17*x^33+(3/81920)*r^13*x^25-(3/2048)*r^9*x^17+(1/32)*x^9*r^5))+(1+h)*(-(1+h) h (x r-1/32 x^9 r^5)+h^6 (-1/45097156608 r^25 x^49+31/5872025600 r^21 x^41-41/73400320 r^17 x^33+3/81920 r^13 x^25-3/2048 r^9 x^17+1/32 x^9 r^5))-h*/200 (1/2033753534126478065664 (-1/203375353412647806566400 (1+h) h^13 r^51-31/13240582904469258240000 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^12 r^46-1540541/197726038040074256384000000 h^18 r^51+31/587202560000 h^6 r^21 (-1/22548578304 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^25-339/6012954214400 h^12 r^30)-1/4509715660800 h^6 r^25 (1/22548578304 h^7 r^26 (1+h)+31/2936012800 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^21+1119/375809638400 h^12 r^26)) h^12 r^50-31/132405829044692582400 (1/219645381685659631091712 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^12 r^50+1661509/17083529686662415751577600000 h^18 r^55-1/4870492913664 h^6 r^25 (-1/22548578304 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^25-339/6012954214400 h^12 r^30)) h^12 r^46-15701517640283/165692679056158781069638940217915666371444736000000000 h^30 r^101+31/557305131435787132506149486592000 h^18 r^71 (-1/22548578304 (1/32 (1+h) h r^5+1/32 h^6 r^5) h^6 r^25-339/6012954214400 h^12 r^30)-1/13574042241325137563395494695141376 h^18 r^75)(75)

-(1+h)*h*(x*(2-r)-(1/32)*x^9*(2-r)^5)-h*(-h^5*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*(2-r)^21*x^41-(41/73400320)*(2-r)^17*x^33+(3/81920)*(2-r)^13*x^25-(3/2048)*(2-r)^9*x^17+(1/32)*x^9*(2-r)^5))

where

=-h*(x*(2-r)-(1/32)*x^9*(2-r)^5)-(1+h)*h*(x*(2-r)-(1/32)*x^9*(2-r)^5)+h^6*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*(2-r)^21*x^41-(41/73400320)*(2-r)^17*x^33+(3/81920)*(2-r)^13*x^25-(3/2048)*(2-r)^9*x^17+(1/32)*x^9*(2-r)^5)+(1+h)*(-(1+h)*h*(x*(2-r)-(1/32)*x^9*(2-r)^5)+h^6*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*(2-r)^21*x^41-(41/73400320)*(2-r)^17*x^33+(3/81920)*(2-r)^13*x^25-(3/2048)*(2-r)^9*x^17+(1/32)*x^9*(2r)^5h*x*((1/208*((1/2033753534126478065664*((1/219645381685659631091712*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^12*(2-r)^50+(1661509/17083529686662415751577600000)*h^18*(2-r)^55-(1/4870492913664)*h^6*(2-r)^25*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2r)^30)))*h^12*(2r)^50+(178007930970977/278363700814346752196993419566098319504027156480000000000)*h^30*(2-r)^105-(1/13574042241325137563395494695141376)*h^18*(2-r)^75*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2-r)^30)))*x^209+(1/200*((1/2033753534126478065664*(-(1/203375353412647806566400*(1+h))*h^13*(2-r)^51-(31/13240582904469258240000*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^12*(2-r)^46-(1540541/197726038040074256384000000)*h^18*(2-r)^51+(31/587202560000)*h^6*(2-r)^21*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2-r)^30)-(1/4509715660800)*h^6*(2-r)^25*((1/22548578304)*h^7*(2-r)^26*(1+h)+(31/2936012800*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^21+(1119/375809638400)*h^12*(2-r)^26)))*h^12*(2-r)^50-(31/132405829044692582400*((1/219645381685659631091712*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^12*(2-r)^50+(1661509/17083529686662415751577600000)*h^18*(2-r)^55-(1/4870492913664)*h^6*(2-r)^25*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2-r)^30)))*h^12*(2-r)^46-(15701517640283/165692679056158781069638940217915666371444736000000000)*h^30*(2-r)^101+(31/557305131435787132506149486592000)*h^18*(2-r)^71*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2-r)^30

-h(x*(2-r)-(1/32)*x^9*(2-r)^5)-(1+h)*h*(x*(2-r)-(1/32)*x^9*(2-r)^5)-h*(-h^5*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*(2-r)^21*x^41-(41/73400320)*(2-r)^17*x^33+(3/81920)*(2-r)^13*x^25-(3/2048)*(2-r)^9*x^17+(1/32)*x^9*(2-r)^5))-h*(x*(2-r)-(1/32)*x^9*(2-r)^5)-(1+h)*h*(x*(2-r)-(1/32)*x^9*(2-r)^5)+h^6*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*(2-r)^21*x^41-(41/73400320)*(2-r)^17*x^33+(3/81920)*(2-r)^13*x^25-(3/2048)*(2-r)^9*x^17+(1/32)*x^9*(2-r)^5)+(1+h)*(-(1+h)*h*(x*(2-r)-(1/32)*x^9*(2-r)^5)+h^6*(-(1/45097156608)*(2 r)^25*x^49+(31/5872025600)*(2-r)^21*x^41-(41/73400320)*(2-r)^17*x^33+(3/81920)*(2-r)^13*x^25-(3/2048)*(2-r)^9*x^17+(1/32)*x^9*(2r)^5))h*x*((1/208*((1/2033753534126478065664*((1/219645381685659631091712*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^12*(2-r)^50+(1661509/17083529686662415751577600000)*h^18*(2-r)^55-(1/4870492913664)*h^6*(2-r)^25*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2r)^30)))*h^12*(2r)^50+(178007930970977/278363700814346752196993419566098319504027156480000000000)*h^30*(2-r)^105-(1/13574042241325137563395494695141376)*h^18*(2-r)^75*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2-r)^30)))*x^209+(1/200*((1/2033753534126478065664*(-(1/203375353412647806566400*(1+h))*h^13*(2-r)^51-(31/13240582904469258240000*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^12*(2-r)^46-(1540541/197726038040074256384000000)*h^18*(2-r)^51+(31/587202560000)*h^6*(2-r)^21*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2-r)^30)-(1/4509715660800)*h^6*(2-r)^25*((1/22548578304)*h^7*(2-r)^26*(1+h)+(31/2936012800*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^21+(1119/375809638400)*h^12*(2-r)^26)))*h^12*(2-r)^50-(31/132405829044692582400*((1/219645381685659631091712*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^12*(2-r)^50+(1661509/17083529686662415751577600000)*h^18*(2-r)^55-(1/4870492913664)*h^6*(2-r)^25*(-(1/22548578304*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^25-(339/6012954214400)*h^12*(2-r)^30)))*h^12*(2-r)^46

Figure 1a. The exact solution .

Figure 1b. The approximate solution .

Figure 2a. The exact solution .

Figure 2b. The approximate solution .

as long as a series solution given by the HAM converges, it must be the exact solution. So it is important to ensure that the homotopy solution series converges. contains the auxiliary parameter h , which provides us a simple way to adjust and control the convergence of the solution series. In general, by means of the so-called h -curve (a curve of a versus h ), it is straightforward to choose an appropriate range for h which ensures the convergence of the solution series. As pointed by Liao [13], the valid region of h is a horizontal line segment. In Fig. (1a), Fig(1b) and Fig(2a), Fig(2b) we plot the h-curves, we could find that if h is about in area [-1,-0.7] the result is convergent. In table (1) we introduce the error between the exact and approximate solutions of HAM for different values of h

Table 1. Computation between the exact and HAM in case 1 and determine the absolute error.

Example 2:

Now we discuss this example by using HAM

Where

with the exact solution

the intimal condition is

By using HAM to solve the system above

-(1+h)*h*(x*r-(1/32)*x^9*(2-r)^5)-h*(-h^5*(-(1/45097156608)*r^25*x^49+(31/5872025600*(2-r))*r^20*x^41-(41/73400320)*(2-r)^2*r^15*x^33+(3/81920)*(2-r)^3*r^10*x^25-(3/2048)*(2-r)^4*r^5*x^17+(1/32)*x^9*(2-r)^5))

(1+h)*(-(1+h)*h*(x*r-(1/32)*x^9*(2-r)^5)+h^6*(-(1/45097156608)*r^25*x^49+(31/5872025600*(2-r))*r^20*x^41-(41/73400320)*(2-r)^2*r^15*x^33+(3/81920)*(2-r)^3*r^10*x^25-(3/2048)*(2-r)^4*r^5*x^17+(1/32)*x^9*(2-r)^5))-h*x*((1/40*((1/4)*(1+h)^3*h^3*(2r)^3*((41/36700160*(1+h))*h^7*(2r)^16*r^2+(3/40960*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^3*(2r)^10+(9/4194304)*h^12*r^8*(2r)^10)+(1/4)*(1+h)^2*h^2*(2r)^2*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)*(-(3/40960*(1+h))*h^7*(2-r)^11*r^3-(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^4*(2-r)^5)+((1/20*(1+h))*h*(2-r)*((3/1024*(1+h))*h^7*(2r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)(1/10)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2*(1+h)*h*(2r)(3/40960)*h^8*r^4*(2r)^7*(1+h)^2)*((3/1024*(1+h))*h^7*(2r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)-(2*((1/28*(1+h))*h*(2r)*((3/40960*(1+h))*h^7*(2r)^11*r^3(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^4*(2-r)^5)+(1/28*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*((3/1024*(1+h))*h^7*(2r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)+(3/28672)*h^7*r^4*(2r)^6*(1+h)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)+(3/2293760)*h^8*r^3*(2r)^12*(1+h)^2))*(1+h)*h*(2r)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)+((1/36*(1+h))*h*(2r)*((41/36700160*(1+h))*h^7*(2r)^16*r^2+(3/40960*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^3*(2r)^10+(9/4194304)*h^12*r^8*(2r)^10)+(1/36*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*(-(3/40960*(1+h))*h^7*(2-r)^11*r^3-(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^4*(2-r)^5)-(1/24576)*h^6*r^4*(2-r)^5*((3/1024*(1+h))*h^7*(2-r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)-(1/491520)*h^7*r^3*(2-r)^11*(1+h)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5

-h*(x*r-(1/32)*x^9*(2-r)^5)-(1+h)*h*(x*r-(1/32)*x^9*(2-r)^5)-h*(-h^5*(-(1/45097156608)*r^25*x^49+(31/5872025600*(2-r))*r^20*x^41-(41/73400320)*(2-r)^2*r^15*x^33+(3/81920)*(2-r)^3*r^10*x^25-(3/2048)*(2-r)^4*r^5*x^17+(1/32)*x^9*(2-r)^5))+(1+h)*(-(1+h)*h*(x*r-(1/32)*x^9*(2-r)^5)+h^6*(-(1/45097156608)*r^25*x^49+(31/5872025600*(2-r))*r^20*x^41-(41/73400320)*(2-r)^2*r^15*x^33+(3/81920)*(2-r)^3*r^10*x^25-(3/2048)*(2-r)^4*r^5*x^17+(1/32)*x^9*(2-r)^5))-h*x*((1/40*(-(1/4)*(1+h)^3*h^3*(2-r)^3*((41/36700160*(1+h))*h^7*(2-r)^16*r^2+(3/40960*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^3*(2-r)^10+(9/4194304)*h^12*r^8*(2-r)^10)+(1/4)*(1+h)^2*h^2*(2-r)^2*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)*(-(3/40960*(1+h))*h^7*(2-r)^11*r^3-(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^4*(2-r)^5)+(-(1/20*(1+h))*h*(2-r)*((3/1024*(1+h))*h^7*(2-r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)-(1/10)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2*(1+h)*h*(2-r)-(3/40960)*h^8*r^4*(2-r)^7*(1+h)^2)*((3/1024*(1+h))*h^7*(2-r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)-(2*(-(1/28*(1+h))*h*(2-r)*(-(3/40960*(1+h))*h^7*(2-r)^11*r^3-(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^4*(2-r)^5)+(1/28*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*((3/1024*(1+h))*h^7*(2-r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)+(3/28672)*h^7*r^4*(2-r)^6*(1+h)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)+(3/2293760)*h^8*r^3*(2-r)^12*(1+h)^2))*(1+h)*h*(2-r)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)+(-(1/36*(1+h))*h*(2-r)*((41/36700160*(1+h))*h^7*(2-r)^16*r^2+(3/40960*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^3*(2-r)^10+(9/4194304)*h^12*r^8*(2-r)^10)+(1/36*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*(-(3/40960*(1+h))*h^7*(2-r)^11*r^3-(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^4*(2-r)^5)-(1/24576)*h^6*r^4*(2-r)^5*((3/1024*(1+h))*h^7*(2-r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)-(1/491520)*h^7*r^3*(2-r)^11*(1+h)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)-(41/2642411520)*h^8*r^2*(2-r)^17*(1+h)^2)*(1+h)^2*h^2*(2-r)^2))*x^40+(1/32*(-(1/4)*(1+h)^3*h^3*(2-r)^3*(-(3/40960*(1+h))*h^7*(2-r)^11*r^3-(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5))*h^6*r^4*(2-r)^5)+(1/4)*(1+h)^2*h^2*(2-r)^2*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)*((3/1024*(1+h))*h^7*(2-r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)-(2*(-(1/20*(1+h))*h*(2-r)*((3/1024*(1+h))*h^7*(2-r)^6*r^4+((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2)-(1/10)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)^2*(1+h)*h*(2-r)-(3/40960)*h^8*r^4*(2-r)^7*(1+h)^2))*(1+h)*h*(2-r)*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5)+(-(1/28*(1+h))*h*(2-r)*(-(3/40960*(1+h))*h^7*(2-r)^11*r^3-(3/1024*((1/32*(1+h))*h*r^5+(1/32)*h^6*r^5

=-(1+h)*h*(x*(2-r)-(1/32)*x^9*r^5)-h*(-h^5*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*r*(2-r)^20*x^41-(41/73400320)*r^2*(2-r)^15*x^33+(3/81920)*r^3*(2-r)^10*x^25-(3/2048)*r^4*(2-r)^5*x^17+(1/32)*x^9*r^5))

= (1+h)*(-(1+h)*h*(x*(2-r)-(1/32)*x^9*r^5)+h^6*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*r*(2-r)^20*x^41-(41/73400320)*r^2*(2-r)^15*x^33+(3/81920)*r^3*(2-r)^10*x^25-(3/2048)*r^4*(2-r)^5*x^17+(1/32)*x^9*r^5))-h*x*((1/136*((1/2033753534126478065664*(-(1/36*(1+h))*h*r*((41/36700160)*h^7*(2-r)^2*r^16*(1+h)+(3/40960*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^3*r^10+(9/4194304)*h^12*(2-r)^8*r^10)+(1/36*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*(-(3/40960)*h^7*(2-r)^3*r^11*(1+h)-(3/1024*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^4*r^5)-(1/24576)*h^6*(2-r)^4*r^5*((3/1024)*h^7*(2-r)^4*r^6*(1+h)+((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5)^2)-(1/491520)*h^7*(2-r)^3*r^11*(1+h)*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5)-(41/2642411520)*h^8*(2-r)^2*r^17*(1+h)^2))*h^12*r^50-(31/132405829044692582400*(-(1/44*(1+h))*h*r*(-(31/2936012800)*h^7*(2-r)*r^21*(1+h)-(41/36700160*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^2*r^15-(9/83886080)*h^12*(2-r)^7*r^15)+(1/44*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*((41/36700160)*h^7*(2-r)^2*r^16*(1+h)+(3/40960*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^3*r^10+(9/4194304)*h^12*(2-r)^8*r^10

h*(x*(2-r)-(1/32)*x^9*r^5)-(1+h)*h*(x*(2-r)-(1/32)*x^9*r^5)-h*(-h^5*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*r*(2-r)^20*x^41-(41/73400320)*r^2*(2-r)^15*x^33+(3/81920)*r^3*(2-r)^10*x^25-(3/2048)*r^4*(2-r)^5*x^17+(1/32)*x^9*r^5))+(1+h)*(-(1+h)*h*(x*(2-r)-(1/32)*x^9*r^5)+h^6*(-(1/45097156608)*(2-r)^25*x^49+(31/5872025600)*r*(2-r)^20*x^41-(41/73400320)*r^2*(2-r)^15*x^33+(3/81920)*r^3*(2-r)^10*x^25-(3/2048)*r^4*(2-r)^5*x^17+(1/32)*x^9*r^5))-h*x*((1/136*((1/2033753534126478065664*(-(1/36*(1+h))*h*r*((41/36700160)*h^7*(2-r)^2*r^16*(1+h)+(3/40960*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^3*r^10+(9/4194304)*h^12*(2-r)^8*r^10)+(1/36*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*(-(3/40960)*h^7*(2-r)^3*r^11*(1+h)-(3/1024*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^4*r^5)-(1/24576)*h^6*(2-r)^4*r^5*((3/1024)*h^7*(2-r)^4*r^6*(1+h)+((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5)^2)-(1/491520)*h^7*(2-r)^3*r^11*(1+h)*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5)-(41/2642411520)*h^8*(2-r)^2*r^17*(1+h)^2))*h^12*r^50-(31/132405829044692582400*(-(1/44*(1+h))*h*r*(-(31/2936012800)*h^7*(2-r)*r^21*(1+h)-(41/36700160*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^2*r^15-(9/83886080)*h^12*(2-r)^7*r^15)+(1/44*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*((41/36700160)*h^7*(2-r)^2*r^16*(1+h)+(3/40960*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^3*r^10+(9/4194304)*h^12*(2-r)^8*r^10)-(3/90112)*h^6*(2-r)^4*r^5*(-(3/40960)*h^7*(2-r)^3*r^11*(1+h)-(3/1024*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)^4*r^5)+(3/3604480)*h^6*(2-r)^3*r^10*((3/1024)*h^7*(2-r)^4*r^6*(1+h)+((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5)^2)+(41/1614807040)*h^7*(2-r)^2*r^16*(1+h)*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5)+(31/258369126400)*h^8*(2-r)*r^22*(1+h)^2))*h^12*(2-r)*r^45+(10891/206884107882332160000*(-(1/52*(1+h))*h*r*((1/22548578304)*h^7*r^26*(1+h)+(31/2936012800*((1/32*(1+h))*h*(2-r)^5+(1/32)*h^6*(2-r)^5))*h^6*(2-r)*r^20+(1119/375809638400)*h^12

Table 2. Computation between the exact and HAM in case 2 and determine the absolute error.

As long as a series solution given by the HAM converges, it must be the exact solution. So it is important to ensure that the homotopy solution series converges. contains the auxiliary parameter h , which provides us a simple way to adjust and control the convergence of the solution series. In general, by means of the so-called h -curve (a curve of a versus h ), it is straightforward to choose an appropriate range for h which ensures the convergence of the solution series. As pointed by Liao [13], the valid region of h is a horizontal line segment. In Fig. (3a), Fig (2b) and Fig (4a), Fig (4b) we plot the h-curves, we could find that if h is about in area [-1,-0.7] the result is different values of h

Figure 3a. The exact solution .

Figure 3b. The approximte solution .

Figure 4a. The exact solution .

Figure 4b. The approoximte solution .

4. Conclusion

The proposed method is a powerful procedure for solving fuzzy nonlinear Volterra integral equation. The examples analyzed illustrate the ability and reliability of the method presented in this paper and severals that the one is very simple and effective. The obtained solutions, in comparision with the exact solution admit a remarkable accuracy. Resultes indicate that the convergence rate is very fast, and lower approximations can achieve high accuracy

References

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