Solving fuzzy differential equations by differential

Information Sciences 179 (2009) 956–966

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Solving fuzzy differential equations by differential transformation method T. Allahviranloo a,*, N.A. Kiani a,1, N. Motamedi b a b

Department of Mathematics, Science and Research Branch, Islamic Azad University, Hesarak, Poonak, Tehran 14778, Iran Department of Mathematics, K.N.Toosi University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 7 January 2007 Received in revised form 10 November 2008 Accepted 12 November 2008

Keywords: Fuzzy differential equations Fuzzy number Fuzzy-valued function Differential transformation method Generalized differentiability

a b s t r a c t An extension of the differential transformation method (DTM), which is an analytical– numerical method for solving the fuzzy differential equation (FDE), is given. The concept of generalised H-differentiability is used. This concept is based on an enlargement of the class of differentiable fuzzy mappings; to define this, the lateral Hukuhara derivatives are considered. The proposed algorithm is illustrated by numerical examples, and some error comparisons are made with other methods for solving a FDE. Ó 2008 Published by Elsevier Inc.

1. Introduction Fuzzy set theory is a powerful tool for modelling uncertainty and for processing vague or subjective information in mathematical models, which has been applied to a wide variety of real problems, for instance, the golden mean [14], practical systems [25], quantum optics and gravity [26], synchronising hyperchaotic systems [32], medicine [1], and engineering problems. In particular, the fuzzy differential equation is a very important topic from the theoretical point of view [16,20–24,27,29] as well as the applied point of view [2–4]; for example, in population models [9] and in hydraulics modelling [8]. The concept of the fuzzy derivative was first introduced by Chang and Zadeh [11], followed by Dubois and Prade [15], who used the extension principle. Other methods were discussed by Puri and Ralescu [28] and Goetschel and Voxman [18]. The use of fuzzy differential equations is a natural way to model dynamical systems under possibilistic uncertainty [31]. The concept of differential equations in a fuzzy environment was formulated by Kaleva [20]. the last few years, several authors have produced a wide range of results in both the theoretical and applied fields [8,13,20,21,29]. 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. However, in some cases this approach suffers from a grave disadvantage: the solution has such property that diam (x(t)) is non-decreasing in t, i.e. the solution is irreversible in possibilistic terms. Therefore, this interpretation is not a good generalisation of the associated crisp case. It is assumed that this problem is due to the fuzzification of the derivative used in the formulation of the FDE. Also, most known methods of solving FDEs are computationally intensive, because they are trial-and-error in nature, or need complicated symbolic computations. In this paper, we are going to get rid of these difficulties by using a more general definition of the derivative for fuzzy mappings,

* Corresponding author. Tel.: +98 9123508816. E-mail address: tofi[email protected] (T. Allahviranloo). 1 Ph.D Student. 0020-0255/$ - see front matter Ó 2008 Published by Elsevier Inc. doi:10.1016/j.ins.2008.11.016

T. Allahviranloo et al. / Information Sciences 179 (2009) 956–966

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enlarging the class of differentiable fuzzy mappings that were introduced by Bede and Gal [5–7], and then used, along with the differential transformation method (DTM), by Chalco and Flores [10] to solve FDEs. DTM is different from the traditional high-order Taylor series method, which requires symbolic computation of the necessary derivatives of the data function and is computationally expensive for high-order. Intrinsically, the differential transformation method evaluates the approximate solution by the finite Taylor series. But, in the differential transformation method, the derivative is not computed directly; instead, the relative derivatives are calculated by an iteration procedure. New equations are obtained from the original equations by applying differential transformations. The proposed method provides Taylor series expansions of the solution between any two adjacent grid points. The concept of differential transformation was at first proposed by Zhou [33], and it was applied to solve linear and non-linear initial value problems in electric circuit analysis. This method provides an iterative procedure to obtain the spectrum of the analytical solutions. During the last seven years, significant progress has been made in applying the differential transformation approach to some linear and non-linear initial value problems. In 1999, Chen and Ho [12] introduced the two-dimensional differential transformation and applied it to solve partial differential equations. Jang et al. [19] introduced the concept of the differential transformation of fixed grid size and adaptive grid size mechanism to approximate solutions of initial value problems. In this paper, we investigate how it should be used in the fuzzy environment. The rest of the paper is organised as follows: Section 2 contains the basic material to be used in the rest of paper; in Section 3, the proposed method for solving FIVP is presented; in the last section, the proposed method is illustrated by numerical examples. 2. Basic concepts There are various definitions for the concept of fuzzy numbers [15,17] A non-empty subset A of R is called convex if and only if ð1 kÞx þ ky 2 A for every x, y 2 A and k 2 [0, 1]. By pk ðRÞ, we denote the family of all non-empty compact convex subsets of R. Definition 2.1. A fuzzy number is a function u: R ? [0, 1] satisfying the following properties: (i) (ii) (iii) (iv)

u is normal, i.e. 9x0 2 R with uðx0 Þ ¼ 1, u is a convex fuzzy set (i.e. uðkx þ ð1 kÞyÞ P minfuðxÞ; uðyÞg8x; y 2 R; k 2 ½0; 1), u is upper semi-continuous on R, fx 2 R : uðxÞ > 0g is compact, where A denotes the closure of A.

The set of all fuzzy real numbers is denoted by E. Obviously R E. For 0 < r 6 1, set ½ur ¼ fx 2 R; uðxÞ P rg and ½u0 ¼ fx 2 R; uðxÞ > 0g. Then it is wellknown that for any r 2 [0, 1], ½ur is a bounded closed interval. Another definition for a fuzzy number is the following: ðrÞÞ, 0 6 r 6 1, Definition 2.2. A fuzzy number in parametric form is represented by an ordered pair of functions ðuðrÞ; u satisfying the following requirements: 1. uðrÞ is a bounded left-continuous non-decreasing function over [0, 1]. ðrÞ is a bounded left-continuous non-increasing function over [0, 1]. 2. u ðrÞ, 0 6 r 6 1. 3. uðrÞ 6 u ðrÞ ¼ a, 0 6 r 6 1. We recall that for a < b < c, a, b, c 2 R, the triangular A crisp number a is simply represented by uðrÞ ¼ u ðrÞ ¼ c ðc bÞr are the endpoints fuzzy number u = (a, b, c) determined by a, b, c is given such that uðrÞ ¼ a þ ðb cÞr and u ðrÞ ¼ b which is denoted by [u]1. For arbitrary fuzzy numbers u ¼ ðuðrÞ; u ðrÞÞ, of the r-level sets, for all r 2 [0, 1]. Here uðrÞ ¼ u v ¼ ðv ðrÞ; v ðrÞÞ and an arbitrary crisp number k we define fuzzy addition and scalar multiplication as 1. 2. 3. 4.

ðu þ v ÞðrÞ ¼ ðuðrÞ þ v ðrÞÞ, ðrÞ þ v ðrÞÞ, ðu þ v ÞðrÞ ¼ ðu ðrÞ, k P 0, ðkuÞðrÞ ¼ kuðrÞ; ðkuÞðrÞ ¼ ku ðrÞ; ðkuÞðrÞ ¼ kuðrÞ, k < 0. ðkuÞðrÞ ¼ ku

Definition 2.3. Consider x, y 2 E. If there exists z 2 E such that x = y + z, then z is called the H-difference of x and y and it is denoted by x y. Definition 2.4. Let E be the set of all fuzzy numbers. A function f : R ! E is called a fuzzy-valued function. if In this paper, following [2], an arbitrary fuzzy number with compact support is represented by a pair of functions ðrÞÞ; 0 6 r 6 1. Also, we use the Hausdorff distance between fuzzy numbers. This fuzzy number space as shown in ðuðrÞ; u [6] can be embedded into Banach space B ¼ c½0; 1 c½0; 1 with the usual metric defined as follows: Let E be the set of all

958


upper semi-continuous normal convex fuzzy numbers with bounded r-level sets. Since the r-levels the fuzzy numbers are ðrÞ, for all r. We denote by x the set of all nonalways closed and bounded, the intervals can be written as u½r ¼ ½uðrÞ; u empty compact subsets of R and by xc the subsets of x consisting of non-empty convex compact sets. Recall that

qðx; AÞ ¼ min kx ak a2A

is the distance of a point x 2 R from A 2 x and the Hausdorff separation q(A, B) of A, B 2 x is defined as

qðA; BÞ ¼ max qða; BÞ: a2A

Note that the notation is consistent, since qða; BÞ ¼ qðfag; BÞ. Now, q is not a metric. In fact, q(A, B) = 0 if and only if A # B. The Hausdorff metric dH on x is defined by

dH ðA; BÞ ¼ maxfqðA; BÞ; qðB; AÞg: The metric dH is defined on E as

d1 ðu; v~ Þ ¼ supfdH ðu½r; v ½rÞ : 0 6 r 6 1g;

u; v 2 E:

for arbitrary ðu; v Þ 2 c½0; 1 c½0; 1. The following properties are wellknown (see e.g. [17,30]): (i) d1 ðu þ w; v þ wÞ ¼ d1 ðu; v Þ, 8u; v ; w 2 E, (ii) d1 ðk:u; k:v Þ ¼ jkjd1 ðu; v Þ, 8k 2 R; u; v 2 E, (iii) d1 ðu þ v ; w þ eÞ 6 d1 ðu; wÞ þ Dðv ; eÞ, 8u; v ; w; e 2 E. Theorem 2.1 ~ 2 E is a neutral element with respect to addition; i.e. u þ 0 ~¼0 ~ þ u ¼ u, for all u 2 E. ~ ¼ v , then 0 (i) If we define 0 0 ~ no u 2 E n R, has an opposite in E. (ii) With respect to 0, (iii) For any a, b 2 R with a, b P 0 or a, b 6 0 and any u 2 E, we have ða þ bÞ u ¼ a u þ b: u; however, this relation does not necessarily hold for general a, b 2 R. (iv) For any k 2 R and any u, v 2 E, we have k ðu þ v Þ ¼ k u þ k v . (v) For any k, l 2 R and any u 2 E, we have k ðl:uÞ ¼ ðk lÞ u.

Remark 2.1. d1 ðu; 0Þ ¼ d1 ð0; uÞ ¼ kuk. Definition 2.5. Let f(x) be a fuzzy-valued function on [a, b]. If f ðx; rÞ and f ðx; rÞ are improper Riemann-integrable on [a, b] then we say that f(x) is improper Riemann-integrable on [a, b]; furthermore, we define

Z

f ðx; rÞdt

a

Z a

!

b

b

¼

b

f ðx; rÞdt; a

! f ðx; rÞdx

Z

¼

Z

b

f ðx; rÞdx:

a

In this paper, the sign ‘‘” always stands for the H-difference. Note that x y–x þ ðyÞ. Let us recall the definition of strongly generalised differentiability introduced in [2, 3]. Definition 2.6. (see [7]). Let f : ða; bÞ ! E and x0 2 ða; bÞ. We say that f is strongly generalised differentiable at x0 (Bede–Gal differentiability), if there exists an element f 0 ðx0 Þ 2 E, such that (i) for all h > 0 sufficiently small, 9f ðx0 þ hÞ f ðx0 Þ, 9f ðx0 Þ f ðx0 hÞ and the following limits hold (in the metric d1 ):

lim h&0

f ðx0 þ hÞ f ðx0 Þ f ðx0 Þ f ðx0 hÞ ¼ lim ¼ f 0 ðx0 Þ h&0 h h

or (ii) for all h > 0 sufficiently small, 9f ðx0 Þ f ðx0 þ hÞ, 9f ðx0 hÞ f ðx0 Þ and the following limits hold (in the metric d1 ):

lim h&0

f ðx0 Þ f ðx0 þ hÞ f ðx0 hÞ f ðx0 Þ ¼ lim ¼ f 0 ðx0 Þ h&0 h h

or (iii) for all h > 0 sufficiently small, 9f ðx0 þ hÞ f ðx0 Þ, 9f ðx0 hÞ f ðx0 Þ and the following limits hold (in the metric d1 ):

959


lim h&0

f ðx0 þ hÞ f ðx0 Þ f ðx0 hÞ f ðx0 Þ ¼ lim ¼ f 0 ðx0 Þ h&0 h h

or (iv) for all h > 0 sufficiently small, 9f ðx0 Þ f ðx0 þ hÞ, 9f ðx0 Þ f ðx0 hÞ and the following limits hold (in the metric d1 ):

lim h&0

f ðx0 Þ f ðx0 þ hÞ f ðx0 Þ f ðx0 hÞ ¼ lim ¼ f 0 ðx0 Þ h&0 h h

(the denominators of h and h denote multiplication by

1 h

and

1 , h

respectively).

Proposition 2.1. If f : ða; bÞ ! E is a continuous fuzzy-valued function then gðxÞ ¼ g 0 ðxÞ ¼ f ðxÞ (see [15]).

Rx a

f ðtÞd is differentiable with derivative

Theorem 2.2. Let f : R ! E be a function and set f ðtÞ ¼ ðf ðt; rÞ; f ðt; rÞÞ, for each r 2 [0, 1]. Then (1) If f is differentiable in the first form (i), then f ðt; rÞ and f ðt; rÞ are differentiable functions and

f 0 ðtÞ ¼ ðf 0 ðt; rÞ; f 0 ðt; rÞÞ: (2) If f is differentiable in the second form (ii), then f ðt; rÞ and f ðt; rÞ are differentiable functions and

f 0 ðtÞ ¼ ðf 0 ðt; rÞ; f 0 ðt; rÞÞ: Lemma 2.1. (see [6]). Here, the equivalence between two equations means that any solution of an equation is also a solution for the other one. For x0 2 R, the fuzzy differential equation y0 ¼ f ðx; yÞ, yðx0 Þ ¼ y0 2 E where f : R E ! E is continuous, is equivalent to one of the integral equations:

yðxÞ ¼ y0 þ

Z

x

8x 2 ½x0 ; x1

f ðt; yðtÞÞdt x0

or

yð0Þ ¼ yðxÞ þ ð1Þ

Z

x

f ðt; yðtÞÞdt

8x 2 ½x0 ; x1

x0

on some interval ðx0 ; x1 Þ R, under the strong differentiability condition (i) or (ii), respectively. Remark 2.2. (see [6]). In the case of strongly generalised differentiability, we may attach to the fuzzy differential equation y0 ¼ f ðx; yÞ two different integral equations, while in the case of H-differentiability, we may attach only one. The second inteRx gral equation in Lemma 2.1 can be written in the form yðxÞ ¼ y0 ð1Þ: x0 f ðt; yðtÞÞdt. The following theorems concern the existence of solutions of a fuzzy initial value problem under generalised differentiability (see [6]). Theorem 2.3. Suppose that the following conditions hold: (a)Let R0 ¼ ½x0 ; x0 þ p Bðy0 ; qÞ, p, q > 0, y0 2 E, where Bðy0 ; qÞ ¼ fy 2 E : d1 ðy; y0 Þ 6 qg denotes a closed ball in E and let f : R0 ! E be a continuous function such that d1 ð0; f ðx; yÞÞ ¼ kf ðx; yÞk 6 M for all ðx; yÞ 2 R0 (b) Let g : ½x0 ; x0 þ p ½0; q ! E, such that gðx; 0Þ 0 and 0 6 gðx; uÞ 6 M 1 , 8x 2 ½x0 ; x0 þ p, 0 6 u 6 q, such that g(x, u) is non-decreasing in u and such that the initial value problem u0 ðxÞ ¼ gðx; uðxÞÞ; uðx0 Þ ¼ 0 has only the solution uðxÞ 0 on ½x0 ; x0 þ p. (c) We have d1 ðf ðx; yÞ; f ðx; zÞÞ 6 gðx; d1 ðy; zÞÞ, 8ðx; yÞ; ðx; zÞ 2 R0 and d1 ðy; zÞ 6Rq. (d) There exists d > 0 such that for x 2 ½x0 ; x0 þ d the sequence yn : ½x0 ; x0 þ d ! E given by nþ1 ðxÞ ¼ y0 ð1Þ: xx f ðt; y n Þdt is defined for any n 2 N. Then the fuzzy initial -value problem 0 ðxÞ ¼ y0 , y y 0

y0 ¼ f ðx; yÞ; yðx0 Þ ¼ y0

has ntwo solutions (one (i) differentiable and the other one o min p; Mq ; Mq1 ; d and the successive iterations

y0 ðxÞ ¼ y0 ; ynþ1 ðxÞ ¼ y0 þ

Z

x

x0

f ðt; yn ðtÞÞdt

and

^nþ1 ðxÞ ¼ y0 ð1Þ: ^0 ðxÞ ¼ y0 ; y y

Z

x

^n ðtÞÞdt f ðt; y

x0

converge to these two solutions respectively.

^ : ½x0 ; x0 þ r ! Bðy0 ; qÞ where r ¼ (ii) differentiable) y; y

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According to Theorem 2.3, we restrict our attention to functions that are (i)- or (ii)-differentiable on their domain except for a finite number of points (see also [6]). The following lemma gives a sufficient condition for the existence of the H-difference of two triangular fuzzy numbers. ð0Þ uð1Þ > 0 and lenðv Þ ¼ ðv ð0Þ v ð0ÞÞ 6 minfuð1Þ Lemma 2.2. (see [6]). Let u, v 2 E be such that uð1Þ uð0Þ > 0, u ð0Þ uð1Þg. Then the H-difference u v exists. uð0Þ; u The following corollary gives a simple sufficient condition for the existence of solutions to fuzzy differential equations under strongly generalised differentiability. ð0; 0Þ yð0; 1Þ. Corollary 2.1. Let f : R0 ! E where R0 ¼ ½x0 ; x0 þ p ðBðy0 ; qÞ \ EÞ, and y0 2 E such that yð0; 1Þ yð0; 0Þ and y ð0; 0Þ yð0; 1Þg. Under the assumptions (a)–(c) of Theorem 2.3, the fuzzy initial value problem Let m ¼ minfyð0; 1Þ yð0; 0Þ; y

y0 ¼ f ðx; yÞ; yðx0 Þ ¼ y0

n o m : ½x0 ; x0 þ r ! Bðy0 ; qÞ where r ¼ min p; Mq ; Mq ; 2M has two solutions y; y and the successive iterations in(2.3) converge to 1 these two solutions. Definition 2.7. Let f : ða; bÞ ! E and x0 2 ða; bÞ. We define the nth-order differential of f as follows: Let f : ða; bÞ ! E and x0 2 ða; bÞ. We say that f is strongly generalised differentiable of the nth-order at x0, if there exists elements, f ðsÞ ðx0 Þ 2 E, 8s ¼ 1; . . . ; n, such that 8s ¼ 1 . . . n one of the following holds: (i) for all h > 0 sufficiently small, 9f ðs1Þ ðx0 þ hÞ f ðs1Þ ðx0 Þ, 9f ðs1Þ ðx0 Þ f ðs1Þ ðx0 hÞ and the following limits hold (in the metric d1 ):

lim h&0

f ðs1Þ ðx0 þ hÞ f ðs1Þ ðx0 Þ f ðs1Þ ðx0 Þ f ðs1Þ ðx0 hÞ ¼ lim ¼ f ðsÞ ðx0 Þ h&0 h h

or (ii) for all h > 0 sufficiently small, 9f ðs1Þ ðx0 Þ f ðs1Þ ðx0 þ hÞ, 9f ðs1Þ ðx0 hÞ f ðs1Þ ðx0 Þ and the following limits hold (in the metric d1 )

lim h&0

f ðs1Þ ðx0 Þ f ðs1Þ ðx0 þ hÞ f ðs1Þ ðx0 hÞ f ðx0 Þ ¼ lim ¼ f ðsÞ ðx0 Þ h&0 h h

or (iii) for all h > 0 sufficiently small, 9f ðs1Þ ðx0 þ hÞ f ðs1Þ ðx0 Þ, 9f ðs1Þ ðx0 hÞ f ðs1Þ ðx0 Þ and the following limits hold (in the metric d1 )

lim h&0

f ðs1Þ ðx0 þ hÞ f ðs1Þ ðx0 Þ f ðs1Þ ðx0 hÞ f ðs1Þ ðx0 Þ ¼ lim ¼ f ðsÞ ðx0 Þ h&0 h h

or (iv) for all h > 0 sufficiently small, 9f ðs1Þ ðx0 Þ f ðs1Þ ðx0 þ hÞ, 9f ðs1Þ ðx0 Þ f ðs1Þ ðx0 hÞ and the following limits hold (in the metric d1 )

lim h&0

f ðk1Þ ðx0 Þ f ðs1Þ ðx0 þ hÞ f ðs1Þ ðx0 Þ f ðs1Þ ðx0 hÞ ¼ lim ¼ f ðsÞ ðx0 Þ h&0 h h

(the denominators h and h denote multiplication by

1 h

and

1 , h

respectively).

Remark 2.3. Note that by the above definition, a fuzzy-valued function is (i)-differentiable (or (ii)-differentiable) of order n if for s = 1, . . ., ,n f ðsÞ is (i)-differentiable (or (ii)-differentiable). It is possible that the different orders have different types (i or ii) of differentiability, but we do not consider this kind of function in this paper. Following [2], we define a first-order fuzzy differentiable equation by

x0 ¼ f ðt; xðtÞÞ; where xðtÞ ¼ ððxðt; rÞ; xðt; rÞ is a fuzzy function of t. f(t, x(t)) is a fuzzy-valued function and the fuzzy variable x0 ðtÞ is the defined derivative of x(t, r). Given an initial value x(t0) = x0 is given, we obtain a fuzzy Cauchy problem of the first-order

x0 ¼ f ðt; xðt; rÞÞ;

xðt 0 Þ ¼ x0 :

So, considering derivatives of type (i) or (ii), we may replace the FIVP by the equivalent system

ð1Þ


x0 ðt; rÞ ¼ hðt; xðt; rÞ; xðt; rÞÞxðt0 ; rÞ ¼ x0 ðrÞ; x0 ðt; rÞ ¼ gðt; xðt; rÞ; xðt; rÞÞxðt 0 ; rÞ ¼ x0 ðrÞ

961

ð2Þ

for r 2[0, 1] or

x0 ðt; rÞ ¼ gðt; xðt; rÞ; xðt; rÞÞ; xðt 0 ; rÞ ¼ x0 ðrÞ; x0 ðt; rÞ ¼ hðt; xðt; rÞ; xðt; rÞÞ; xðt 0 ; rÞ ¼ x0 ðrÞ

ð3Þ

for r 2 [0, 1], For any fixed r, the system represents an ordinary Cauchy problem, to which any convergent classical numerical procedure can be applied. In the next section, a differential transformation method is proposed for solving the problem. 3. Differential transformation method Definition 3.1. If x(t, r) is strongly generalised differentiable of order k in the time domain T then If f is (i)-differentiable,

ðt; k; rÞ ¼ u

k d ðxðt; rÞÞ

dt

8t 2 T;

k

ðti ; k; rÞ ¼ X i ðk; rÞ ¼ u

# k d ðxðt; rÞÞ dt

8k 2 K;

k t¼t i

k

uðt; k; rÞ ¼

d ðxðt; rÞÞ dt

k

8t 2 T;

# d ðxðt; rÞÞ k

X i ðkÞ ¼ uðt i ; kÞ ¼

dt

8k 2 K

k t¼t i

and if f is (ii)-differentiable, k

ðt; k; rÞ ¼ u

d ðxðt; rÞÞ dt

X i ðk; rÞ ¼ uðti ; k; rÞ ¼

uðt; k; rÞ ¼

8t 2 T;

k


k d ðxðt; rÞÞ

dt

t¼t i

8t 2 T;

k

ðti ; kÞ ¼ X i ðk; rÞ ¼ u

k is odd;

k


k is odd;

k t¼t i

where Xðk; rÞ and Xðk; rÞ are called the lower and the upper spectrum of x(t, r) at t ¼ t i in the domain K, respectively. So, if f is (i)-differentiable, then x(t, r) can be represented as

xðt; rÞ ¼

1 X ðt t i Þk Xðk; rÞ; k! k¼0

xðt; rÞ ¼

1 X ðt t i Þk Xðk; rÞ k! k¼0

or if f is (ii)-differentiable, as

xðt; rÞ ¼

1 1 X X ðt t i Þk ðt t i Þk Xðk; rÞ þ Xðk; rÞ; k! k! k¼1;odd k¼0;even

xðt; rÞ ¼

1 1 X X ðt t i Þk ðt t i Þk Xðk; rÞ þ Xðk; rÞ: k! k! k¼1;odd k¼0;even

The above set of equations is known as the inverse transformation of X(k). If X(k) is defined as

Xðk; rÞ ¼ MðkÞ

Xðk; rÞ ¼ MðkÞ

" k # d ðqðtÞxðt; rÞÞ dt

k

" k # d ðqðtÞxðt; rÞÞ dt

;

k ¼ 0; 1; 2; . . . ; 1;

;

k ¼ 0; 1; 2; . . . ; 1

t¼0

k t¼0

962


or

Xðk; rÞ ¼ MðkÞ Xðk; rÞ ¼ MðkÞ

" k # d ðqðtÞxðt; rÞÞ k

dt " #t¼0 k d ðqðtÞxðt; rÞÞ dt

k

;

k ¼ 1; 3; 5; . . . ; 1;

;

k ¼ 0; 2; 4; . . . ; 1:

t¼0

Then the function x(t, r) can be described as

or

xðt; rÞ ¼

1 1 X ðt ti Þk Xðk; rÞ ; qðtÞ k¼0 MðkÞ k!

xðt; rÞ ¼

1 1 X ðt ti Þk Xðk; rÞ ; qðtÞ k¼0 MðkÞ k!

! 1 1 X X ðt t i Þk Xðk; rÞ ðt ti Þk Xðk; rÞ þ ; MðkÞ MðkÞ k! k! k¼1;odd k¼0;even ! 1 1 X X 1 ðt t i Þk Xðk; rÞ ðt ti Þk Xðk; rÞ ; xðt; rÞ ¼ þ qðtÞ k¼1;odd MðkÞ MðkÞ k! k! k¼0;even

xðt; rÞ ¼

1 qðtÞ

where M(K) > 0 and q(t) > 0. M(k) is called the weighting factor and q(t) is regarded as a kernel corresponding to x(t, r). In this k paper, the transformation with MðkÞ ¼ Hk! and q(t) = 1 is applied, where H is the time horizon of interest. If is (i)-differentiable, then k

Hk d xðt; rÞ ; k k! dt k Hk d xðt; rÞ Xðk; rÞ ¼ : k k! dt Xðk; rÞ ¼

If f is (ii)-differentiable, then k

Hk d xðt; rÞ ; k k! dt k Hk d xðt; rÞ Xðk; rÞ ¼ ; k k! dt Xðk; rÞ ¼

k is odd; k is odd:

If k is even, then u is considered as in the first form (i). Using the differential transformation, a differential equation in the domain of interest can be transformed to an algebraic equation in the domain K, and x(t, r) can be obtained as the finite-term Taylor series plus a remainder, as 1 1 X ðt t0 Þk Xðk; rÞ þ Rnþ1 ðtÞ qðtÞ k¼0 MðkÞ k! k 1 X t t0 Xðk; rÞ þ Rnþ1 ðtÞ; ¼ H k¼0

xðt; rÞ ¼

1 1 X ðt t0 Þk Xðk; rÞ þ Rnþ1 ðtÞ qðtÞ k¼0 MðkÞ k! k 1 X t t0 Xðk; rÞ þ Rnþ1 ðtÞ ¼ H k¼0

xðt; rÞ ¼

or

! 1 1 X X ðt t 0 Þk Xðk; rÞ ðt t 0 Þk Xðk; rÞ þ þ Rnþ1 ðtÞ MðkÞ k¼0;even MðkÞ k! k! k¼1;odd k k 1 1 X X t t0 t t0 ¼ Xðk; rÞ þ Xðk; rÞ þ Rnþ1 ðtÞ; H H k¼1;odd k¼0;even ! 1 1 X X 1 ðt t 0 Þk Xðk; rÞ 1 ðt t 0 Þk Xðk; rÞ xðt; rÞ ¼ þ Rnþ1 ðtÞ þ qðtÞ k¼1;odd MðkÞ qðtÞ k¼0;even MðkÞ k! k! k k 1 1 X X t t0 t t0 Xðk; rÞ þ Xðk; rÞ þ Rnþ1 ðtÞ: ¼ H H k¼1;odd k¼0;even 1 xðt; rÞ ¼ qðtÞ

963


The objective of this section is to find the solution of (1) at the equally spaced grid points ½t 0 ; t 1 ; . . . ; tN where t i ¼ a þ ih for .that is, the domain of interest [a, b] is divided in to N sub-domains and the approximation each i ¼ 0; 1; . . . ; N and h ¼ ðbaÞ N function in each sub-domain are xi ðt; rÞ, for i ¼ 0; 1; . . . ; N 1, respectively. Taking the differential transformation of (2) or (3), the transformed equation describes the relationship between the spectrum x(t, r) as

ðk þ 1ÞXðk þ 1; rÞ ¼ Hðt; Xðk; rÞ; Xðk; rÞÞ; ðk þ 1ÞXðk þ 1; rÞ ¼ Gðt; Xðk; rÞ; Xðk; rÞÞ or

ðk þ 1ÞXðk þ 1; rÞ ¼ Gðt; Xðk; rÞ; Xðk; rÞÞ ðk þ 1ÞXðk þ 1; rÞ ¼ Hðt; Xðk; rÞ; Xðk; rÞÞ; ðk þ 1ÞXðk þ 1; rÞ ¼ Hðt; Xðk; rÞ; Xðk; rÞÞðk þ 1ÞXðk þ 1; rÞ ¼ Gðt; Xðk; rÞ; Xðk; rÞÞ;

k is odd; k is even;

where H() denotes the transformed function of hðt; xðt; rÞ; xðt; rÞÞ and G() denotes the transformed function of xðt; rÞÞ. From the initial conditions the following can be obtained: gðt; xðt; rÞ;

Xð0; rÞ ¼ x0 ðrÞ;

Xð0; rÞ ¼ x0 ðrÞ:

In the first sub-domain, xðt; rÞ, xðt; rÞ can be described by x0 ðt; rÞ and x0 ðt; rÞ, respectively. They can be represented in terms of their nth-order Taylor polynomials with respect to a, that is

x0 ðt; rÞ ¼ X 0 ð0; rÞ þ X 0 ð1; rÞðt aÞ þ X 0 ð2; rÞðt aÞ2 þ þ X 0 ðn; rÞðt aÞn ;

ð4Þ

x0 ðt; rÞ ¼ X 0 ð0; rÞ þ X 0 ð1; rÞðt aÞ þ X 0 ð2; rÞðt aÞ2 þ þ X 0 ðn; rÞðt aÞn ;

ð5Þ

where the subscript 0 denotes that the Taylor Polynomial is expanded to t0 = a. Once the Taylor Polynomial is obtained xðt 1 ; rÞ can be evaluated as

xðt1 ; rÞ ¼ X 0 ð0; rÞ þ X 0 ð1; rÞðt1 aÞ þ X 0 ð2; rÞðt 1 aÞ2 þ þ X 0 ðn; rÞðt 1 aÞn 2

n

¼ X 0 ð0; rÞ þ X 0 ð1; rÞh þ X 0 ð2; rÞh þ þ X 0 ðn; rÞh ¼

n X

j

X 0 ðj; rÞh ;

j¼0

xðt1 ; rÞ ¼ X 0 ð0; rÞ þ X 0 ð1; rÞðt1 aÞ þ X 0 ð2; rÞðt 1 aÞ2 þ þ X 0 ðn; rÞðt 1 aÞn 2

n


n X

j

X 0 ðj; rÞh :

j¼0

The first value, x0 ðt 1 ; rÞ of the first sub-domain is the initial value of the second sub-domain, i.e. x1 ðt1 ; rÞ ¼ x0 ðt1 ; rÞ. In a similar manner xðt2 ; rÞ can be represented as

xðt2 ; rÞ x1 ðt2 ; rÞ ¼ X 1 ð0; rÞ þ X 1 ð1; rÞðt2 t 1 Þ þ X 1 ð2; rÞðt 2 t 1 Þ2 þ þ X 1 ðn; rÞðt2 t 1 Þn 2

n


n X

j

X 1 ðj; rÞh ;

j¼0

xðt2 ; rÞ x1 ðt2 ; rÞ ¼ X 1 ð0; rÞ þ X 1 ð1; rÞðt2 t 1 Þ þ X 0 ð2; rÞðt 2 t 1 Þ2 þ þ X 0 ðn; rÞðt2 t 1 Þn 2

n


n X

j

X 1 ðj; rÞh :

j¼0

Hence, the solution on the grid points ðtiþ1 Þ can be obtained as follows:

Xðtiþ1 ; rÞ ¼ X i ðtiþ1 ; rÞ ¼ X i ð0; rÞ þ X i ð1; rÞðt iþ1 ti Þ þ X i ð2; rÞðt iþ1 t i Þ2 þ þ X i ðn; rÞðtiþ1 t i Þn n X j ¼ X i ðj; rÞh ; j¼0

964


Xðtiþ1 ; rÞ ¼ X i ðtiþ1 ; rÞ ¼ X i ð0; rÞ þ X i ð1; rÞðt iþ1 t i Þ þ X i ð2; rÞðtiþ1 t i Þ2 þ þ X i ðn; rÞðtiþ1 t i Þn n X j ¼ X i ðj; rÞh : j¼0

From Definition 3.1, it can easily be proven that the transformation function has the basic mathematical properties shown in Table 1. Theorem 3.1 [10]. If f is non-decreasing with respect to the second argument then, in terms of either (i)- or (ii)-differentiability, the fuzzy solution of the FIVP and the solution via differential inclusions are identical. 4. Numerical examples Example 4.1. Consider the following FIVP x0 ðtÞ ¼ kxðt; rÞ;

xð0Þ ¼ ð1 þ r; 1 rÞ. 0

If we consider x in the first form, we have to solve the following differential system: 0

hð0; rÞ ¼ 1 þ r;

0

gð0; rÞ ¼ 1 r:

h ðt; rÞ ¼ kgðt; rÞ; g ðt; rÞ ¼ khðt; rÞ;

The solution of the system is xðt; rÞ ¼ ½ð1 þ rÞekt ; ð1 rÞekt which is an analytical solution. Remark 4.1. Note that the function of the FIVP, considering the derivative x0 in the first form, has the property that diamxðtÞ is unbounded as t ! 1. But if we use the result obtained by using the second form is much more intuitive for the FIVP since in this case diamxðtÞ ! 0 as t ! 1. Example 4.2. Consider the following FIVP:

y0 ðtÞ ¼ yðtÞ t 2 þ 1;

06t62

y0 ðtÞ þ t 2 1 ¼ yðtÞ;

0 6 t 6 2:


0 6 t 6 2;

yð0Þ ¼ ð3 þ 2r; 1 2rÞ


0 6 t 6 2;

yð0Þ ¼ ð3 þ 2r; 1 2rÞ:

and

and

They are the same in crisp arithmetic, but when they are investigated under uncertainty, they have two solutions, by Theorem 2.3, depending on how the two crisp equations are written and how they are fuzzified. Based on this application, different forms of an ordinary differential equation should be considered; for more details, see [7]. Consider the following equations:


0 6 t 6 2;

yð0Þ ¼ ð3 þ 2r; 1 2rÞ


0 6 t 6 2;

yð0Þ ¼ ð3 þ 2r; 1 2rÞ:

and

Table 1 Basic mathematical properties of transformation functions. Functional form

Differential transform

yðxÞ ¼ uðxÞ v ðxÞ yðxÞ ¼ awðxÞ m m yðxÞ ¼ ðd zðxÞ=dx Þ

YðkÞ ¼ uðkÞ yðkÞ YðkÞ ¼ awðkÞ YðkÞ ¼ ðmþkÞ! zðk þ mÞ Pk! YðkÞ ¼ ki¼0 uðlÞv ðk lÞ YðkÞ ¼ dðk mÞ YðkÞ ¼ kk =k! YðkÞ ¼ mðm1Þðmk1Þ k! k YðkÞ ¼ xk! sinðp 2!k þ aÞ k x YðkÞ ¼ k! cosðp 2!k þ aÞ

yðxÞ ¼ uðxÞ v ðxÞ yðxÞ ¼ xm yðxÞ ¼ expðkxÞ yðxÞ ¼ ð1 þ xÞm yðxÞ ¼ sinðxx þ aÞ yðxÞ ¼ cosðxx þ aÞ

965

T. Allahviranloo et al. / Information Sciences 179 (2009) 956–966 Table 2 Bounds of computational errors. Bounded error

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Rkm4 DT4 Dk5 Dk10

0.07 0.035 0 0

0.1 0.063 0 0

0.23 0.094 0 0

0.4 0.186 0 0

0.53 0.28 0 0

0.6 0.4 0.01 0

0.73 0.66 0.027 0

1.08 0.84 0.03 0

Assume N = 10 and h = 0.2. For the first solution, using the first equation form, the differential equation of the system between t i and tiþ1 can be represented as

y0 ðt Þ ¼ yðt Þ t 2 2t i t þ ð1 t 2i Þ; where t ¼ t ti . Since we know that y(0) 2 E, it follows that y0 ðtÞ 2 E and so on. If we represent the first equation in the parametric form, we have

0 ðt ; rÞ ¼ y ðt ; rÞ t 2t i t þ ð1 ti Þ2 ; y y0 ðt ; rÞ ¼ yðt ; rÞ t 2t i t þ ð1 ti Þ2 : Taking the differential transformation of the above set of equations, we see that

Y i ðk þ 1; rÞ ¼ Y i ðk; rÞ dðk 2Þ 2ti ðdðk 1ÞÞ þ ð1 t2i ÞdðkÞ=ðk þ 1Þ; Y i ðk þ 1; rÞ ¼ Y i ðk; rÞ dðk 2Þ 2ti ðdðk 1ÞÞ þ ð1 t2i Þdðk þ 1Þ=ðk þ 1Þ ðt; rÞ on the grid point can be obtained by the with Y 0 ð0Þ ¼ 1 2r and Y 0 ð0Þ ¼ 3 þ 2r. The approximations of yðt; rÞ and y above equations. The exact solution of the problem is

yðt; rÞ ¼ ð2rÞet þ ðt þ 1Þ2 ; ðt; rÞ ¼ ð2 þ 2rÞet þ ðt þ 1Þ2 : y Table 2 shows the errors in the solution by the differential transformation method of various orders, along with the result obtained by the Runge–Kutta method of order 4. ðt; rÞg. Bound of Error = max{Error of yðt; rÞ, Error of y For the second solution, using the second equation form, the differential equation of the system between ti and tiþ1 can be represented as

y0 ðt Þ þ t 2 þ 2t i t ð1 t 2i Þ ¼ yðt Þ; where t ¼ t ti . Since we know that y(0) 2 E, then y0 ðtÞ 2 E and so on. If we represent the second form of the equation in parametric form, we have

ðt ; rÞ; 0 ðt ; rÞ þ t þ 2t i t ð1 t i Þ2 ¼ y y y0 ðt :rÞ þ t þ 2ti t ð1 t i Þ2 ¼ yðt ; rÞ: Taking the differential transformation of the above set of equations, we see that

Y i ðk þ 1; rÞ þ dðk 2Þ þ 2ti ðdðk 1ÞÞ ð1 t2i ÞdðkÞ=ðk þ 1Þ ¼ Y i ðk; rÞ; Y i ðk þ 1; rÞ þ dðk 2Þ þ 2ti ðdðk 1ÞÞ ð1 t2i Þdðk þ 1Þ=ðk þ 1Þ ¼ Y i ðk; rÞ ðt; rÞ on the grid point can be obtained by the with Y 0 ð0Þ ¼ 1 2r and Y 0 ð0Þ ¼ 3 þ 2r. The approximation of yðt; rÞ and y above equations.

Table 3 Bounds of computational errors. Bounded error

0.25

0.5

0.75

1

1.25

1.5

1.75

2

DT4 Dk5 Dk10

0.045 0 0

0.069 0 0

0.1 0 0

0.196 0 0

0.31 0.02 0

0.43 0.03 0

0.76 0.043 0.018

0.93 0.047 0.01

966


The exact solution of the problem in the second form is

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