The Numerical Solution of Full Fuzzy Algebraic Equations

In this article, we try to solve a full fuzzy algebraic equation with a fuzzy variable and fuzzy coefficients. This can be done by a numerical iterative process. We offer an algorithm to produce a sequence that may converge to a root of such an equation.


Introduction
In many purposes of a system, we may need to solve a fuzzy equation. Zadeh introduced some properties of fuzzy equations in [1] and some other types of such equations were discussed by some researchers [2,3] and they tried to solve them analytically. Solving an algebraic equation of degree at most 3 has no analytical method even in crisp form. So it is necessary to find the numerical solution of an equation. In [4], fuzzy nonlinear equations with fuzzy unknown were considered. A research effort has been done on algebraic fuzzy equations by using neural network [5]. Two numerical iterative methods have been presented in [6,7] to find the roots of an algebraic fuzzy equation of degree n, one with fuzzy coefficients and crisp unknown and the other one with crisp coefficients and fuzzy unknown. In the present essay, we introduce a numerical method to solve a full fuzzy algebraic equation, such that all numbers are fuzzy, both the unknown variable and coefficients. The organisation of the present paper is as follows. In Section 2, we represent a full fuzzy algebraic equation and we discuss solving a nonlinear system of equations. In Section 3, an algorithm is introduced to numerically solving a full fuzzy algebraic fuzzy equation. Some examples are given in Section 4, and Section 5 contains the conclusion of this work.

Preliminaries
The presentation of a fuzzy numberf by a pair of functions as (f , f ) is called the parametric form off , such that over [0, 1], we have: f is a left continuous and monotonically increasing function, f is a left continuous and monotonically decreasing function, and f ≤ f [8][9][10][11][12]. We consider the set of all fuzzy numbers by F. Supposef = (f , f ) andg = (g, g) are both belonging to F. Then some operations are defined as follows: It is obvious that for non-negative fuzzy numbersf andg we havefg = (f .g, f .g), and for non-positive fuzzy numbersf andg we havẽ fg = (f .g, f .g). Definition 2.1: r−cut of a fuzzy numberṽ is the interval [v(r), v(r)], and we use the notation [ṽ] r for it. Definition 2.2:P n (x) is a fuzzy polynomial of degree at most n with fuzzy coefficients, if there are some fuzzy numbersã 0 , . . . ,ã n , such that For a positive integer n, an algebraic equation with full fuzzy feature of degree n, can be defined asP whereã 0 , . . .ã n ,x ∈ F,ã n = 0 andb ∈ F.

Definition 2.3:
A 'm-degree polynomial form' fuzzy numberṽ is defined by a pairṽ = (p m , q m ), such that p m and q m are two polynomials of degree at most m [13][14][15].
We use PF m for the set of all m-degree polynomial form fuzzy numbers [13,14,16]. Let where d ≥ s. To solve the system of equations by Gauss-Newton method [17], we consider A(Z) as the Jacobian of F: ∂y j ], j = 1, . . . , s and i = 1, . . . , d. Considering Z (0) as an initial vector, we try to improve this guess, so the system A(Z (k) )H (k) = −F(Z (k) ) should be solved in each iteration and then the approximated solution will be improved by considering Z (k+1) = Z (k) + H (k) . This system of equation can be solved by a least square method as the following: Some conditions on convergence and uniqueness of the solution of (3) are proposed [18][19][20]. Letb = (b, b) andã j = (a j , a j ) for j = 0, . . . , n are known fuzzy numbers. We try to solve the following full fuzzy algebraic equation from degree n: in which we haveã n = 0. We considerx = (x, x) as the unknown fuzzy root of Equation (5).

Solving the Algebraic Fuzzy Equation
Let the unknownx be a fuzzy number belonging to PF k . Also let we haveã j ∈ PF m and b ∈ PF l , in which l ≤ nk + m. We discuss about Equation (5), by four cases: • First we consider the case that the unknown fuzzy numberx be non-negative and for all j = 0, . . . , n, we haveã j ≥ 0. By these assumptions, Equation (5) changes to the two following equations: n j=0 a j x j = b.
Now by the assumptions on fuzzy numbers, we consider that By these assumptions, Equation (6) changes to Sinceã j ∈ PF k ,x ∈ PF m , we havẽ By rewriting (8) by an ordering on the power of r and by considering (9), for i = 0, . . . , nk + m, the coefficient of r i is a functionL i and we have After taking the corresponding coefficients in both sides equal, for i = 0, . . . , nk + m, we have an equation:L Similarly, by rewriting (7) by an ordering on the power of r and by considering (9), for i = 0, . . . , nk + m, the coefficient of r i is a functionÛ i and we have and by taking the corresponding coefficients in both sides equal, for i = 0, . . . , nk + m, we have an equation:Û • Letx be non-negative and for some j,ã j be negative. Thus Equation (5) changes to the two following equations: • In the case thatã j 's are all non-negative andx is negative, Equation (5) changes to the following equations: j even j even • Ifx be negative and for some j,ã j be negative, then Equation (5) changes to the following equations: and a j ≥0 j even In the last three cases, rewriting the equations by an ordering on the power of r leads us to introduce two functionsL i andÛ i for i = 0, . . . , nk + m, which are the coefficients of r i (L i respect to (14), (16) or (18) andÛ i respect to (15), (17) or (19)). For i = 0, . . . , nk + m both functionsL i andÛ i are functions of α j 's and β j 's. Thus we havê andÛ Let us introduce two functions L i and U i as follows: and Therefore we have Equation (24) is a nonlinear system of equations with d = 2(nk + m + 1) equations and s = 2k + 2 unknowns, so it can be solved by an iterative Gauss-Newton method. Suppose that t = nk + m Defining and we have A and B are used instead of A(Z) and B(Z), respectively. By solving this system with an initial vector Z (0) , a sequence {Z (k) } will be obtained as follows: we consider an initial vector Z (0) . For k = 1, 2, . . ., we compute A (k) and B (k) and then we solve A (k) H (k) = B (k) by a least square method: and the obtaining vector will be improved by Again A (k) and B (k) are used instead of A(Z (k) ) and B(Z (k) ), respectively. If {Z (k) } converges to a vector Z * ; thenx * is a solution of (5) where Therefore the algorithm will be as follows:
(2) Do the following steps until convergence condition is yield .

Corollary 3.2:
The algebraic fuzzy equation of degree n (5) has at most n fuzzy roots with distinct cores.

Theorem 3.3: If the equation is crisp then the sequence is given by the following recurrence equation:
x ν+1 = x ν − a n x n ν + . . .

Example 4.7:
Suppose that a car is moving in a straight line with the velocityṽ. If this car breaks, after a timet it will stop. The distance satisfies in the following relation: in whichã is the constant negative acceleration andt is the stopping time, and all variables are linguistic (Figure 1). If acceleration is about −6 m s 2 and the velocity is about 30 m s , then by solving the equation, it will be clear that the car will stop after about 5 s (Figures 2 and 3).   Example 4.8: Consider the following fuzzy linear differential equation of order n: a n d nx dt n + · · · +ã 1 dx dt +ã 0x =0, where t is the independent crisp variable. In order to find the solution of this ODE one must solve the following fuzzy algebraic equation of degree n: a nw 2 + · · · +ã 1w +ã 0 =0.  For example by consideringã 2 ,ã 1 andã 0 as shown in Figure 4, the solution of quadratic equation is the fuzzy number which is shown in Figure 5 and the solution of related differential equation isx(t) = exp((1 + 0.35r, 1.6 − 0.25r)t) (Figures 4 and 5).

Conclusion
In this work, we presented a method to find a numerical solution of a full fuzzy algebraic equation. The method can be used for any kind of fuzzy equations, with non-positive or non-negative roots and with non-positive or non-negative coefficients. Some possible future research directions may consist of the more general case than the four cases which considered in this paper, and using the proposed for multiple roots of an equation.