Adomian Method for Solving Fuzzy Fredholm-Volterra Integral Equations

Available online at www.ispacs.com/jfsva Volume 2013, Year 2013 Article ID jfsva-00114, 7 Pages

doi:10.5899/2013/jfsva-00114 Research Article

Adomian Method for Solving Fuzzy Fredholm-Volterra

Integral Equations

M. Barkhordari Ahmadi1_{, M. Mosleh}2_{, M. Otadi}2∗

(1)Department of Mathematics, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran

(2)Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

Copyright 2013 c⃝M. Barkhordari Ahmadi, M. Mosleh and M. Otadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, Adomian method has been applied to approximate the solution of fuzzy volterra-fredholm integral equa-tion. That, by using parametric form of fuzzy numbers, a fuzzy volterra-fredholm integral equation has been converted to a system of volterra-fredholm integral equation in crisp case. Finally, the method is explained with illustrative ex-amples.

Keywords: Fuzzy numbers, Fuzzy volterra-fredholm integral equation, Adomian method, System of Volterra-Fredholm integral

equation

1 Introduction

The fuzzy mapping function was introduced by Cheng and Zadeh [5]. Later, Dubois and Prade [7] presented an elementary fuzzy calculus based on the extension principle [20]. Puri and Ralescu [19] suggested two definitions for fuzzy function. The concept of integration of fuzzy functions was first introduced by Dubois and prade [7]. Alternative approaches were later suggested by Goetschel and Voxman [12], Kaleva [14], Matloka [17], Nanda [18] and others, while Goetschel and Voxman [12] and later Matloka [17] preferred a Riemann integral type approach, Kaleva [14] chose to define the integral of fuzzy function, using the Lebesgue type concept for integration. Numerical method for solving fuzzy integral by Allahviranloo and Otadi [2] introduced.

One of the first applications of fuzzy integration was given by Wu and Ma [6] who investigated the fuzzy Fredholm integral equation of the second kind.

The topics of fuzzy integral equations which attracted growing interest for some time, in particular in relation to fuzzy control, have been rapidly developed in recent years.

we consider approximate solution of fuzzy Volterra-Fredholm integral equation by using Adomian method. In section 2, the basic concept of fuzzy number is discussed, In section 3, fuzzy Volterra-Fredholm integral equation is converted to a system of Fredholm integral equation. The Adomian method is applied for solving a system of Volterra-Fredholm integral equation in section 4. Finally, conclusion is drown in section 6.

2 Preliminaries and notations

We now recall some definitions needed through the paper. The basic definition of fuzzy numbers is given in [7, 13].

ByR, we denote the set of all real numbers. A fuzzy number is a mappingu:R→[0,1]with the following properties: (a)uis upper semi-continuous,

(b)uis fuzzy convex, i.e.,u(λx+ (1−λ)y)≥min{u(x),u(y)}for allx,y∈R,λ∈[0,1], (c)uis normal, i.e.,∃x0∈Rfor whichu(x0) =1,

(d) suppu={x∈R|u(x)>0}is the support of theu, and its closure cl(supp u) is compact.

Let E be the set of all fuzzy number onR. Ther-level set of a fuzzy numberu∈E,o≤r≤1, denoted by[u]r , is

defined as

[u]r= {

{x∈R|u(x)≥r} i f 0≤r≤1 cl(supp u) i f r=0

It is clear that ther-level set of a fuzzy number is a closed and bounded interval[u(r),u(r)], whereu(r)denotes the left-hand endpoint of[u]r andu(r)denotes the right-hand endpoint of[u]r. Since each y∈Rcan be regarded as a

fuzzy number_eydefined by

e

y(t) = {

1 i f t=y o i f t̸=y

Rcan be embedded inE.

Remark 2.1. (See [21]) Let X be Cartesian product of universes X=X1×...×Xn, and A1, . . . ,Anbe n fuzzy numbers

in X1, . . . ,Xn, respectively. f is a mapping from X to a universe Y , y= f(x1, ...,xn). Then the extension principle

allows us to define a fuzzy set B in Y by

B={(y,u(y))|y=f(x1, ...,xn),(x1, ...,xn)∈X}

where

uB(y) = {

sup_{(x1,...,xn)∈f}−1_(y)min{uA1(x1), ...,uAn(xn))}, i f f−1(y)̸=0,

0 i f otherwise.

where f−1is the inverse of f .

For n=1, the extension principle, of course, reduces to

B={(y,uB(y))|y=f(x),x∈X}

where

uB(y) = {

supx∈f−1_(y)uA(x), i f f−1(y)̸=0,

0 i f otherwise.

According to Zadeh,s extension principle, operation of addition onEis defined by (u⊕v)(x) =supy∈Rmin{u(y),v(x−y)}, x∈R

and scalar multiplication of a fuzzy number is given by

(k⊙u)(x) =

{ _{u(x/k), k>}

0,

e

0, k=0,

where ˜0∈E.

It is well known that the following properties are true for all levels

[u⊕v]r= [u]r+ [v]r, [k⊙u]r=k[u]r

From this characteristic of fuzzy numbers, we see that a fuzzy number is determined by the endpoints of the intervals [u]r. This leads to the following characteristic representation of a fuzzy number in terms of the two ”endpoint”

Definition 2.1. A fuzzy number u in parametric form is a pair(u,u)of functions u(r), u(r),0≤r≤1, which satisfy the following requirements:

1. u(r)is a bounded non-decreasing left continuous function in(0,1], and right continuous at 0,

2. u(r)is a bounded non-increasing left continuous function in(0,1], and right continuous at 0,

3. u(r)≤u(r),0≤r≤1.

A crisp number α is simply represented by u(r) =u(r) =α, 0 ≤r≤1. We recall that for a<b <c which a,b,c∈R, the triangular fuzzy numberu= (a,b,c)determined bya,b,cis given such thatu(r) =a+ (b−a)rand u(r) =c−(c−b)rare the endpoints of the r-level sets, for allr∈[0,1].

For arbitraryu= (u(r),u(r)),v= (v(r),v(r))andk>0 we define additionu⊕v, subtractionu⊖vand scaler multi-plication bykas (See [10, 15])

(a) Addition:

u⊕v= (u(r) +v(r),u(r) +v(r))

(b) Subtraction:

u⊖v= (u(r)−v(r),u(r)−v(r))

(c) Multiplication:

u⊙v= (min{u(r)v(r),u(r)v(r),u(r)v(r),u(r)v(r)},max{u(r)v(r),u(r)v(r),u(r)v(r),u(r)v(r)})

(d) Scaler multiplication:

k⊙u= {

(ku,ku), k≥0, (ku,ku), k<0.

3 Fuzzy Volterra-Fredholm Integral Equation

Integral equations which are used in this section are fuzzy Volterra-Fredholm integral equations.

Definition 3.1. consider the following linear fuzzy Volterra-Fredholm integral equation

e

u(x,y) =ef(x,y) +

∫ y

c

∫ b

a

k(x,y,s,t)ue(s,t)dsdt (3.1)

where k(x,y,s,t)is a known function andu_e(x,y),fe(x,y)are unknown and known fuzzy valued functions,respectively.

Now, parametric form of a linear fuzzy Volterra-Fredholm integral equation can be written as the following:

{

u(x,y,r) =f(x,y,r) +∫_cy∫_abg1(x,y,s,t,u(s,t,r),u(s,t,r))dsdt, u(x,y,r) =f(x,y,r) +∫_cy∫_abg2(x,y,s,t,u(s,t,r),u(s,t,r))dsdt,

(3.2)

whereef(x,y) = (f(x,y,r),f(x,y,r)),u_e(x,y) = (u(x,y,r),u(x,y,r)),

g1(x,y,s,t,u(s,t,r),u(s,t,r)) =

{

k(x,y,s,t)u(s,t,r), k(x,y,s,t)≥0 k(x,y,s,t)u(s,t,r), k(x,y,s,t)<0, and

g2(x,y,s,t,u(s,t,r),u(s,t,r)) =

{

k(x,y,s,t)u(s,t,r), k(x,y,s,t)≥0 k(x,y,s,t)u(s,t,r), k(x,y,s,t)<0, for each 0≤r≤1, a≤x≤b and c≤y≤d.

4 Adomian Method for SolvingFuzzy Fredholm-Volterra Integral Equations

we explain the Adomian method to compute numerical solutions of Volterra-Fredholm integral equations as fol-lows:

U(x,y) =F(x,y) +

∫ y

c

∫ b

a

k(x,y,s,t)U(s,t)dsdt (4.3)

where

U(x,y) = (u1(x,y), ...,un(x,y))T,

F(x,y) = (f1(x,y), ...,fn(x,y))T

andk(x,y,s,t) = [ki j(x,y,s,t)] i=1, ...,n, j=1, ...,nconsider theith equation of (4.3) as:

ui(x,y) = fi(x,y) +

∫ y c ∫ b a n

∑

ki j(x,y,s,t)uj(s,t)dsdt (4.4)

ifNi(u1,u2, ...,un)(x,y) =

∫y c

∫b

a∑nj=1ki j(x,y,s,t)uj(s,t)dsdtthen, we have:

ui(x,y) = fi(x,y) +Ni(u1,u2, ...,un)(x,y) (4.5)

to use the Adomian method, letui(x,y) =∑∞j=0ui j(x,y)andNi(x,y) =∑∞j=0Ai jwhereAi j, i=0, ...,nare polynomials

depending onu10, ...,u1j, ...,un0, ...,un jthat called Adomian polynomials and then has been approximated the solution

by

ϕik(x,y) =∑kj=−10ui j(x,y)and by substituting in (4.4), we have:

ui(x,y) = ∞

∑

ui j(x,y) =fi(x,y) + ∞

∑

Ai j(u10, ...,u1j, ...,un0, ...,un j) (4.6)

to use Adomian method, let

u_iλ(x,y) =

∞

∑

ui j(x,y)λj (4.7)

N_iλ(x,y) =

∞

∑

Ai jλj (4.8)

whereλ is a parameter introduced for convenience. by substituting (4.7) and (4.8) in (4.5), we have:

∞

∑

uim(x,y)λm=fi(x,y) +

∫ y c ∫ b a n

∑

ki j(x,y,s,t)( ∞

∑

uim(x,y)λm)dsdt (4.9)

Equating powers ofλ on both sides of Eq.(4.9) gives: ui0(x,y) = fi(x,y)ui,k+1=

∫y c

∫b

a∑nj=1ki j(x,y,s,t)ujkdsdt.

We practice, of course, the sum of the infinite series has to be truncated at some orderk. The quantityϕik, can thus be reasonable approximation of the exact solution, providedkis sufficiently large. Ask−→∞, the series converge smoothly toward the exact solution for 0≤r≤1 [1].

5 Numerical examples

Example 5.1. Consider the linear fuzzy Volterra-Fredholm integral equation as

e

u(x,y) = (x2+13 15xy−

2 15xy

4₎₍

2r,r−3) +

∫ y

0

∫ 1 −1xys

2_t2

e

u(s,t)dsdt,0≤r≤1.

By Eqs. (4.4)-(4.9) for n=2, we obtain

e

u(x,y) = (u(x,y;r),u(x,y;r)) = (x2+xy)(2r,3−r),0≤r≤1.

0 0.5 1 1.5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact n=2

Figure 1: Compares the exact solution and obtained solutions of Adomian method atx=0.5 andy=0.5.

The exact and obtained solution of Adomian method in this example atx=0.5 andy=0.5 forn=2, are shown in figure 1.

Example 5.2. Consider the linear fuzzy Volterra-Fredholm integral equation as

e

u(x,y) = (x+sin y+2 3−

2

3exp(y))( r2+r

9 ,

4−r3−r 9 ) +

∫ y

0

∫ 1 −1

s.exp(t)eu(s,t)dsdt, 0≤r≤1.

By Eqs. (4.4)-(4.9) for n=3, we obtain

e

u(x,y) = (u(x,y;r),u(x,y;r)) =1

9(x+y− y3

3)(r 2_+r,

4−r3−r),0≤r≤1.

0 0.5 1 1.5 2 2.5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact n=3

Figure 2: Compares the exact solution and obtained solutions of Adomian method atx=0.5 andy=0.5.

6 Conclusion

In this work, we tried to obtain solution of fuzzy Volterra-Fredholm integral equations by Adomian method. FVFIE was converted to a system of Volterra-Fredholm integral equations The efficiency of method was illustrated by numerical examples.

Acknowledgements

We would like to present our sincere thanks to the Editor in Chief, Prof. T. Allahviranloo for their valuable suggestions.

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