Solution of the Fuzzy Boundary Value Differential Equations Under Generalized Differentiability By Shooting Method

Available online at www.ispacs.com/jfsva Volume 212, Year 212 Article ID jfsva-136, 19 pages doi:1.5899/212/jfsva-136 Research Article Solution of the Fuzzy Boundary Value Differential Equations Under Generalized Differentiability By Shooting Method L. Jamshidi 1, L. Avazpour 2 (1)Department of Mathematics, Science and Research Branch, Islamic Azad University, Hamedan, Iran. (2)Department of Mathematics, Yasouj Branch, Islamic Azad University, Yasouj, Iran. Copyright 212 c L. Jamshidi and L. Avazpour. 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, we apply the shooting method for solving the Fuzzy Boundary Value Differential Equations (FBVDEs) of the second order under generalized differentiability. By this method an FBVDE of the second order will be replaced with two fuzzy initial value differential equations and the answers of each of them are obtained by the Adomian method. Finally via linear combination of their solutions, the fuzzy solution will be obtained. Keywords : Shooting method; Fuzzy Boundary Value Differential Equations ; Generalized differentiability ; Adomian method. 1 Introduction The theory of fuzzy differential equations (FDEs) has attracted much attention in recent times because this theory represents a natural way to model dynamical systems under uncertainty. The concept of the fuzzy derivative was first introduced by Chang and Zadeh [12]; it was followed up by Dubois and Prade [16], who used the extension principle in their approach. The study of fuzzy differential equations has been initiated as an independent subject in conjunction with fuzzy valued analysis [17] and [22] and set-valued differential equations [21]. Initially, the derivative for fuzzy valued mappings was developed by Puri and Ralescu [24], that generalized and extended the concept of Hukuhara differentiability (H-derivative) for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative, Kaleva [19] started to develop a theory for FDE. In the last few Corresponding author. Email address: avazpour.l@gmail.com, Tel:+989173414337. 1

years, many works have been done by several authors in theoretical and applied fields (see [3, 1, 8, 1, 4, 13, 19, 23]. A variety of exact, approximate, and purely numerical methods are available to find the solution of a fuzzy initial value problem (FIVP). The pioneers in the field have usually pursued methods based on the Hukuhara derivative. First order linear fuzzy differential equations or systems are studied under different interpretations by several papers (see [18, 14, 6, 7, 2]). It is an important problem that we know a differential equation has a unique solution. There are many theorems and reasonable conditions for this aim. The shooting method, can approximate the unique solution for a linear FBVDE. In this paper, we want propose the shooting method for solving these equations. Therefore, this process would be developed through four sections. In section 2, we recall some fundamental results on fuzzy numbers. In section 3 we define the fuzzy boundary value problem and then, we obtain a solution by shooting method of it. Numerical examples are proposed in section 4 and last section is devoted to conclusions. 2 Preliminaries The basic definition of fuzzy numbers is given in [28]. We denote the set of all real numbers by R and the set of all fuzzy numbers on R is indicated by R F. A fuzzy number is a mapping u : R [, 1] with the following properties: (a) u is upper semi-continuous, (b) u is fuzzy convex, i.e., u(λx + (1 λ)y) min{u(x), u(y)} for all x, y R, λ [, 1], (c) u is normal, i.e., x R for which u(x ) = 1, (d) supp u = {x R u(x) > } is the support of the u, and its closure cl(supp u) is compact. Definition 2.1. [15]. An arbitrary fuzzy number is represented by an ordered pair of functions (u(α), ū(α)), α 1 that, satisfies the following requirements: u(α) is a bounded left continuous nondecreasing function over [,1], with respect to any α. ū(α) is a bounded left continuous nonincreasing function over [,1], with respect to any α. u(α) ū(α), α 1 Then the α-level set [u] α of a fuzzy set u on R is defined as: [u] α = {x R : u(x) α}, for each α (, 1], (2.1) and for α = [u] = [u] α, (2.2) α (,1] where A denotes the closure of A. For u, v R F and λ R the sum u + v and the product λu are defined by [u + v] α = [u] α + [v] α, [λu] α = λ[u] α, α [, 1] where [u] α + [v] α means the usual addition of two 2

intervals (subsets) of R and λ[u] α means the usual product between a scalar and a subset of R. The metric structure is given by the Hausdorff distance D : R F R F R + {}, D(u, v) = sup max{ u(α) v(α), ū(α) v(α) }. (2.3) α [,1] where [u] α = [u(α), ū(α)] and [v] α = [v(α), v(α)]. (R F, D) is a complete metric space and the following properties are well known:(see e.g. [17]) D(u + w, v + w) = D(u, v) u, v, w R F, D(ku, kv) = k D(u, v) k R, u, v R F, D(u + v, w + e) D(u, w) + D(v, e) u, v, w, e R F. Definition 2.2. Consider x, y R F. If there exists z R F such that x = y + z, then z is called the H-difference of x and y and it is denoted by x y. In this paper, the sign always stands for H-difference and note that x y x + ( y). Also throughout of paper is assumed the Hukuhara difference and Hukuhara generalized differentiability are existed. Let us recall the definition of strongly generalized differentiability introduced in [5]. Definition 2.3. [5]. Let F : I R F. For t I, we say F is differentiable at t, if there exists an element F (t ) R F such that either (i) For all h > sufficiently close to, the H-differences F (t h) exist and the limits (in the metric D ) lim F (t +h) F (t ) h + h = lim F (t ) F (t h) h + h = F (t ), or F (t + h) F (t ), F (t ) (ii) For all h > sufficiently close to, the H-differences F (t ) F (t + h), F (t h) F (t ) exist and the limits (in the metric D ) lim F (t ) F (t +h) h + h = lim F (t h) F (t ) h + h = F (t ). or (iii) For all h > sufficiently close to, the H-differences F (t + h) F (t ), F (t h) F (t ) exist and the limits (in the metric D ) lim F (t +h) F (t ) h + h = lim F (t h) F (t ) h + h = F (t ), or 3

(iv) For all h > sufficiently close to, the H-differences F (t ) F (t + h), F (t ) F (t h) exist and the limits (in the metric D ) lim h + F (t ) F (t +h) h = lim h + F (t ) F (t h) h = F (t ). Remark 2.1. In [5], the authors consider four cases for derivatives. Here we only consider the two first cases of Definition 2.3 in [5]. In the other cases, the derivative is trivial because it is reduced to a crisp element (for details see Theorem 7 in [5]). Definition 2.4. Let F : I = (a, b) R F. We say F is (1)-differentiable on I if F be differentiable in the sense (1) of Definition 2.3 and similarly F is (2)-differentiable on I if F be differentiable in the sense (2) of Definition 2.3. Theorem 2.1. [11]. Let F : I R F and [F ] α = [f(t, α), g(t, α)] for each α [, 1]. (i) If F is (1)-differentiable then f(t, α) and g(t, α) are differentiable functions and [F ] α = [f (t, α), g (t, α)]. (ii) If F is (2)-differentiable then f(t, α) and g(t, α) are differentiable functions and [F ] α = [g (t, α), f (t, α)]. Proof. see[11]. Definition 2.5. [1] Let F : (a, b) R F and t (a, b). We define the nth-order differential of f as follows: Let F : (a, b) R F and t (a, b). We say F is strongly generalized differentiable of the nth-order at t, if there exists an element F (t ) R F such that s = 1,..., n (i) For all h > sufficiently close to, the H-differences F (s 1) (t + h) F (s 1) (t ), F (s 1) (t ) F (s 1) (t h) exist and the limits (in the metric D ) or lim h + F (s 1) (t +h) F (s 1) (t ) h = lim h + F (s 1) (t ) F (s 1) (t h) h = F (s) (t ), (ii) For all h > sufficiently close to, the H-differences F (s 1) (t ) F (s 1) (t + h), F (s 1) (t h) F (s 1) (t ) exist and the limits (in the metric D ) or lim h + F (s 1) (t ) F (s 1) (t +h) h = lim h + F (s 1) (t h) F (s 1) (t ) h = F (s) (t ), (iii) For all h > sufficiently close to, the H-differences F (s 1) (t + h) F (s 1) (t ), F (s 1) (t h) F (s 1) (t ) exist and the limits (in the metric D ) or lim h + F (s 1) (t +h) F (s 1) (t ) h = lim h + F (s 1) (t h) F (s 1) (t ) h = F (s) (t ), 4

(iv) For all h > sufficiently close to, the H-differences F (s 1) (t ) F (s 1) (t + h), F (s 1) (t ) F (s 1) (t h) exist and the limits (in the metric D ) lim h + F (s 1) (t ) F (s 1) (t +h) h = lim h + F (s 1) (t ) F (s 1) (t h) h = F (s) (t ). 3 The linear shooting method In this section, we study the linear inhomogeneous fuzzy boundary value differential equation of the second order as following y = p(t)y + q(t)y + r(t), a t b, y(a) = β, y(b) = γ, (3.4) so that p(t), q(t) and r(t) are continuous polynomials and the values of q(t) are positive and also β, γ R F. By using the procedure of the shooting method, we can obtain two fuzzy initial value differential equations from Eq. (3.4) as follows y = F (t, y(t), y (t)) = p(t)y + q(t)y + r(t), y(a) = β, y (a) =, (3.5) y = p(t)y + q(t)y, y(a) =, y (a) = 1. (3.6) Let y 1 denote the solution to Eq. (3.5), y1 denote the solution to Eq. (3.6) and assume that y(b) then y(x) = y 1 (x) + γ y 1(b) y1 (b) y1(x) (3.7) is the unique solution to the linear boundary-value problem (3.4), [9]. Our technique for solving every one of Eqs. (3.5) and (3.6) is based on the selection of derivative type in the fuzzy differential equation that mention in Definition 2.5. In proceeding we consider the polynomials in two cases. 3.1 p(t), q(t), r(t) > It means that, t [a, b] the quantity of the polynomials are positive. We can translate (3.5) to a system of second-order linear differential equations and based on the type of derivatives consider 4 cases. (We act about (3.6) the spitting image with (3.5)). 5

Definition 3.1. [27]. Let y : [a, b] R F be a fuzzy-valued function and n, m = 1, 2. One says y is an (n, m)-solution for problem (3.5) if D n (1) y(t), D n,my(t) (2) exist and D n,my(t) (2) = p(t)d n (1) y(t) + q(t)y(t) + r(t), y(a) = β, D n (1) y(a) =. Let y(t, α) = [y(t, α), y(t, α)] be an (n, m)-solution for (3.5). To find it, utilizing theorem 2.1 and considering the initial values, we can translate (3.5) to a system of second-order linear ordinary differential equation, namely corresponding (n,m) system. (1,1)system y (t, α) = p(t) y (t, α) + q(t) y(t, α) + r(t), y (a, α) =, y (a, α) =. (1,2)system y (a, α) =, y (a, α) =. (2,1)system y (a, α) =, y (a, α) =. (2,2)system y (a, α) =, y (a, α) =. 6

3.2 p(t) <, r(t) <, q(t) > Now, t [a, b] the quantity of the polynomial q(t) is positive and P (t), r(t) are negative. As same as the previous subsection we have the following systems: (1,1)system y (a, α) =, y (a, α) =. (1,2)system y (a, α) =, y (a, α) =. (2,1)system y (a, α) =, y (a, α) =. (2,2)system y (t, α) = p(t) y (t, α) + q(t) y(t, α) + r(t), y (a, α) =, y (a, α) =. In fact we translate a fuzzy initial value differential equation into a system of ODEs. By the following theorem we show that the solution of the fuzzy initial value differential 7

equation will be equivalent to the system of ODEs. Theorem 3.1. Let us consider FIVP (3.5) where F : [t, t + a] R F R F R F such that is (i) [F (t, x(t), x (t))] α = [f(t, x, x )(α), f(t, x, x )(α)] for each α [, 1], (ii) The functions f(t, x, x )(α) and f(t, x, x )(α) are equi-continuous, i.e. for any ε > there exists δ > such that for any (t, y, z), (t 1, y 1, z 1 ) [t, t + a, t + b] R 2, we have f(t, y, z)(α) f(t 1, y 1, z 1 )(α) < ε and f(t, y, z)(α) f(t 1, y 1, z 1 )(α) < ε whenever (t 1, y 1, z 1 ) (t, y, z) < δ. (iii) [F (t, x(t), x (t))] α is named uniformly bounded on any bounded set, if exist L > such that f(t, y 1, z 1 )(α) f(t, y 2, z 2 )(α) < L max { t 1 t 2, y 1 y 2, z 1 z 2 }, and also f(t, y 1, z 1 )(α) f(t, y 2, z 2 )(α) < L max { t 1 t 2, y 1 y 2, z 1 z 2 }. Then, for (n,m)-differentiability, the FIVP (3.5) and the corresponding (n,m) system are equivalent. Proof. The equi-continuity of f(t, x, x )(α) and f(t, x, x )(α) implies the continuity of the function F. Further, the Lipschitz property in (iii) ensures that F satisfies a Lipschitz property as follows sup max{ f(t, x, x )(α) f(t, y, y )(α), f(t, x, x )(α) f(t, y, y )(α) } α [,1] i.e., L sup max{ y(α) x(α), y (α) x (α) } α [,1] D(F (t, x, x ), F (t, y, y )) L D(x, y, x, y ). By the continuity of F, from this last Lipschitz condition and the bounded condition in (i) and (ii) it follows that fuzzy initial value differential equation has a unique solution. The solution of the fuzzy initial value differential equation is Hukuhara differentiable, so by Theorem 2.1, the functions x(t, α) and x(t, α) are differentiable and this conclude that (x(t, α), x(t, α)) is a solution of system of ODEs. Conversely, Let us suppose a solution x(t, α) and x(t, α), with fixed α [, 1], for the system of ODEs, (this solution exists by (ii)). Also, the Lipschitz condition implies the existence and uniqueness of the fuzzy solution x(t). Now, since x(t) is Hukuhara-differentiable, x(t, α) with the endpoints representation x(t, α) and x(t, α) (which are obviously valid level 8

sets of a fuzzy-valued function) is a solution of system of ODEs. Since the solution of system of ODEs is unique, we have x(t, α) = [x(t, α), x(t, α)], which implies that the fuzzy initial value differential equation and the system of ODEs are equivalent. Now we solve the obtained ordinary differential equation systems. In this article, the Adomian method will be applied, which is discussed in the next section. 4 Adomian Method for fuzzy differential equations In this section, we only consider the (1,1) system and solve it by Adomain decomposition method, since the other cases are analogous. For the sake of simplicity, we have y(t, α) = y and y(t, α) = y. Let y 1 = y, y 2 = y and also y 1 = y, y 2 = y. By using the operator L = d dt we have { Ly1 = y 2, Ly 2 = p(t) y 2 + q(t) y 1 + r(t). (4.8) Thus y 1 = t a y 2ds + y 1 (a), y 2 = t a (p(s) y 2 + q(s) y 1 + r(s))ds + y 2 (a). (4.9) And also { Ly1 = y 2, Ly 2 = p(t) y 2 + q(t) y 1 + r(t). (4.1) Thus y 1 = t a y 2ds + y 1 (a), y 2 = t a (p(s) y 2 + q(s) y 1 + r(s))ds + y 2 (a). (4.11) Where y 1 (a) = β, y 2 (a) =, y 1 (a) = β and y 2 (a) =. Adomian decomposition method gives the solutions y and y as infinite series y i = j= y i,j, y i = j= y i,j, i = 1, 2. In this paper, we denote the n-th order approximate solution of y and y by y in = n j= y i,j, y in = n j= y i,j, i = 1, 2, and n >. We can calculate these series term by term. Thus we have 9

y 1, + y 1,1 +... = t a (y 2, + y 2,1 +...)ds + y 1 (a), y 2, + y 2,1 +... = t a [p(s)(y 2, + y 2,1 +...) + q(s)(y 1, + y 1,1 +...) + r(s)]ds + y 2 (a), y 1, + y 1,1 +... = t a (y 2, + y 2,1 +...)ds + y 1 (a), y 2, + y 2,1 +... = t a [p(s)(y 2, + y 2,1 +...) + q(s)(y 1, + y 1,1 +...) + r(s)]ds + y 2 (a). (4.12) With comparing the both side of the every one of the equations in the above system the first term will be obtained as follow: y 1, = y 1 (a), y 1,(n+1) = t a y 2,nds, y 2, = t a r(s)ds + y 2(a), y 2,(n+1) = t a [p(s)y 2,n + q(s)y 1,n ]ds, y 1, = y 1 (a), y 1,(n+1) = t a y 2,nds, y 2, = t a r(s)ds + y 2(a), y 2,(n+1) = t a [p(s)y 2,n + q(s)y 1,n ]ds. (4.13) Notice the following examples. 5 Examples Example 5.1. Consider the following fuzzy boundary value differential equation of the second order y (t) = y (t) + y(t) + t, t 1 y() =, y(1) = 1, (5.14) so that and 1 are the triangular fuzzy number having α-level sets [] α = [α 1, 1 α] and [1] α = [α, 2 α], respectively. By the shooting method, Eq. (5.14) is replaced with two following fuzzy initial value differential equations: (i) y (t) = y (t) + y(t) + t [y()] α = [α 1, 1 α], [y ()] α = [, ], α [, 1] 1

(ii) y (t) = y (t) + y(t) [y()] α = [, ], [y ()] α = [1, 1], α [, 1]. Suppose that the solution of (i) be [y 1, y 1 ] and the solution of (ii) be [y1, y 1 ], thus the (1,1) solution for n = 2 is as follows: In the case (i) for n = we have: y 1, = α 1, y 2, = t sds + = t2 2, s 2 2 ds = t3 6, y 2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ s2 2 + α 1]ds = t3 6 y 1,1 = t a y 2,ds = t + (α 1)t, y 1, = 1 α, y 2, = t a r(s)ds + y 2(a) = t sds + = t2 2, y 1,1 = t a y 2,ds = t s 2 2 ds = t3 6, y 2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ s2 2 + 1 α]ds = t3 6 + (1 α)t. For n = 1 we have: y 1,2 = t a y 2,1ds = t s 3 6 + (α 1)sds = t4 24 + (α 1) t2 2, y 2,2 = t a [p(s)y 2,1 + q(s)y 1,1 ]ds = t [ s3 6 + (α 1)s + s3 6 ]ds = t4 12 + (α 1) t2 2, y 1,2 = t a y 2,1ds = t s 3 t4 6 + (1 α)sds = 24 + (1 α) t2 2, y 2,2 = t a [p(s)y 2,1 + q(s)y 1,1 ]ds = t [ s3 6 + (1 α)s + s3 t4 6 ]ds = 12 + (1 α) t2 2. For n = 2 we have: y 12 = (α 1) + t3 6 + t4 24 + (α 1) t2 2, y 12 = (1 α) + t3 6 + t4 24 + (1 α) t2 2. In the case (ii) for n = we have: y 1, =, y2, = t ds + 1 = 1, y1,1 = t a y 2, ds = t 1ds = t, y2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [1]ds = t, y1, =, y2, = t a r(s)ds + y 2 (a) = t ds + 1 = 1, y1,1 = t a y 2, ds = t 1ds = t, y2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [1]ds = t. For n = 1 we have: 11

y 1,2 = t sds = t2 2, y 2,2 = t [s + s]ds = t2, y1,2 = t a y 2,1 ds = t sds = t2 2, y2,2 = t a [p(s)y 2,1 + q(s)y 1,1 ]ds = t [s + s]ds = t2. For n = 2 we have: y 1 2 = y 1 2 = + t + t2 2. Now by the Eq. (3.7) the fuzzy solution will be y = y(t, α) = (α 1) + t3 6 + t4 24 + (α 1) t2 2 + ( α [(α 1)+ 1 3 y = y(t, α) = (1 α) + t3 6 + t4 24 + (1 α) t2 2 + ( 2 α [(1 α)+ 1 3 24 +(α 1) 12 2 ] 1+ 12 2 24 +(1 α) 12 2 ] 1+ 12 2 The (1,2) solution for Eq. (5.14) in the case (i) for n = is: y 1, = α 1, y 2, = t sds + = t2 2, y 1,1 = t a y 2,ds = t s 2 2 ds = t3 6, y 2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ s2 2 + 1 α]ds = t3 6 + (1 α)t, )(t + t2 2 ), )(t + t2 2 ). y 1, = 1 α, y 2, = t a r(s)ds + y 2(a) = t sds + = t2 2, y 1,1 = t a y 2,ds = t s 2 2 ds = t3 6, y 2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ s2 2 + α 1]ds = t3 6 + (α 1)t. For n = 1 we have: { y 1,2 = t a y 2,1ds = t s 3 t4 6 + (1 α)sds = 24 + (1 α) t2 2, y 1,2 = t a y 2,1ds = t s 3 t4 6 + (α 1)sds = 24 + (α 1) t2 2. For n = 2 we have: y 12 = (α 1) + t3 6 + t4 24 + (1 α) t2 2, y 12 = (1 α) + t3 6 + t4 24 + (α 1) t2 2. In the case (ii) for n = we have: 12

y1, =, y2, = t ds + 1 = 1, y1,1 = t a y 2, ds = t 1ds = t, y2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [1]ds = t, y1, =, y2, = t a r(s)ds + y 2 (a) = t ds + 1 = 1, y1,1 = t a y 2, ds = t 1ds = t, y2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [1]ds = t. For n = 1 we have: y 1,2 = t sds = t2 2, y 2,2 = t [s + s]ds = t2, y1,2 = t a y 2,1 ds = t sds = t2 2, y2,2 = t a [p(s)y 2,1 + q(s)y 1,1 ]ds = t [s + s]ds = t2. For n = 2 we have: y 1 2 = y 1 2 = + t + t2 2. Now by the Eq. (3.7) the fuzzy solution will be y = y(t, α) = (α 1) + t3 6 + t4 24 + (1 α) t2 2 + ( α [(α 1)+ 1 3 y = y(t, α) = (1 α) + t3 6 + t4 24 + (α 1) t2 2 + ( 2 α [(1 α)+ 1 3 We obtain the (2,1) solution for (i) and (ii) in the same way: y 1 = (α 1) + (α 1) t2 2 + t3 6 + t4 24, y 1 = (1 α) + (1 α) t2 2 + t3 6 + t4 24. y 1 = y 1 = t + t2 2. 24 +(1 α) 12 2 ] 1+ 12 2 24 +(α 1) 12 2 ] 1+ 12 2 )(t + t2 2 ), )(t + t2 2 ). y = y(t, α) = (α 1) + t3 6 + t4 24 + (α 1) t2 2 + ( α [(α 1)+ 1 3 y = y(t, α) = (1 α) + t3 6 + t4 24 + (1 α) t2 2 + ( 2 α [(1 α)+ 1 3 24 +(α 1) 12 2 ] 1+ 12 2 24 +(1 α) 12 2 ] 1+ 12 2 )(t + t2 2 ), Finally the (2,2) solution for (i) and (ii) and also the solution of Eq. (5.14) is )(t + t2 2 ). 13

y 12 = (α 1) + t3 6 + t4 24 + (1 α) t2 2, y 12 = (1 α) + t3 6 + t4 24 + (α 1) t2 2. y 1 2 = y 1 2 = + t + t2 2. Now by the Eq. (3.7) the fuzzy solution will be y = y(t, α) = (α 1) + t3 6 + t4 24 + (1 α) t2 2 + ( α [(α 1)+ 1 3 y = y(t, α) = (1 α) + t3 6 + t4 24 + (α 1) t2 2 + ( 2 α [(1 α)+ 1 3 24 +(1 α) 12 2 ] 1+ 12 2 24 +(α 1) 12 2 ] 1+ 12 2 )(t + t2 2 ), )(t + t2 2 ). Example 5.2. Consider the following fuzzy boundary value differential equation of the second order y (t) = y (t) + y(t) t, t 1 y() =, y(1) = 1, (5.15) so that and 1 are the triangular fuzzy number having α-level sets [] α = [α 1, 1 α] and [1] α = [α, 2 α], respectively. By the shooting method Eq. (5.15) is replaced with two fuzzy initial value differential equations as follows (i) y (t) = y (t) + y(t) t [y()] α = [α 1, 1 α], [y ()] α = [, ], α [, 1] (ii) y (t) = y (t) + y(t) [y()] α = [, ], [y ()] α = [1, 1], α [, 1]. Suppose that the solution of (i) be [y 1, y 1 ] and the solution of (ii) be [y1, y 1 ], thus the (1,1) solution for n = 2 is as follows: In the case (i) for n = we have: y 1, = α 1, y 2, = t sds + = t2 2, y 1,1 = t a y 2,ds = t s2 2 ds = t3 6, y 2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ ( s2 2 ) + α 1]ds = t3 6 + (α 1)t, y 1, = 1 α, y 2, = t a r(s)ds + y 2(a) = t sds + = t2 2, s2 2 ds = t3 6, y 1,1 = t a y 2,ds = t y 2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ ( s2 2 ) + 1 α]ds = t3 6 + (1 α)t. 14

For n = 1 we have: { y 1,2 = t a y 2,1ds = t s 3 t4 6 + (α 1)sds = 24 + (α 1) t2 2, y 1,2 = t a y 2,1ds = t s 3 t4 6 + (1 α)sds = 24 + (1 α) t2 2. For n = 2 we have: y 12 = (α 1) t3 6 + t4 24 + (α 1) t2 2, y 12 = (1 α) t3 6 + t4 24 + (1 α) t2 2. In the case (ii) for n = we have: y1, =, y2, = t ds + 1 = 1, y1,1 = t a y 2, ds = t 1ds = t, y2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ 1]ds = t, y1, =, y2, = t a r(s)ds + y 2 (a) = t ds + 1 = 1, y1,1 = t a y 2, ds = t 1ds = t, y2,1 = t a [p(s)y 2, + q(s)y 1, ]ds = t [ 1]ds = t. For n = 1 we have: y 1,2 = t sds = t2 2, y 2,2 = t a [p(s)y 2,1 + q(s)y 1,1 ]ds = t [ ( s) + s]ds = t2, y1,2 = t a y 2,1 ds = t sds = t2 2, y2,2 = t a [p(s)y 2,1 + q(s)y 1,1 ]ds = t [s + s]ds = t2. For n = 2 we have: y 1 2 = y 1 2 = + t t2 2. Now by the Eq. (3.7) the fuzzy solution will be y = y(t, α) = (α 1) t3 6 + t4 24 + (α 1) t2 2 + ( α [(α 1) 1 3 y = y(t, α) = (1 α) t3 6 + t4 24 + (1 α) t2 2 + ( 2 α [(1 α) 1 3 24 +(α 1) 12 2 ] 1 12 2 24 +(1 α) 12 2 ] 1 12 2 The (1,2) solution for (i) and (ii) and also the solution of Eq. (5.15) is: )(t t2 2 ), )(t t2 2 ). 15

y 12 = (α 1) t3 6 + t4 24 + (1 α) t2 2, y 12 = (1 α) t3 6 + t4 24 + (α 1) t2 2. y 1 2 = y 1 2 = + t t2 2. Now by the Eq. (3.7) the fuzzy solution will be y = y(t, α) = (α 1) t3 6 + t4 24 + (1 α) t2 2 + ( α [(α 1) 1 3 y = y(t, α) = (1 α) t3 6 + t4 24 + (α 1) t2 2 + ( 2 α [(1 α) 1 3 24 +(1 α) 12 2 ] 1 12 2 24 +(α 1) 12 2 ] 1 12 2 The (2,1) solution for (i) and (ii) and also the solution of Eq. (5.15) is: y 12 = (α 1) t3 6 + t4 24 + (α 1) t2 2, y 12 = (1 α) t3 6 + t4 24 + (1 α) t2 2. y 1 2 = y 1 2 = + t t2 2. Now by the Eq. (3.7) the fuzzy solution will be: y = y(t, α) = (α 1) t3 6 + t4 24 + (α 1) t2 2 + ( α [(α 1) 1 3 y = y(t, α) = (1 α) t3 6 + t4 24 + (1 α) t2 2 + ( 2 α [(1 α) 1 3 24 +(α 1) 12 2 ] 1 12 2 24 +(1 α) 12 2 ] 1 12 2 )(t t2 2 ), )(t t2 2 ). )(t t2 2 ), Finally the (2,2) solution for (i) and (ii) and also the solution of Eq. (5.15) is: y 12 = (α 1) t3 6 + t4 24 + (1 α) t2 2, y 12 = (1 α) t3 6 + t4 24 + (α 1) t2 2. y 1 2 = y 1 2 = + t t2 2. Now by the Eq. (3.7) the fuzzy solution will be y = y(t, α) = (α 1) t3 6 + t4 24 + (1 α) t2 2 + ( α [(α 1) 1 3 y = y(t, α) = (1 α) t3 6 + t4 24 + (α 1) t2 2 + ( 2 α [(1 α) 1 3 24 +(1 α) 12 2 ] 1 12 2 24 +(α 1) 12 2 ] 1 12 2 )(t t2 2 ). )(t t2 2 ), )(t t2 2 ). 16

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