Soft Comput (2010) 14:235–243 DOI 10.1007/s00500-008-0397-6 ORIGINAL PAPER Fuzzy Laplace transforms Tofigh Allahviranloo Æ M. Barkhordari Ahmadi Published online: 17 February 2009 Ó Springer-Verlag 2009 Abstract In this paper we propose a fuzzy Laplace transform and under the strongly generalized differentiability concept, we use it in an analytic solution method for some fuzzy differential equations (FDEs). The related theorems and properties are proved in detail and the method is illustrated by solving some examples. Keywords Fuzzy number
Fuzzy Laplace transform
Strongly generalized differential
Fuzzy differential equation
Fuzzy valued function 1 Introduction A natural way to model dynamic systems under uncortainty is to use fuzzy differential equations (FDEs). First order linear fuzzy differential equations are one of the simplest FDEs which may appear in many applications. The topic of FDEs has been rapidly growing in recent years. The concept of the fuzzy derivative was first introduced by Chang and Zadeh (1972); it was followed up by Dubois and Prade (1982), who used the extension principle in their approach. Other methods have been discussed by Puri and Ralescu (1983) and Goetschel and Voxman (1986). Kandel and Byatt (1980, 1978) applied T. Allahviranloo (&) Department of Mathematics, Science and Research Branch, Islamic Azad University, 14515/775 Tehran, Iran e-mail: tofigh@allahviranloo.com; allahviranloo@yahoo.com M. B. Ahmadi Science and Research Branch, Islamic Azad University (IAU), 14515/775 Tehran, Iran the concept of FDEs to the analysis of fuzzy dynamical problems. The FDE and the fuzzy initial value problem (Cauchy problem) were rigorously treated by Kaleva (1987, 1990), Seikkala (1987), He and Yi (1989), Kloeden (1991) and Menda (1988), and by other researchers (see Bede et al. 2006; Buckley 2006; Buckley and Feuring 1999, 2003; Congxin and Shiji 1998; Ding et al. 1997; Jowers et al. 2007). The numerical methods for solving FDEs are introduced in Abbasbandy and Allahviranloo (2002), Abbasbandy et al. (2004), and Allahviranloo et al. (2007). A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva (1987), Seikkala (1987), He and Yi (1989), and Kloeden (1991) and Wu (2000). Kaleva (1987) discussed the properties of differentiable fuzzy set-valued functions by means of the concept of H-differentiability due to Puri and Ralescu (1983), gave the existence and uniqueness theorem for a solution of the FDE y0 ¼ f ðt; yÞ; yðt0 Þ ¼ y0 when f satisfies the Lipschitz condition. Further, Song and Wu (2000) investigate FDEs, and generalize the main results of Kaleva (1987). Seikkala (1987), defined the fuzzy derivative which is the generalization of Hukuhara derivative, and showed that fuzzy initial value problem y0 ¼ f ðt; yÞ; yðt0 Þ ¼ y0 has a unique solution, for the fuzzy process of a real variable whose values are in the fuzzy number space (E, D), where f satisfies the generalized Lipschitz condition. Strongly generalized differentiability was introduced in Bede and Gal (2005) and studied in Bede et al. (2006). The strongly generalized derivative is defined for a larger class of fuzzy-valued function than the H-derivative, and FDEs can have solutions which have a decreasing length of their support. So we use this differentiability concept in the present paper. The fuzzy Laplace transform method solves FDEs and corresponding fuzzy initial and boundary value problems. In this way 123 236 fuzzy Laplace transforms reduce the problem of solving a FDE to an algebraic problem. This switching from operations of calculus to algebraic operations on transforms is called operational calculus, a very important area of applied mathematics, and for the engineer, the fuzzy Laplace transform method is practically the most important operational method. The fuzzy Laplace transform also has the advantage that it solves problems directly, fuzzy initial value problems without first determining a general solution, and non-homogeneous differential equations without first solving the corresponding homogeneous equation. The paper is organized as follows: In Sect. 2 we present the basic notions of fuzzy number, fuzzy valued function, fuzzy derivative, and fuzzy integral. In Sect. 3 fuzzy Laplace transform is defined and its basic properties are investigated. Procedure for solving FDEs by fuzzy Laplace transform is proposed in Sect. 4. Several examples are given in Sect. 5, and conclusions are drawn in Sect. 6. T. Allahviranloo, M. B. Ahmadi Remark 2.1 (see Zimmermann 1991) Let X be Cartesian product of universes X ¼ X1 


 Xn , and A1 ; . . .; An be n fuzzy numbers in X1 ; . . .; Xn , respectively. f is a mapping from X to a universe Y; y ¼ f ðx1 ; . . .; xn Þ. Then the extension principle allows us to define a fuzzy set B in Y by B ¼ fðy; uðyÞÞ j y ¼ f ðx1 ; . . .; xn Þ; ðx1 ; . . .; xn Þ 2 Xg where 8 < supðx1 ;...;xn Þ2f 1 ðyÞ uB ðyÞ ¼ minfuA1 ðx1 Þ; . . .; uAn ðxn ÞÞg; : 0 if f 1 ðyÞ 6¼ 0; if otherwise: where f 1 is the inverse of f. For n = 1, the extension principle, of course, reduces to B ¼ fðy; uB ðyÞÞ j y ¼ f ðxÞ; x 2 Xg where uB ðyÞ ¼  supx2f 1 ðyÞ uA ðxÞ; 0 if f 1 ðyÞ 6¼ 0; if otherwise: According to Zadeh’s extension principle, operation of addition on E is defined by 2 Preliminaries ðu  vÞðxÞ ¼ supy2R minfuðyÞ; vðx  yÞg; x 2 R We now recall some definitions needed through the paper. The basic definition of fuzzy numbers is given in Xu et al. (2007) and Song and Wu (2000). By R, we denote the set of all real numbers. A fuzzy number is a mapping u : R ! ½0; 1 with the following properties: and scalar multiplication of a fuzzy number is given by  uðx=kÞ; k [ 0; ðk  uÞðxÞ ¼ e 0; k ¼ 0; (a) u is upper semi-continuous, (b) u is fuzzy convex, i.e., uðkx þ ð1  kÞyÞ  minfuðxÞ; uðyÞg for all x; y 2 R; k 2 ½0; 1 , (c) u is normal, i.e., 9x0 2 R for which uðx0 Þ ¼ 1 , (d) supp u ¼ fx 2 R j uðxÞ [ 0g is the support of the u, and its closure cl(supp u) is compact. Let E be the set of all fuzzy number on R. The r-level set of a fuzzy number u 2 E; 0  r  1 , denoted by ½ur , is defined as  fx 2 R j uðxÞ  rg if 0  r  1 ½ur ¼ : clðsupp uÞ if r ¼ 0 It is clear that the r-level set of a fuzzy number is a closed and bounded interval ½uðrÞ; uðrÞ; where uðrÞ denotes the left-hand endpoint of ½ur and uðrÞ denotes the right-hand endpoint of ½ur : Since each y 2 R can be regarded as a fuzzy number ye defined by  1 if t ¼ y yeðtÞ ¼ 0 if t 6¼ y R can be embedded in E. 123 where ~0 2 E. It is well known that the following properties are true for all levels ½u  vr ¼ ½ur þ ½vr ; ½k  ur ¼ k½ur From this characteristic of fuzzy numbers, we see that a fuzzy number is determined by the endpoints of the intervals ½ur . This leads to the following characteristic representation of a fuzzy number in terms of the two ‘‘endpoint’’ functions uðrÞ and uðrÞ: An equivalent parametric definition is also given in Friedman et al. (1999), Ma et al. (1999) as: Definition 2.1 A fuzzy number u in parametric form is a pair ðu; uÞ of functions uðrÞ, uðrÞ; 0  r  1, which satisfy the following requirements: 1. 2. 3. uðrÞ is a bounded non-decreasing left continuous function in ð0; 1, and right continuous at 0, uðrÞ is a bounded non-increasing left continuous function in ð0; 1, and right continuous at 0, uðrÞ  uðrÞ; 0  r  1. A crisp number a is simply represented by uðrÞ ¼ uðrÞ ¼ a; 0  r  1. We recall that for a\b\c which a; b; c 2 R, the triangular fuzzy number u ¼ ða; b; cÞ Fuzzy Laplace transforms determined by a, b,c is given such that uðrÞ ¼ a þ ðb  cÞr and uðrÞ ¼ c  ðc  bÞr are the endpoints of the r-level sets, for all r 2 ½0; 1. For arbitrary u ¼ ðuðrÞ; uðrÞÞ, v ¼ ðvðrÞ; vðrÞÞ and k [ 0 we define addition u  v, subtraction u v and scalar multiplication by k as (See Friedman et al. 1999; Ma et al. 1999) (a) Addition: u  v ¼ ðuðrÞ þ vðrÞ; uðrÞ þ vðrÞÞ (b) u Subtraction: v ¼ ðuðrÞ  vðrÞ; uðrÞ  vðrÞÞ (c) Scalar multiplication:  k  0; ðku; kuÞ; ku¼ ðku; kuÞ; k\0: If k ¼ 1 then k  u ¼ u. The Hausdorff distance between fuzzy numbers given S by D : E  E ! Rþ 0, Dðu; vÞ ¼ sup maxfjuðrÞ  vðrÞj; juðrÞ  vðrÞjg; r2½0;1 where u ¼ ðuðrÞ; uðrÞÞ; v ¼ ðvðrÞ; vðrÞÞ R is utilized in Bede and Gal (2006). Then, it is easy to see that D is a metric in E and has the following properties (See Puri and Ralescu 1986) (i) Dðu  w; v  wÞ ¼ Dðu; vÞ; 8u; v; w 2 E; (ii) Dðk  u; k  vÞ ¼ jkjDðu; vÞ; 8k 2 R; u; v 2 E; (iii) Dðu  v; w  eÞ  Dðu; wÞ þ Dðv; eÞ; 8u; v; w; e 2 E; (iv) ðD; EÞ is a complete metric space. Definition 2.2 (see Friedman et al. 1999) Let f : R ! E be a fuzzy-valued function. If for arbitrary fixed t0 2 R and
[ 0, a d [ 0 such that jt  t0 j\d ) Dðf ðtÞ; f ðt0 ÞÞ\
; f is said to be continuous. Definition 2.3 (see Song and Wu 2000) A mapping f : R  E ! E is called continuous at point ðt0 ; x0 Þ 2 R  E provided for any fixed r 2 ½0; 1 and arbitrary
[ 0, there exists an dð
; rÞ such that Dð½f ðt; xÞr ; ½f ðt0 ; x0 Þr Þ\
whenever jt  t0 j\dð
; rÞ and Dð½xr ; ½x0 r Þ\dð
; rÞ for all t 2 R; x 2 E. Theorem 2.1 (see Wu 1999) Let f(x) be a fuzzy value function on ½a; 1Þ and it is represented by ðf ðx; rÞ; f ðx; rÞÞ. For any fixed r 2 ½0; 1, assume f ðx; rÞ and f ðx; rÞ are Riemann-integrable on ½a; b for every b  a, and assume 237 there are two positive RMðrÞ and MðrÞ such that Rb b f ðx; rÞjdx  MðrÞ and a j f ðx; rÞjdx  MðrÞ for every j a b  a. Then f(x) is improper fuzzy Riemann-integrable on ½a; 1Þ and the improper fuzzy Riemann-integral is a fuzzy number. Further more, we have: 0 1 1 Z1 Z1 Z f ðxÞdx ¼ @ f ðx; rÞdx; f ðx; rÞdxA: a a a Proposition 2.1 (see Wu 2000) If each of f(x) and g(x) is fuzzy value function and fuzzy Riemman integrable on ½a; 1Þ then f ðxÞ  gðxÞ is fuzzy Riemman-integrable on ½a; 1Þ. Moreover, we have Z Z Z ðf ðxÞ  gðxÞÞdx ¼ f ðxÞdx  gðxÞdx: I I I It is well-known that the H-derivative (differentiability in the sense of Hukuhara)for fuzzy mappings was initially introduced by Puri and Ralescu (1983) and it is based in the H-difference of sets, as follows. Definition 2.4 Let x; y 2 E. If there exists z 2 E such that x ¼ y  z, then z is called the H-difference of x and y, and it is denoted by x h y. In this paper, the sign ‘‘h ’’ always stands for H-difference, and also note that x h y 6¼ x y. In this paper we consider the following definition which was introduced by Bede and Gal (2005) and Bede et al. (2006). Definition 2.5 Let f : ða; bÞ ! E and x0 2 ða; bÞ. We say that f is strongly generalized differential at x0 (Bede–Gal differential) if there exists an element f 0 ðx0 Þ 2 E, such that (i) for all h [ 0 sufficiently small, 9f ðx0 þ hÞ h f ðx0 Þ; 9f ðx0 Þ h f ðx0  hÞ and the limits(in the metric D) f ðx0 þ hÞ h f ðx0 Þ f ðx0 Þ h f ðx0  hÞ ¼ lim ¼ f 0 ðx0 Þ h&0 h&0 h h lim or (ii) for all h [ 0 sufficiently small, 9f ðx0 Þ h f ðx0 þ hÞ; 9f ðx0  hÞ h f ðx0 Þ and the limits(in the metric D) f ðx0 Þ h f ðx0 þ hÞ f ðx0  hÞ h f ðx0 Þ ¼ lim ¼ f 0 ðx0 Þ h&0 h&0 h h lim or (iii) for all h [ 0 sufficiently small, 9f ðx0 þ hÞ h f ðx0 Þ; 9f ðx0  hÞ h f ðx0 Þ and the limits(in the metric D) f ðx0 þ hÞ h f ðx0 Þ f ðx0  hÞ h f ðx0 Þ ¼ lim ¼ f 0 ðx0 Þ h&0 h&0 h h lim 123 238 or (iv) T. Allahviranloo, M. B. Ahmadi h for all h [ 0 sufficiently small, 9f ðx0 Þ  f ðx0 þ hÞ; 9f ðx0 Þ h f ðx0  hÞ and the limits(in the metric D) f ðx0 Þ h f ðx0 þ hÞ f ðx0 Þ h f ðx0  hÞ 0 ¼ lim ¼ f ðx0 Þ h&0 h&0 h h lim In the special case when f is a fuzzy-valued function, we have the following result. Theorem 2.3 (see e.g. Chalco-Cano and Roman-Flores 2006) Let f : R ! E be a function and denote f ðtÞ ¼ ðf ðt; rÞ; f ðt; rÞÞ, for each r 2 ½0; 1. Then (1) (h and -h at denominators mean 1 h and 1 h, respectively) (2) Theorem 2.2 (see e.g. Chalco-Cano and Roman-Flores 2006) Let y : ½0; a  R ! R be continuous and f : ½0; a  E ! E, be the Zadehs extension of y, i.e., ½f ðt; xÞr ¼ f ðt; ½xr Þ. If y is nonincreasing with respect to the second argument, using the derivative in Definition 2.5, case (ii), the fuzzy solution of y0 ¼ f ðt; yÞ; yðt0 Þ ¼ y0 whenever it exists, coincides with the solution obtained via differential inclusions. Remark 2.2 (see e.g. Bede and Gal 2006) These cases (iii) and (iv) introduced in Bede and Gal (2005), in order to ensure a differentiable switch the case (i) and case (ii) in Definition 2.5. Of course, as the authors in Bede and Gal (2005) and Chalco-Cano and Roman-Flores (2006) have stated, the cases (i) and (ii) in Definition 2.5, are more important since cases (iii) and (iv) in Definition 2.5 occur only on a discrete set of points. As an example supporting these comments, let us consider c 2 E n R be any fuzzy(non-real) constant and let f : ½0; a  E ! E; f ðtÞ ¼ c  cos t; t 2 ½0; a. It is natural to expect that f is differentiable everywhere in its domain. Let us observe that f is differentiable according to Definition 2.5 (ii), on each sub interval ð2kp; ð2k þ 1ÞpÞ and differentiable according to Definition 2.5 (i), on each interval of the form ð2k þ 1Þp; ð2kÞpÞ; k 2 Z. But, at the points fkpg; k 2 Z, none of the cases (i) and (ii) in Definition 2.5 are fulfilled. Namely, at these points the H-differences f ðkp þ hÞ h f ðkpÞ and f ðkpÞ h f ðkp  hÞ may not exist simultaneously. Also, the H-differences f ðkpÞ h f ðkp þ hÞ and f ðkp  hÞ h f ðkpÞ cannot exist simultaneously, so f is not differentiable at kp in none of the cases (i) and (ii) of differentiability in Definition 2.5. Instead, it will be differentiable as in the cases (iii) and (iv) in Definition 2.5. Another argument for the importance of the cases (iii) and (iv) in Definition 2.5, is in the Theorem 2.2. Indeed, above stated theorem dose not cover the case when f ðt; xÞ has not constant monotonicity. In these cases we will have to switch between the cases (i) and (ii) of differentiability in Definition 2.5, so the cases (iii) and (iv) in Definition 2.5 may become important as switch points. 123 If f is (i)-differentiable, then f ðt; rÞ and f ðt; rÞ are 0 differentiable functions and f 0 ðtÞ ¼ ðf 0 ðt; rÞ; f ðt; rÞÞ If f is (ii)-differentiable, then f ðt; rÞ and f ðt; rÞ are 0 differentiable functions and f 0 ðtÞ ¼ ðf ðt; rÞ; f 0 ðt; rÞÞ. Lemma 2.1 (see Bede and Gal 2005) For x0 2 R, the FDE y0 ¼ f ðx; yÞ; yðx0 Þ ¼ y0 2 E where f : R  E ! E is supposed to be continuous, is equivalent to one of the integral equations: Zx yðxÞ ¼ y0  f ðt; yðtÞÞdt; 8x 2 ½x0 ; x1  x0 or 1 1 y ð0Þ ¼ y ðxÞ  ð1Þ  Zx f ðt; y1 ðtÞÞdt; 8x 2 ½x0 ; x1  x0 on some interval ðx0 ; x1 Þ R, depending on the strongly differentiability considered, (i) or (ii), respectively. Here the equivalence between two equations means that any solution of an equation is a solution too for the other one. Remark 2.3 (see Bede and Gal 2005) In the case of strongly generalized differentiability, to the FDE y0 ¼ f ðx; yÞ we may attach two different integral equations, while in the case of differentiability in the sense of definition H-differentiable, we may attach only one. The second integral equation in Lemma R x 2.1 can be written in the form y1 ðxÞ ¼ y10 h ð1Þ  x0 f ðt; y1 ðtÞÞdt. The following theorems concern the existence of solutions of a fuzzy initial-value problem under generalized differentiability (see Bede and Gal 2005). Theorem 2.4 Let us suppose that the following conditions hold: (a) Let R0 ¼ ½x0 ; x0 þ s  Bðy0 ; qÞ; s; q [ 0; y0 2 E, where Bðy0 ; qÞ ¼ fy 2 E : Dðy; y0 Þ  qg denote a closed ball in E and let f : R0 ! E be a continuous function such that Dð0; f ðx; yÞÞ  M for all ðx; yÞ 2 R0 and 0 2 E (b) Let g : ½x0 ; x0 þ s  ½0; q ! R, such that gðx; 0Þ 0 and 0  gðx; uÞ  M1 ; 8x 2 ½x0 ; x0 þ s; 0  u  q, such that gðx; uÞ is non-decreasing in u and g is such that the initialvalue problem u0 ðxÞ ¼ gðx; uðxÞÞ; uðx0 Þ ¼ 0 has only the solution uðxÞ 0 on ½x0 ; x0 þ s. (c) We have Dðf ðx; yÞ; f ðx; zÞÞ  gðx; Dðy; zÞÞ; 8ðx; yÞ; ðx; zÞ 2 R0 and Fuzzy Laplace transforms 239 Dðy; zÞ  q. (d) There exists d [ 0 such that for x 2 ½x0 ; x0 þ d the sequence y1n : ½x0 ; x0Rþ d ! E given by y10 ðxÞ ¼ y0 ; y1nþ1 ðxÞ ¼ y0 h ð1Þ  xx f ðt; y1n Þdt is defined for any 0 n 2 N. Then the fuzzy initial value problem  0 y ¼ f ðx; yÞ; yðx0 Þ ¼ y0 has two solutions (one (i)-differentiable and the other one (ii)-differentiable) y; y1 : ½x0 ; x0 þ r ! Bðy0 ; qÞ where r ¼ minfs; Mq ; Mq1 ; dg and the successive iterations y0 ðxÞ ¼ y0 , Zx ð2:1Þ ynþ1 ðxÞ ¼ y0  f ðt; yn ðtÞÞdt; L½f ðxÞ ¼ Z1 From Theorem 2.1, we have Z1 f ðxÞ  e px 0 0 dx ¼ @ Z1 f ðx; rÞe px dx; 0 Z1 px f ðx; rÞe 0 1 dxA; also by using the definition of classical Laplace transform: Z1 ‘½f ðx; rÞ ¼ f ðx; rÞepx dx and ‘½f ðx; rÞ 0 ¼ and y10 ðxÞ ¼ y0 , y1nþ1 ðxÞ ¼ y0 h ð1Þ  ðp [ 0 and integerÞ: 0 x0 Zx f ðxÞ  epx dx Z1 f ðx; rÞepx dx 0 f ðt; y1n ðtÞÞdt; ð2:2Þ x0 converge to these two solutions, respectively. According to Theorem 2.4, we restrict our attention to functions which are (i) or (ii)-differentiable on their domain except on a finite number of points (see also Bede and Gal 2005). Remark 2.4 The disadvantage of strongly generalized differentiability of a function with respect to H-differentiability and Hukuhara differentiability is that a FDE has no unique solution. This phenomenon appears in the present paper too, so a FDE has several solutions (two solutions locally). The advantage of the existence of these solutions is that we can choose the solution that reflects better the behaviour of the modelled real-world system. then, we follow: L½ f ðxÞ ¼ 𑽠f ðx; rÞ; ‘½ f ðx; rÞÞ: Theorem 3.1 Let f0 (x) be an integrable fuzzy-valued function, and f(x) is the primitive of f0 (x) on ½0; 1Þ. Then L½f 0 ðxÞ ¼ p  L½f ðxÞ h f ð0Þ where f is (i)-differentiable or L½ f 0 ðxÞ ¼ ðf ð0ÞÞ h ðp  L½f ðxÞÞ where f is (ii)-differentiable: Proof For arbitrary fixed r 2 ½0; 1 we have: ðp  L½ f ðxÞÞ h f ð0Þ ¼ ðp‘½ f ðx; rÞ  f ð0; rÞ; p‘½ f ðx; rÞ  f ð0; rÞÞ since ‘½ f 0 ðx; rÞ ¼ p‘½ f ðx; rÞ  f ð0; rÞ and ‘½ f 0 ðx; rÞ ¼ p‘½ f ðx; rÞ  f ð0; rÞ then 3 Fuzzy Laplace transform ðp  L½ f ðxÞÞ h f ð0Þ ¼ 𑽠f 0 ðx; rÞ; ‘½ f 0 ðx; rÞÞ; The fuzzy Laplace transform method solves FDEs and corresponding fuzzy initial and boundary value problems. In this way fuzzy Laplace transforms reduce the problem of solving a FDE to an algebraic problem. This switching from operations of calculus to algebraic operations on transforms is called operational calculus, a very important area of applied mathematics, and for the engineer, the fuzzy Laplace transform method is practically the most important operational method. In this section we are going to define fuzzy Laplace transform. by linearity of L, Definition 3.1 Let f(x) be continuous fuzzy-value function. Suppose that f ðxÞ  eRpx is improper fuzzy Rimann1 integrable on ½0; 1Þ, then 0 f ðxÞ  epx dx is called fuzzy Laplace transforms and is denoted as ðp  L½ f ðxÞÞ h f ð0Þ ¼ ‘½ðf 0 ðx; rÞ; f 0 ðx; rÞÞ: Since f is (i)-differentiable, it follows that ðp  L½ f ðxÞÞ h f ð0Þ ¼ L½ f 0 ðxÞÞ: Now we assume that f is the (ii)-differentiable, for arbitrary fixed r 2 ½0; 1 we have: ðf ð0ÞÞ h ðp  L½ f ðtÞÞ ¼ ðf ð0; rÞ þ p‘½ f ðx; rÞ; f ð0; rÞ þ p‘½ f ðx; rÞÞ this is equivalent to the following: ðp‘½f ðx; rÞ  f ð0; rÞ; p‘½f ðx; rÞ  f ð0; rÞÞ 123 240 T. Allahviranloo, M. B. Ahmadi since ‘½f 0 ðx; rÞ ¼ p‘½f ðx; rÞ  f ð0; rÞ and ‘½f 0 ðx; rÞ ¼ p‘½f ðx; rÞ  f ð0; rÞ then ðf ð0ÞÞ h ðp  L½ f ðtÞÞ ¼ 𑽠f 0 ðx; rÞ; ‘½ f 0 ðx; rÞÞ h ðf ð0ÞÞ  ðp  L½ f ðtÞÞ ¼ 0 0 ‘½ð f 0 ðx; rÞ; f 0 ðx; rÞÞ @ since f is (ii)-differentiable then it follows that ðf ð0ÞÞ h ðp  L½f ðtÞÞ ¼ L½f 0 ðxÞÞ: Theorem 3.2 Let f ðxÞ; gðxÞ be continuous fuzzy-valued functions suppose that c1 ; c2 are constant, then L½ðc1  f ðxÞÞ  ðc2  gðxÞÞ Z1 ¼ ðc1  f ðxÞ  c2  gðxÞÞ  epx dx ¼ c1  f ðxÞ  e dx  0 0 ¼ @c 1  0 Z1 f ðxÞ  e 0 px 1 Z1 gðxÞf ðx; rÞe px 0 1 dxA: Theorem 3.3 Let f is continuous fuzzy-value function and L½f ðxÞ ¼ FðpÞ , then L½eax  f ðxÞ ¼ Fðp  aÞ ax L½e  f ðxÞ ¼ Z1 eaxpx  f ðxÞ 0 0 ¼ c2  gðxÞ  epx dx Z1 eaxpx f ðx; rÞdx; 0 0 Z1 Z1 1 eaxpx f ðx; rÞdxA eðpaÞx  f ðxÞ ¼ Fðp  aÞ 0 0 dxA  @c2  ¼c1  L½f ðxÞ  c2  L½gðxÞ: dx; 0 ¼@ 0 Z1 gðxÞf ðx; rÞe px Proof Proof By using Remark 2.2 L½ðc1  f ðxÞÞ R1 ðc2  gðxÞÞ ¼ 0 ððc1  f ðxÞÞ  ðc2  gðxÞÞÞ  epx dx px Z1 where eax is real value function and p - a [ 0. L½ðc1  f ðxÞÞ  ðc2  gðxÞÞ ¼ ðc1  L½f ðxÞÞ  ðc2  L½gðxÞÞ: Z1 Z1 ðf ðxÞ  gðxÞÞ  epx dx ¼ Z1 0 1 gðxÞ  epx dxA Hence L½ðc1  f ðxÞÞ  ðc2  gðxÞÞ ¼ ðc1  L½f ðxÞÞ  ðc2  L½gðxÞÞ: 4 Constructing solutions via fuzzy initial value problem In this section, we consider the fuzzy initial value problem  0 y ðtÞ ¼ f ðt; yðtÞÞ ð4:3Þ yð0Þ ¼ ðyð0; rÞ; yð0; rÞÞ 0\r  1: where f : Rþ  E ! E is a continuous fuzzy mapping. By using fuzzy Laplace transform method we have: Remark 3.1 Let f(x) be continuous fuzzy-value function on ½0; 1Þ and k  0 , then L½y0 ðtÞ ¼ L½f ðt; yðtÞÞ: L½k  f ðxÞ ¼ k  L½f ðxÞ: Then, we have the following alternatives for solving (4.3): Fuzzy Laplace transform k  f ðxÞ is denoted as Z1 L½k  f ðxÞ ¼ k  f ðxÞ  epx dx ðp [ 0 and integerÞ Proof 0 and also we have: Z1 Z1 px k  f ðxÞ  e dx ¼ k  f ðxÞ  epx dx 0 0 then L½k  f ðxÞ ¼ k  L½f ðxÞ Remark 3.2 Let f(x) be continuous fuzzy-value function and gðxÞ  0 . Suppose that ðf ðxÞ  gðxÞÞ  epx is improper fuzzy Rimann-integrable on ½0; 1Þ , then 123 ð4:4Þ Case I If we consider y0 (t) by using (i)-differentiable, then from Theorem 2.3 in (i)-differentiable we have y0 ðtÞ ¼ ðy0 ðt; rÞ; y0 ðt; rÞÞ and L½y0 ðtÞ ¼ ðp  L½yðtÞÞ h yð0Þ: Equation 4.4 can be written as follows:  ‘½f ðt; yðtÞ; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ; ‘½f ðt; yðtÞ; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ: ð4:5Þ ð4:6Þ where f ðt; yðtÞ; rÞ ¼ minff ðt; uÞju 2 ðyðt; rÞ; yðt; rÞÞg and f ðt; yðtÞ; rÞ ¼ maxff ðt; uÞju 2 ðyðt; rÞ; yðt; rÞÞ:g To solve the linear system (4.6), for simplicity we assume that: ‘½yðt; rÞ ¼ H1 ðp; rÞ ‘½yðt; rÞ ¼ K1 ðp; rÞ Fuzzy Laplace transforms 241 where H1(p, r) and K1(p, r) are solutions of system (4.6). By using inverse Laplace transform, yðt; rÞ and yðt; rÞ is computed as follows: yðt; rÞ ¼ ‘1 ½H1 ðp; rÞ yðt; rÞ ¼ ‘1 ½K1 ðp; rÞ: Case II If we consider y0 (t) by using (ii)-differentiable, then from Theorem 2.3 in the second form we have y0 ðtÞ ¼ ðy0 ðt; rÞ; y0 ðt; rÞÞ and L½y0 ðtÞ ¼ ðyð0ÞÞ h ðp  L½yðtÞÞ: Equation 4.4 can be written as follows:  ‘½f ðt; yðtÞ; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ; ‘½f ðt; yðtÞ; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ: Hence solution of system (5.10) is as follows:     p 1  yð0; rÞ 2 ‘½yðt; rÞ ¼ yð0; rÞ 2 p 1 p 1     p 1 ‘½yðt; rÞ ¼ yð0; rÞ 2  yð0; rÞ 2 p 1 p 1 thus 1 yðt; rÞ ¼ yð0; rÞ‘    yð0; rÞ‘ 1   yðt; rÞ ¼ yð0; rÞ‘1    yð0; rÞ‘1   ð4:7Þ ð4:8Þ f ðt; yðtÞ; rÞ ¼ minff ðt; uÞju 2 ðyðt; rÞ; yðt; rÞÞg and f ðt; yðtÞ; rÞ ¼ maxff ðt; uÞju 2 ðyðt; rÞ; yðt; rÞÞg to solve the linear system (4.8), we, for the sake of simplicity, are considering: p 2 p 1 p 2 p 1 1 2 p 1 1 2 p 1 where     yð0; rÞ þ yð0; rÞ yð0; rÞ  yð0; rÞ t yðt; rÞ ¼ e þe 2 2 t and     yð0; rÞ þ yð0; rÞ yð0; rÞ  yð0; rÞ þ et : 2 2 ‘½yðt; rÞ ¼ H2 ðp; rÞ yðt; rÞ ¼ et ‘½yðt; rÞ ¼ K2 ðp; rÞ Now, if we consider y0 (t) in (ii)-differentiable, then by using Case II we have where H2(p, r) and K2(p, r) are solutions of system (4.8). By using inverse Laplace transform, yðt; rÞ and yðt; rÞ be computed as follows: yðt; rÞ ¼ ‘1 ½H2 ðp; rÞ Example 5.1 Consider the initial value problem  0 y ðtÞ ¼ yðtÞ; 0  t  T; yð0Þ ¼ ðyð0; rÞ; yð0; rÞÞ: ð5:9Þ by using fuzzy Laplace transform method we have L½y0 ðtÞ ¼ L½yðtÞ and L½y ðtÞ ¼ therefore 0 yðt; rÞ ¼ yð0; rÞ‘1  1 1þp  yðt; rÞ ¼ yð0; rÞ‘1  1 1þp  in (i)-differentiable, then by using Case I, we have: yð0Þ therefore L½yðtÞ ¼ ðpL½yðtÞÞ ð5:11Þ Hence solution of (5.11) is as follows:   1 ‘½yðt; rÞ ¼ yð0; rÞ 1þp   1 ‘½yðt; rÞ ¼ yð0; rÞ 1þp thus y0 ðtÞ  ept dt L½y0 ðtÞ ¼ ðpL½yðtÞÞ ðpL½yðtÞÞ then, by relations (4.8), can be written as  ‘½yðt; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ : ‘½yðt; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ 5 Examples Z1 ðpL½yðtÞÞ L½yðtÞ ¼ ðyð0ÞÞ yðt; rÞ ¼ ‘1 ½K2 ðp; rÞ: 0 L½y0 ðtÞ ¼ ðyð0ÞÞ where yð0Þ then, by relations (4.6), can be written as  ‘½yðt; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ : ‘½yðt; rÞ ¼ p‘½yðt; rÞ  yð0; rÞ yðt; rÞ ¼ et yð0; rÞ and ð5:10Þ yðt; rÞ ¼ et yð0; rÞ: 123 242 T. Allahviranloo, M. B. Ahmadi If initial condition have been presented as a symmetric triangular fuzzy number for instance yð0Þ ¼ ðað1  rÞ; að1  rÞÞ , then case (I): ‘½yðt; rÞ ¼ yð0; rÞ       1 1 1 þ þ 2 1þp pð1 þ pÞ p ð1 þ pÞ thus yðt; rÞ ¼ að1  rÞet yðt; rÞ ¼ yð0; rÞet þ t yðt; rÞ ¼ að1  rÞet : and And case (II): yðt; rÞ ¼ að1  rÞe yðt; rÞ ¼ yð0; rÞet þ t: t yðt; rÞ ¼ að1  rÞet : Remark 5.1 Note that the solution of the FDE (5.9) considering the derivative y0 in (i)-differentiable, has the property that diam(supp y(t)) = 2aet is unbounded as t ! 1, demonstrating that this interpretation does not generalize in an appropriate way the crisp case and gives counterintuitive result. However, if in (5.9) the derivative y0 is interpreted in (ii)-differentiable, then the result is much more intuitive for (5.9) since now diam(supp yðtÞ)=2aet ! 0 as t ! 1: From remark, we see that the solution of a FDE is dependent of the election of the derivative: in the (i)-differentiable or in the (ii)-differentiable. Thus, as in the above example, the solution can be adequately chosen. On the other hand, it is clear that in this new procedure the unicity of the solution is lost, but it is a expected situation in the fuzzy context. Example 5.2 Consider the initial value problem  0 y ðtÞ ¼ yðtÞ þ t þ 1; 0  t  T; yð0Þ ¼ ðyð0; rÞ; yð0; rÞÞ: by using fuzzy Laplace transform method we have L½yðtÞ ¼ ðp  L½yðtÞÞ h L½yð0Þ and 0 L½y ðtÞ ¼ Z1 y0 ðtÞept dt 0 then, from Case II we have: L½y0 ðtÞ ¼ ðyð0ÞÞ ðp  L½yðtÞÞ therefore L½yðtÞ þ ‘½t þ ‘½1 ¼ ðyð0ÞÞ ðp  L½yðtÞÞ then, from relation (4.8), can be written as  ‘½yðt; rÞ þ ‘½t þ ‘½1 ¼ p‘½yðt; rÞ  yð0; rÞ : ‘½yðt; rÞ þ ‘½t þ ‘½1 ¼ p‘½yðt; rÞ  yð0; rÞ ð5:12Þ Hence solution of (5.12) is as follows:       1 1 1 ‘½yðt; rÞ ¼ yð0; rÞ þ þ 2 1þp pð1 þ pÞ p ð1 þ pÞ 123 6 Conclusion Developing fuzzy Laplace transform, we provided solutions to fuzzy initial-value problems for first order linear FDE which is interpreted by using the strongly generalized differentiability concept. This may confer solutions which have a decreasing length of their support. References Abbasbandy S, Allahviranloo T (2002) Numerical solution of fuzzy differential equation by Tailor method. 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