Semi Analytical-Solution of Nonlinear Two Points Fuzzy Boundary Value Problems by Adomian Decomposition Method

In this paper the Adomian Decomposition Method (ADM) is employed to solve n th order (n>2) non linear two point fuzzy boundary value problems (TPFBVP). The Adomian decomposition method can be used for solving nth order fuzzy differential equations directly without reduction to first order system. We illustrate the method in numerical experiment including fourth order nonlinear TPFBVP to show the capabilities of ADM.


Introduction
Many dynamical real life problems may be formulated as a mathematical model.Many of them can be formulated either as a system of ordinary or partial differential equations.Fuzzy differential equations (FDEs) are a useful tool to model a dynamical system when information about its behavior is inadequate.FDE appears when the modeling of these problems was imperfect and its nature is under uncertainty.FDEs are suitable mathematical models to model dynamical systems in which there exist uncertainties or vagueness.These models are used in various applications including, population models [1][2][3], mathematical physics [4], and medicine [5,6].In recent year's semi -analytical methods such as the Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), Optimal Homotopy asymptotic method (OHAM) and Homotopy Analysis Method (HAM) have been used to solve fuzzy first and n th order ordinary differential equations.For n th order fuzzy initial value problems, The ADM was employed in [7] to solve second order linear fuzzy initial value problems.Abbasbandy et al. [8] used the VIM to solve linear system of first order fuzzy initial value problems.Moreover, some of these methods have been also used to obtain a semi-analytical solution of TPFBVP.VIM has been used in [9] to solve linear TPFBVP.Other method like undetermined fuzzy coefficients method has been introduced in [10] in order to obtain an approximate solution of second order linear TPFBVP.
The ADM have been introduced in [11,12] and has been applied to a wide class of deterministic and stochastic problems of mathematical and physical sciences [13][14][15].This method provides the solution as a rapidly convergent series with components that are elegantly computed.This method can be used to solve all types of linear and nonlinear equations such as differential and integral equations, so it is known as a powerful method.Another important advantage of this method is that it can reduce the size of computations, while increases the accuracy of the approximate solutions so it is known as a powerful method In this paper, our aim is to formulate ADM from crisp into fuzzy case in order to solve nonlinear n th order TPFBVP directly.To the best of our knowledge, this is the first attempt at solving the n th order TPFBVP using the ADM.The structure of this paper is as follows: In section 2, some basic definitions and notations are given about fuzzy numbers that will be used in other sections we discussed.In section 3, the structure of ADM is formulated for solving high order TPFBVP.In section 4, we present a numerical example and finally, in section 5, we give the conclusion of this study Figure 1.

Fuzzy Numbers
Fuzzy numbers are a subset of the real numbers set, and represent uncertain values.Fuzzy numbers are linked to degrees of membership which state how true it is to say if something belongs or not to a determined set Figure 2. A fuzzy number [16] µ is called a triangular fuzzy number if defined by three numbers α < β < γ where the graph of (x) µ is a triangle with the base on the interval [α,β] and vertex at x , = β and its membership function has the following form: In this paper the class of all fuzzy subsets of R will be denoted by RF and satisfy the following properties [16,17]: . µ upper semi-continuous on R, and RF is called the space of fuzzy numbers and R is a proper subset of RF.
Define the r-level set x R

Fuzzification and Defuzzification of ADM
The general structure of ADM for solving crisp n th order two point boundary value problems involving ordinary differential equations are mentioned in [18][19][20].To solve n th order TPFBVP, we need to fuzzify ADM and then defuzzify it Figure 3.
Consider the following general n th order TPFBVP subject to two point boundary conditions where 0 k n 2 ≤ ≤ − is an integer( For cases when n=4 and n=6, [21,   22]), ( ) Where W(t) is crisp or fuzzy inhomogeneous term such that , such that t; r t; r , t; r y t; r , y t; r , , y t; r , y t; r , y t; r , , y t; r by using Zadeh extension principles [24,25] we have Then we have , for all r < [0,1], Esq.( 4) -( 5) can be written as follows  and applying it on Eqs .( 6) and ( 7) we have ( ) y t; r; a r y t; r; a r L w t; r L t, t; r y t; r; a r , and y t; r; a r are initials guessing and can be defining as follows: a r a r y t; r; a r a r a r t t t 2! n 1 a r a r y t; r; a r a r a r t t t 2! n 1 where ( ) where ( ) i A t; r  are the so-called Adomian polynomials [19,20].Now let Ø t; r; a r y t; r; a r w t; r Ø t; r; a r y t; r; a r w t; r then substituting (11) and ( 12) into ( 8) and ( 9) respectively we obtain y t r a r t r a r A t r (16) According to the ADM, the components ( ) ( ) ; ; i i y t r a r   can be determined as (t; r; a (r))] y (t; r; a (r)) 2 L A (y (t; r; a (r))) y (t; r; a (r)) L A ( y (t; r; a (r))) and for the upper bound y (t; r;a (r)) [ (t; r; a (r))] y (t; r;a (r)) 2 L A (y (t; r;a (r))) y (t; r; a (r)) L A ( y (t; r; a (r))) Since the approximate solution series

Numerical Example
In this section employ OHAM on two examples of a linear and non-linear fourth order TPFBVP and present their approximate results with absolute errors.These examples were presented in [26] for the crisp case (non-fuzzy).
According to the ADM in section 3, the n components of ( ) ( ) j i y t; r;a r   can be determined as y (t; r;a (r)) [ (t; r;a (r))] y (t; r;a (r)) 2 L [y (t; r;a (r))y (t; r;a (r))] y (t; r;a (r)) L A ( y (t; r;a (r))) After ten terms of the ADM approximate solution series we have From Tables 1-3 and Figure 4 one can see that the numerical results are satisfies the convex symmetric triangular fuzzy number and we have the same results for all 0 ≤ t ≤ 2.

Conclusions
In this study, a semi-analytical method ADM was applied to obtain an approximate solution n th order nonlinear TPFBVP.A scheme based ADM approximate the solution of n th order TPFBVP has been formulated to obtain the approximate solution directly with our reduced in to first order system.From the nonlinear TPFIVP the accuracy of ADM can be determined even these equations without exact analytical solution.ADM give more accurate solution when the number of terms of the series solution was increased.Numerical examples which included nonlinear TPFBVP show the efficiency of the implemented of this semi-analytical method.The problem results satisfied the fuzzy numbers properties by taking the triangular fuzzy numbers shape.( ) y 0.8; r , we define residual absolute error [27] as follows

.
In the parametric form, a fuzzy number is represented by an ordered pair of functions ( bounded left continuous non-decreasing function over [0.1]. 2. ( ) t µ is a bounded left continuous non-increasing function over [0.1] .

4   with the inverse operator 1 4 −
Define the linear operator of Eq. (18)  , then the initial approximation guesses

.
the boundary conditions of Eq.(4.1) with the ADM solution series(23) in(18), we obtained Substitute these values in Eq. (12) to get 10-order ADM approximate solution series.The following tables show ADM approximate solution series Since Eq. (18) is without exact analytical solution, so to show the accuracy of 10-order ADM absolute errors [ ]

Table 1 :
Lower bound