A Fuzzy Newton-Cotes method For Integration of Fuzzy Functions

Available online at www.ispacs.com/jfsva

Volume 2015, Issue 2, Year 2015 Article ID jfsva-00247, 8 Pages doi:10.5899/2015/jfsva-00247

Research Article

A Fuzzy Newton-Cotes method For Integration of Fuzzy

Functions

N. Ahmady∗

Department of Mathematics, Varamin-Pishva Branch, Islamic Azad University, Varamin, Iran

Copyright 2015 c⃝N. Ahmady. 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, a numerical method for integration of fuzzy functions is considered. Fuzzy Newton-Cotes formula, such as fuzzy trapezoidal method and fuzzy Simpson method are calculated by integration of fuzzy functions on two and three equally space points. Also the composite fuzzy trapezoidal and composite fuzzy Simpson method are proposed fornequally space points. The proposed method are illustrated by numerical examples.

Keywords:Keywords: Fuzzy integration, fuzzy Newton-Cotes method, Fuzzy trapezoidal Fuzzy Simpson’s rule.

1 Introduction

In numerical analysis, the integration problem plays a major role in various areas such as mathematics, physics, statistics, engineering and social sciences. In many real-world problems, not all of the data can be precisely assessed. When information is easily measurable or accessible, the information should be coded in crisp numbers. Fuzzy num-bers theory makes it possible to incorporate unquantifiable information, incomplete information and non-obtainable information in to mathematics models. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh[11]. The topic of Newton-Cotes methods with positive coefficient for integration of fuzzy function by Allahviranloo [1] were discussed. Bede and Gal,[4] proposed quadrature rules for integrals of valued, they introduced some quadrature rules for the Henstock integral of fuzzy-number-valued mappings by giving error bounds for mappings of bounded variation and of Lipschitz type. They considered generalizations of classical quadrature rules, such as midpoint-type, trapezoidal and three-point-type quadrature. In this paper, the fuzzy Simpson’s rule is obtained by using fuzzy lagrange interpolation fit to the fuzzy function at three equally spaced points.This paper is organized as follows:

In section 2, some basic definitions and results which will be used later are brought. Fuzzy Newton-Cotes rule for solving fuzzy integral is introduced in section 3. Numerical examples are presented in section 4, and the final section contains a conclusion.

2 Preliminaries

First, we introduce the notation that will be used in this paper.

2.1 _{Notation and definitions}

Definition 2.1. [6] A fuzzy number u is a fuzzy subset of the real line with a normal, convex and upper semicontinuous

membership function of bounded support. The family of fuzzy numbers will be denoted by RF.

An arbitrary fuzzy numberuis represented by an ordered pair of functions(u(r),u(r)), 0≤r≤1 that, satisfies the following requirements:

• u(r)is a bounded left continuous nondecreasing function over [0,1], with respect to anyr.

• u(r)is a bounded left continuous nonincreasing function over [0,1], with respect to anyr.

• u(r)≤u(r), 0≤r≤1. Then ther-level set

[v]r_{={s|v(s)≥r}, 0<r≤1,}

is a closed bounded interval, denoted by

[v]r= [v(r),v(r)].

LetIbe a real interval. A mappingy:I→Eis called a fuzzy process, and itsr−level set is denoted by

[y(t)]r= [y(t,r),y(t,r)], t∈I, r∈(0,1].

Definition 2.2. [6]A triangular fuzzy number is a fuzzy set U in E that is characterized by an ordered triple(xl,xc,xr)∈

R3_{with x}

l≤xc≤xrsuch that[U]0= [xl,xr]and[U]1={xc}.

The r-level set of a triangular fuzzy number U is given by,

[U]r_{= [x}

c−(1−r)(xc−xl),xc+ (1−r)(xr−xc)], (2.1)

for any r∈I .

Definition 2.3. [6]. For arbitrary fuzzy numbers u= (u,u)and v= (v,v)the quantity

D(u,v) = [

∫ 1 0 (

u(r)−v(r))2dr+

∫ 1 0 (

u(r)−v(r))2dr]12_{, (2.2)}

is the distance between u and v.

2.2 _{Interpolation for fuzzy number}

The problem of interpolation for fuzzy sets is as follows:

suppose that at various time instanttinformationf(t)is presented as fuzzy set. The aim is to approximate the function f(t), for alltin the domain off. Lett0<t1< . . . <tnben+1 distinct points inRand letu0,u1, . . . ,unben+1 fuzzy

sets inE. A fuzzy polynomial interpolation of the data is a fuzzy-value function f:R→Esatisfying:

• For alli=1, . . . ,n, f(ti) =eui,

• f is continuous,

• If the data is crisp, then the interpolation f is a crisp polynomial.

A function f fulfilling these condition may be constructed as follows. For each r∈[0,1] andi=0,1, . . . ,n, let C_αi = [uei]α. For eachx= (x0,x1, . . . ,xn)∈Rn+1, denote byPX the unique polynomial of degree≤nsuch that

That is, by the crisp Lagrange interpolation formola

PX(t) = n

∑

i=0 xi(

∏

i̸=j

t−tj

ti−tj

).

Finally, for eacht∈Rand allξ∈Rdefinef(t)∈Eby

(f(t))(ξ) =sup{r∈[0,1]:∃X∈C0_r×. . .×C_rn such that PX(t) =ξ}.

The interpolation polynomial can be written level set wise as

[f(t,r)] ={y∈R:y=PX(t),x∈[ui(r)],i=1,2, . . . ,n}, f or0≤r≤1.

When the datauipresents as triangular fuzzy numbers, values of the interpolation polynomial are also triangular fuzzy

numbers. Thenf(t)has a particular simple form that is well suited to computation.

Theorem 2.1. Let(ti,ui),i=0,1,2, . . . ,n be the observed data and suppose that each of the ui= (uli,uci,uri)is an

element of E. Then for each t∈[t0,tn], f(t) = (fl(t),fc(t),fr(t))∈E,

fl(t) =∑ℓi(t)≥0ℓi(t)u l

i+∑ℓi(t)<0ℓi(t)u r i,

fc(t) =∑ni=0ℓi(t)uci,

fr(t) =∑ℓi(t)≥0ℓi(t)u r

i+∑ℓi(t)<0ℓi(t)u l i,

such thatℓi(t) =∏nj̸=i t−tj ti−tj.

Proof. see [7].

3 Fuzzy Newton-Cots method

In this paper we are going to explore various ways for approximating the integral of a fuzzy function over a given domain. The basic method involved in approximating∫_abf(x)dxis called numerical quadrature. The basic idea is to select a set of distinct nodesx0,x1,· · ·xnfrom the interval[a,b], then integrate the fuzzy Lagrange interpolating

polynomial, in order to gain some insight on numerical integration. In this paper we introduce two formulas that produced by using first and second fuzzy Lagrange polynomials with equally-spaced nodes. This gives the fuzzy Trapezoidal rule and Fuzzy Simpson rule.

3.1 _{Trapezoidal rule for fuzzy integral}

To derive the Trapezoidal rule for approximating∫_abf(x)dx,letxi=a,xi+1=b,h=b−a.We showxi,xi+1with two fuzzy valuesf(xi)and f(xi+1)by:

{(xi,f(xi)),(xi+1,f(xi+1))}

we will use fuzzy Linear Lagrange polynomialP(x)in order to interpolate the fuzzy function f(x)at two equally spaced pointsxi,xi+1. Lagrange coefficientsℓi(x), ℓi+1(x)are obtained as follows forxi≤x≤xi+1.

ℓi(x) =

(x−xi+1)

(xi−xi+1)

≥0, (3.3)

ℓi+1(x) =

(x−xi)

(xi+1−xi)

≥0, (3.4)

Also we have:

∫ xi+1

xi

ℓi(x)dx=

∫ xi+1

xi

(x−xi+1)

(xi−xi+1) dx=

∫ 1 0

(θ−1)h

−h hdθ=

h

∫ xi+1

xi

ℓi+1(x)dx=

∫ xi+1

xi

(x−xi)

(xi+1−xi)

dx=

∫ 1 0

θ

hhdθ= h

2, (3.6)

wherex=xi+θhanddx=hdθ.

By theorem (2.1), the fuzzy Lagrange polynomialP(x)with triple form(Pl(x),Pc(x),Pr(x))is obtain as follows:

Pl(x) =ℓi(x)fil(x) +ℓi+1(x)fil+1(x), (3.7)

Pr(x) =ℓi(x)fir(x) +ℓi+1(x)fir+1(x), (3.8)

Pc(x) =ℓi(x)fic(x) +ℓi+1(x)fic+1(x). (3.9)

For solving fuzzy integral∫xi+1

xi f(x)dx, we use fuzzy linear Lagrange polynomialP(x)instant of fuzzy function f(x),

∫ xi+1

xi

f(x)dx=

∫ xi+1

xi

P(x)dx

For upper and lower bound of fuzzy Lagrange polynomialP(x)we have:

∫ xi+1

xi

P(x,r)dx=

∫ xi+1

xi

Pl(x) +r(Pc(x)−Pl(x))dx, (3.10)

∫ xi+1

xi

P(x,r)dx=

∫ xi+1

xi

Pr(x)−r(Pr(x)−Pc(x))dx. (3.11)

In order to obtain lower bound of fuzzy Trapezoidal rule we substitute (3.7) and (3.8) in (3.10) and by integration of (3.10), we easily obtain:

∫ xi+1

xi

f(x,r)dx=

∫ xi+1

xi

P(x,r)dx=h

2{f(xi,r) +f(xi+1,r)} (3.12)

and by similar way we obtain:

∫ xi+1

xi

f(x,r)dx=

∫ xi+1

xi

P(x,r)dx=h

2{f(xi,r) +f(xi+1,r)} (3.13)

The composite fuzzy trapezoidal rule is obtained by applying the fuzzy trapezoidal rule in each subinterval[xi,xi+1], i=0,· · ·,n−1, i.e.,

∫ b

a

f(x,r)dx=

i=n−1

∑

i=0

∫ xi+1

xi

f(x,r)dx=1

2

i=n−1

∑

i=0

(xi+1−xi){f(xi,r) +f(xi+1,r)} (3.14)

∫ b

a

f(x,r)dx=

i=n−1

∑

i=0

∫ xi+1

xi

f(x,r)dx=1

2

i=n−1

∑

i=0

(xi+1−xi){f(xi,r) +f(xi+1,r)} (3.15)

A particular case is when these points are uniformly spaced,when all intervals have an equal length. For example, if xi=a+ih, whereh=b−_na,the fuzzy trapezial rule for fuzzy function f(x)is obtain as follows:

∫ b

a

f(x,r)dx=h

2{f(a) +

n−1

∑

i=1

f(a+ih) +f(b)}, (3.16)

∫ b

a

f(x,r)dx=h

2{f(a) +

n−1

∑

i=1

3.2 _{Fuzzy Simpson rule}

Now we recall three equally spaced points,xi,xi+1,xi+2,with fuzzy values f(xi),f(xi+1)andf(xi+2)denote by:

{(xi,f(xi)),(xi+1,f(xi+1)),(xi+2,f(xi+2))}

we will use fuzzy second Lagrange polynomialP(x)in order to interpolate the fuzzy function f(x)at three equally spaced pointsxi,xi+1,xi+2.

First we suppose thatxi≤x≤xi+1≤xi+2, then the Lagrange coefficients are obtained as follows :

ℓi(x) =

(x−xi+1)(x−xi+2)

(xi−xi+1)(xi−xi+2)

≥0, (3.18)

ℓi+1(x) =

(x−xi)(x−xi+2)

(xi+1−xi)(xi+1−xi+2)

≥0, (3.19)

ℓi+2(x) =

(x−xi)(x−xi+1)

(xi+2−xi)(xi+2−xi+1)

≤0, (3.20)

By theorem (2.1) the fuzzy Lagrange polynomialP1(x)with triple form(P1l(x),P1c(x),P1r(x))is obtain:

P₁l(x) =ℓi(x)fil(x) +ℓi+1(x)fil+1(x) +ℓi+1(x)fir+2(x), (3.21)

P₁r(x) =ℓi(x)fir(x) +ℓi+1(x)fir+1(x) +ℓi+1(x)fil+2(x), (3.22)

P₁c(x) =ℓi(x)fic(x) +ℓi+1(x)fic+1(x) +ℓi+1(x)fic+2(x). (3.23) Also the Lagrange coefficientsℓi(x), ℓi+1(x)andℓi+2(x)are obtained as follows, when we suppose thatxi≤xi+1≤ x≤xi+2,

ℓi(x) =

(x−xi+1)(x−xi+2)

(xi−xi+1)(xi−xi+2)

≤0, (3.24)

ℓ_i+1(x) = (x−xi)(x−xi+2) (xi+1−xi)(xi+1−xi+2)

≥0, (3.25)

ℓi+2(x) =

(x−xi)(x−xi+1)

(xi+2−xi)(xi+2−xi+1)

≥0, (3.26)

then fuzzy Lagrange polynomialP2(x)with triple form(P2l(x),P2c(x),P2r(x))is denote by:

P₂l(x) =ℓi(x)fir(x) +ℓi+1(x)fil+1(x) +ℓi+1(x)fil+2(x), (3.27)

P₂r(x) =ℓi(x)fil(x) +ℓi+1(x)fir+1(x) +ℓi+1(x)fir+2(x), (3.28)

P₂c(x) =ℓi(x)fic(x) +ℓi+1(x)fic+1(x) +ℓi+1(x)fic+2(x). (3.29) For solving fuzzy integral∫xi+2

xi f(x)dx, we have

∫ xi+2

xi

f(x)dx=

∫ xi+1

xi

f(x)dx+

∫ xi+2

xi+1

then we use fuzzy interpolationP(x)instant of fuzzy functionf(x), forxi≤x≤xi+1,

∫ xi+2

xi

P(x)dx=

∫ xi+1

xi

P1(x)dx+

∫ xi+2

xi+1

P2(x)dx,

For upper and lower bound ofP(x)we have:

∫ xi+2

xi

P(x)dx=

∫ xi+2

xi

P₁l(x) +r(P₁c(x)−P₁l(x))dx+

∫ xi+2

xi

P₂l(x) +r(P₂c(x)−P₂l(x))dx, (3.30)

∫ xi+2

xi

P(x)dx=

∫ xi+2

xi

P₁r(x)−r(P₁r(x)−P₁c(x))dx+

∫ xi+2

xi

P₂r(x)−r(P₂r(x)−P₂c(x))dx. (3.31)

IfP₁l(x),P₁c(x),P₂l(x)andP₂c(x)are substituted in (3.30) and by substitutingP₁r(x),P₁c(x),P₂r(x),P₂c(x)in (3.31) and by calculating the integral of Lagrange Coefficients,

∫xi+1

xi ℓi(x)dx=512h,

∫xi+1

xi ℓi+1(x)dx=23h,

∫xi+1

xi ℓi+2(x)dx=−12h,

∫xi+2

xi+1 ℓi(x)dx=

−h

12,

∫xi+2

xi+1 ℓi+1(x)dx=

2h

3,

∫xi+2

xi+1 ℓi+2(x)dx=

5h

12,

(3.32)

we obtain:

∫ xi+2

xi

f(x,r)dx=h

3(( 5

4fi(x,r)−

1

4fi(x,r)) +4fi+1(x,r) + ( 5

4fi+2(x,r)− 1

4fi+2(x,r)) (3.33)

clearly for upper boundf(x,r)we obtain

∫ xi+2

xi

f(x,r)dx=h

3(( 5

4fi(x,r)− 1

4fi(x,r)) +4fi+1(x,r) + (

5

4fi+2(x,r)− 1

4fi+2(x,r)) (3.34)

The composite fuzzy Simpson rule is obtained by applying the fuzzy Simpson rule in each subinterval[xi,xi+1], i=0,· · ·,n−1, i.e.,

∫ xn

x0

f(x,r)dx=h

3{( 5

4f(x0,r)− 1

4f(x0,r)) +4(f(x1,r) +f(x3,r) +· · ·+f(xn−1,r))

+2[(5

4f(x2,r)− 1

4f(x2,r)) + ( 5

4f(x4,r)− 1

4f(x4,r)) +· · ·

+(5

4f(xn−2,r)− 1

4f(xn−2,r))] + ( 5

4f(xn,r)− 1

4f(xn,r))}

and _∫

xn x0

f(x,r)dx=h

3{( 5

4f(x0,r)− 1

4f(x0,r)) +4(f(x1,r) +f(x3,r) +· · ·+f(xn−1,r))

+2[(5

4f(x2,r)− 1

4f(x2,r)) + ( 5

4f(x4,r)− 1

4f(x4,r)) +· · ·

+(5

4f(xn−2,r)− 1

4f(xn−2,r))] + ( 5

4f(xn,r)− 1

4f(xn,r))}

4 Numerical Example

Example 4.1. Consider the following fuzzy integral,

∫ 1 0

f(x)dx, f(x,r) = ( 2r

the exact solution is1₄π(2r,4−2r). Byh=0.1 and fuzzy trapezoidal method we obtain:

∫ 1 0 (

2r 1+x2,

4−2r

1+x2)dx= (1.469962994r+0.1,3.039925988−1.469962994r), the error of fuzzy Trapezoidal method(E(FT(h)))forh=0.1, is:

E(FT(h))) = (0.100833333r−0.1,0.101666667−0.100833333r), obviously

d(E(FT(h),0) =0.001666667, By using fuzzy Simpson rule and h=0.1, we obtain:

∫ 1 0

( 2r

1+x2, 4−2r

1+x2)dx= (1.964053351r−0.3932570440,3.534849657−1.964053351r), and the error of fuzzy Simpson rule(E(Fs(h)))is:

E(FS(h)) = (−0.393257024r+0.3932570440,−0.393257002+0.393257024r), therefore for fuzzy Simpson rule we have:

d(ES(h),0) =4.2×10−8.

Example 4.2. Consider the following fuzzy integral,

∫ 1.4 −1 f(x)dx

, f(x,r) = ((0.5+0.5r)x3,(1.5−0.5r)x3)

the exact solution is 0.7104(0.5+0.5r,1.5−0.5).By Fuzzy Trapezoidal rule andh=0.1 we obtain:

∫ 1.4 −1 ((0

.5+0.5r)x3,(1.5−0.5r)x3)dx= (0.1+0.62r,1.34−0.62r), the error term for fuzzy Trapezoidal rule is:

E(FT(h)) = (0.2552−.2648r,−0.2744+0.2648r), and distance of error to zero is:

d(E(FT(h)),0) =−0.0192, By using fuzzy Simpson rule andh=0.1, we have:

∫ 1.4 −1 ((0

.5+0.5r)x3,(1.5−0.5r)x3)dx= (−0.0056+.716r,1.426400000−0.716r),

the error term is obtain as follows

E(FS(h)) = (0.3608−0.3608r,−0.3608+.36080r), therefore we have:

5 Conclusion

Recently, the Newton Cotes methods with positive coefficient for integration of fuzzy function by allahviranloo [1] were discussed. In this paper fuzzy Newton Cotes method for integration of all type of fuzzy function is presented.

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