A Fuzzy Newton-cotes Method for Integration of Fuzzy Functions

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 for n equally space points. The proposed method are illustrated by numerical examples.


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 numbers 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 fuzzy-number-valued, they introduced some quadrature rules for the Henstock integral of fuzzy-numbervalued 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.

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

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 R F .
An arbitrary fuzzy number u is 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 any r.
• u(r) is a bounded left continuous nonincreasing function over [0,1], with respect to any r.
Then the r-level set Let I be a real interval.A mapping y : I → E is called a fuzzy process, and its r−level set is denoted by

[6]
A triangular fuzzy number is a fuzzy set U in E that is characterized by an ordered triple The r-level set of a triangular fuzzy number U is given by, for any r ∈ I .
Definition 2.3.[6].For arbitrary fuzzy numbers u = (u, u) and v = (v, v) the quantity is the distance between u and v.

Interpolation for fuzzy number
The problem of interpolation for fuzzy sets is as follows: suppose that at various time instant t information f (t) is presented as fuzzy set.The aim is to approximate the function f (t), for all t in the domain of f .Let t 0 < t 1 < . . .< t n be n + 1 distinct points in R and let u 0 , u 1 , . . ., u n be n + 1 fuzzy sets in E. A fuzzy polynomial interpolation of the data is a fuzzy-value function f : R → E satisfying: • 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] and i = 0, 1, . . ., n, let , denote by P X the unique polynomial of degree ≤ n such that

International Scientific Publications and Consulting Services
That is, by the crisp Lagrange interpolation formola Finally, for each t ∈ R and all ξ ∈ R define f (t) ∈ E by The interpolation polynomial can be written level set wise as When the data u i presents as triangular fuzzy numbers, values of the interpolation polynomial are also triangular fuzzy numbers.Then f (t) has a particular simple form that is well suited to computation.

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 ∫ b a f (x)dx is called numerical quadrature.The basic idea is to select a set of distinct nodes x 0 , x 1 , • • • x n from 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.

Trapezoidal rule for fuzzy integral
To derive the Trapezoidal rule for approximating we will use fuzzy Linear Lagrange polynomial P(x) in order to interpolate the fuzzy function f (x) at two equally spaced points x i , x i+1 .Lagrange coefficients ℓ i (x), ℓ i+1 (x) are obtained as follows for Also we have: International Scientific Publications and Consulting Services where x = x i + θ h and dx = hdθ .By theorem (2.1), the fuzzy Lagrange polynomial P(x) with triple form (P l (x), P c (x), P r (x)) is obtain as follows: ) ) For solving fuzzy integral x i f (x)dx, we use fuzzy linear Lagrange polynomial P(x) instant of fuzzy function f (x), x i

P(x)dx
For upper and lower bound of fuzzy Lagrange polynomial P(x) we have: x i P l (x) + r(P c (x) − P l (x))dx, (3.10) 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: and by similar way we obtain: The composite fuzzy trapezoidal rule is obtained by applying the fuzzy trapezoidal rule in each subinterval A particular case is when these points are uniformly spaced,when all intervals have an equal length.For example, if x i = a + ih, where h = b−a n , the fuzzy trapezial rule for fuzzy function f (x) is obtain as follows:

.17)
International Scientific Publications and Consulting Services

Fuzzy Simpson rule
Now we recall three equally spaced points, x i , x i+1 , x i+2 , with fuzzy values f (x i ), f (x i+1 ) and f (x i+2 ) denote by: we will use fuzzy second Lagrange polynomial P(x) in order to interpolate the fuzzy function f (x)at three equally spaced points x i , x i+1 , x i+2 .First we suppose that x i ≤ x ≤ x i+1 ≤ x i+2 , then the Lagrange coefficients are obtained as follows : ) ) By theorem (2.1) the fuzzy Lagrange polynomial P 1 (x) with triple form (P l 1 (x), P c 1 (x), P r 1 (x)) is obtain: Also the Lagrange coefficients ℓ i (x), ℓ i+1 (x) and ℓ i+2 (x) are obtained as follows, when we suppose that then fuzzy Lagrange polynomial P 2 (x) with triple form (P l 2 (x), P c 2 (x), P r 2 (x)) is denote by: For solving fuzzy integral x i f (x)dx, we have International Scientific Publications and Consulting Services then we use fuzzy interpolation P(x) instant of fuzzy function f (x), for x i ≤ x ≤ x i+1 , For upper and lower bound of P(x) we have: we obtain: clearly for upper bound f (x, r) we obtain