Numerical solutions of fuzzy initial value problem by Adams-Moulton method

In this paper we introduce the fuzzy interpolation polynomial by forward g-difference to solve the fuzzy ordinary differential equations under generalized differentiability by Adams-Moulton methods. And then we present the general expression of these solutions. Then one example is solved with proposed method.


Introduction
The past decades, several numerical solutions of Fuzzy differential equations are studied and analyzed under the Hukuhara and Seikkala or gh-derivative [1]- [5].A new differentiability concept, the so-called gderivative, was described and studied by Bede and Stefanini [1].The Seikkala-derivative used in [6] for numerical solution of Fuzzy differential equations by predictor-corrector method.The aim of this research is to study Adams-Moulton method to solve fuzzy differential equations with the concept of gdifferentiability.
Production and hosting by ISPACS GmbH.http://www.ispacs.com/journals/cacsa/2017/cacsa-00082/International Scientific Publications and Consulting Services The paper is arranged as follows: in Section 2 we give the main definitions that will be used in the following sections, and we presen, in section 3 we obtain the fuzzy solutions of first order linear fuzzy differential equations, y ′ (t) = f(t, y(t)), y(t 0 ) = y 0 , using Adams-Moulton method, analytically.The concept of convergence defined and proved in section 3. an example is given for demonstration in the last section.

Preliminaries and notations
Let us denote by R  the class of fuzzy subsets of the real axis : R → [0,1] , satisfying the following properties: (iv) {x ∈ R; u(x) > 0}is compact, where Adenotes the closure of A. Then R  is called the space of fuzzy numbers .Obviously R ∈ R  .Here R ∈ R  is understood as = {χ x : is usual real number } .For 0 ≤ r ≤ 1 , denote [u] r = {x ∈ R|u(x) ≥ r} and [u] 0 = {x ∈ R|u(x) ≥ 0}.Then it is well-known that for any r ∈ [0,1], [u] r is a bounded closed interval.For u, v ∈ R  , and λ ∈ R, the sum u + vand the product λ.u are defined by[u means the usual addition of two intervals (subsets) of R and λ[u] r = {λx: x ∈ [u] r } means the usual product between a scalar and a subset of R.
The following properties are well-known ∀ u, v, w, e ∈ R F and (R F , D) is a complete metric space [7].Also are known the following results and concepts.
by using the forward g-difference ∆ g f ̃n = f ̃n+1 ⊝ g f ̃n , for n ≥ 1, and Y ̃( t i ) as approximation of y ̃( t i ) and thensubstituting t = t i + θ h we will get According to definition 2.1 we have Four different cases will happen in these relation for selecting the infmin and supmax magnitude.since selection of each cases causes similar interpretations, as result only one of them will be considered.One of the cases: So that By considering the initial-value problem y ̃′(t) = f ̃ (t, y ̃(t)),t 0 ≤ t ≤ T, y ̃(t 0 ) = α ̃0, Y ̃0 = α ̃0, Y ̃1 = α ̃1, ... , Y ̃m−1 = α ̃m−1 wich the (i + 1)st step in a multistep method is we know the concept of convergence for multistep method is provided that lim h→0 |Y ̃i ⊝ g y ̃(t i )| = 0.
For the two-step fuzzy Adams-Moulton method, we have seen that According to assumptions of paper, f ̃((t i , y ̃(t i )) ∈ R F , and by definition g-differentiability So we see that, defference method is convergent.

Table 1 :
Adams-Moulton three-step method with N = 10 and approximation errors at t = 0.1.