Soft Comput (2014) 18:2461–2469DOI 10.1007/s00500-014-1224-x

METHODOLOGIES AND APPLICATION

Laplace transform formula on fuzzy nth-order derivativeand its application in fuzzy ordinary differential equations

M. Barkhordari Ahmadi · N. A. Kiani ·N. Mikaeilvand

Published online: 12 February 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper, the Laplace transform formula onthe fuzzy nth-order derivative by using the strongly general-ized differentiability concept is investigated. Also, the relatedtheorems and properties are proved in detail and, it is used inan analytic method for fuzzy two order differential equation.The method is illustrated by solving some examples.

Keywords Fuzzy number · Fuzzy valued function ·Generalized differentiability · Fuzzy differential equation ·Laplace transform

1 Introduction

The topic of fuzzy differential equations (FDEs) has beenrapidly growing in recent years. The concept of the fuzzyderivative was first introduced by Chang and Zadeh (1972);it was followed up by Dubios and Prade (1982), who usedthe extension principle in their approach. Other methods havebeen discussed by Puri and Ralescu (1983) and Goetschel andVoxman (1986). Kandel (1980), Kandel and Byatt (1978)

Communicated by T. Allahviranloo.

M. B. Ahmadi (B)Department of Mathematics, Bandar Abbas Branch, Islamic AzadUniversity, Bandar Abbas, Irane-mail: mahnazbarkhordarii@gmail.com

N. A. KianiViroquant Research Group Modeling, Bioquant,Heidelberg University, Im Neuenheimer Feld 267,69120 Heidelberg, Germany

N. MikaeilvandDepartment of Mathematics, Ardabil Branch, Islamic Azad University,Ardabil, Iran

applied the concept of FDEs to the analysis of fuzzy dynami-cal problems. The FDE and the initial value problem (Cauchyproblem) were rigorously treated by Kaleva (1987, 1990),Seikkala (1987), Ouyang and Wu (1989), Kloeden (1991)and Menda (1988), and by other researchers (see Bede et al.2006a; Buckly and Feuring 2003, 1999; Buckly 2006; Wuand Shiji 1998; Ding 1997; Jowers et al. 2007). The numer-ical methods for solving fuzzy differential equations areintroduced in Abbasbandy and Allahviranloo (2002, 2004),Allahviranloo (2002), Ghanbari (2009).

A thorough theoretical research of fuzzy Cauchy prob-lems was given by Kaleva (1987), Seikkala (1987), Ouyangand Wu (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-differentia-bility due to Puri and Ralescu (1983), gave the existence anduniqueness theorem for a solution of the fuzzy differentialequation y′ = f (t; y); y(t0) = y0 when f satisfies theLipschitz condition. Further, Song and Wu (2000) investi-gate fuzzy differential equations, and generalize the mainresults of Kaleva (1987). Seikkala (1987), defined the fuzzyderivative which is the generalization of Hukuhara deriv-ative, and showed that fuzzy initial value problem y′ =f (t; y); y(t0) = y0 has a unique solution, for the fuzzyprocess of a real variable whose values are in the fuzzy num-ber space (E, D), where f satisfies the generalized Lipschitzcondition.

Strongly generalized differentiability was introduced inBede and Gal (2005) and studied in Bede et al. (2006a). Theexistence and uniqueness theorem of solution of Nth-orderfuzzy differential equations under Nth-order generalized dif-ferentiability was studied by Salahshour (2011). The stronglygeneralized derivative is defined for a larger class of fuzzy-valued function than the H-derivative, and fuzzy differentialequations can have solutions which have a decreasing length

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2462 M. B. Ahmadi et al.

of their support. So we use this differentiability concept inthe present paper.

The Laplace transform method on fuzzy nth-order deriva-tive solves FTDEs in Allahviranloo and Barkhordari (2010)and corresponding fuzzy nth order and boundary value prob-lems and partial. In this way Laplace transforms reduce theproblem to an algebraic problem. This switching from oper-ations of calculus to algebraic operations on transforms iscalled operational calculus, a very important area of appliedmathematics, and for the engineer, the fuzzy Laplace trans-form method is practically the most important operationalmethod. The fuzzy Laplace transform also has the advantagethat it solves problems directly, fuzzy two order value prob-lems without first determining a general solution, and nonhomogeneous differential equations without first solving thecorresponding 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. InSect. 3 Laplace transform on fuzzy nth-order derivative andfor fuzzy two order differential equation are defined. Severalexamples are given in Sect. 4, and conclusions are drawn inSect. 5.

2 Preliminaries

We now recall some definitions needed through the paper.The basic definition of fuzzy numbers is given in (Friedmanet al. 1999; Ma et al. 1999) as:

Definition 2.1 A fuzzy number u in parametric form is a pair(u, u) of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfy thefollowing requirements:

1. u(r) is a bounded non-decreasing left continuous func-tion in (0, 1], and right continuous at 0,

2. u(r) is a bounded non-increasing left continuous functionin (0, 1], and right continuous at 0,

3. u(r) ≤ u(r), 0 ≤ r ≤ 1.

A crisp number α is simply represented by u(r) = u(r) =α, 0 ≤ r ≤ 1. We recall that for a < b < c which a, b, c ∈R, the triangular fuzzy number u = (a, b, c) determined bya, b, c is given such that u(r) = a + (b − a)r and u(r) =c − (c − b)r are the endpoints of the r-level sets, for allr ∈ [0, 1].

For arbitrary u = (u(r), u(r)), v = (v(r), v(r)) and k >

0 we define addition u ⊕ v, subtraction u � v, multiplicationand scaler multiplication by k as (see Friedman et al. 1999;Ma et al. 1999).

(a) Addition:

u ⊕ v = (u(r) + v(r), u(r) + v(r))

(b) Subtraction:

u � v = (u(r) − v(r), u(r) − v(r))

(c) Multiplication:

u � v = (min{u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)},max{u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)})

(d) Scaler multiplication:

k � u ={

(ku, ku), k ≥ 0,

(ku, ku), k < 0.

The Hausdorff distance between fuzzy numbers given byD : E × E −→ R+

⋃0,

D(u, v) = supr∈[0,1]

max{|u(r) − v(r)|, |u(r) − v(r)|},

where u = (u(r), u(r)), v = (v(r), v(r)) ⊂ R is utilized(see Bede and Gal 2005). Then, it is easy to see that D is ametric in E and has the following properties (see Puri andRalescu 1986)

(1) D(u ⊕ w, v ⊕ w) = D(u, v), ∀u, v, w ∈ E ,(2) D(k � u, k � v) = |k|D(u, v), ∀k ∈ R, u, v ∈ E ,(3) D(u⊕v,w⊕e) ≤ D(u, w)+D(v, e), ∀u, v, w, e ∈ E ,(4) (D, E) is a complete metric space.

Theorem 2.1 (See Allahviranloo and Gal 2004)

(1) If we define 0̃ = χ0, then 0̃ ∈ E is a neutral element withrespect to addition, i.e. u ⊕ 0̃ = 0̃⊕u = u, for all u ∈ E.

(2) With respect to 0̃, none of u ∈ E\R, has opposite in E.(3) For any a, b ∈ R with a, b ≥ 0 or a, b ≤ 0 and any

u ∈ E, we have (a + b) � u = a � u ⊕ b � u; for thegeneral a, b ∈ R, the above property does not necessarilyhold.

(4) For any λ ∈ R and any u, v ∈ E, we have λ� (u ⊕ v) =λ � u ⊕ λ � v;

(5) For any λ,μ ∈ R and any u ∈ E, we have λ�(μ�u) =(λ.μ) � u;

Definition 2.2 Let E be a set of all fuzzy numbers, we saythat f is fuzzy-valued-function if f : R → E .

It is well-known that the H-derivative (differentiabilityin the sense of Hukuhara) for fuzzy mappings was initiallyintroduced by Puri and Ralescu (1983) and it is based in theH-difference of sets, as follows.

Definition 2.3 Let x, y ∈ E . If there exists z ∈ E such thatx = y ⊕ z, then z is called the H-difference of x and y, andit is denoted by x −h y.

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Laplace transform formula on fuzzy nth-order derivative 2463

In this paper, the sign “−h” always stands for H-difference,and also note that x −h y �= x � y.

In this paper we consider the following definition whichwas introduced by Bede and Gal (2005), Bede et al. (2006b).

Definition 2.4 Let f : (a, b) → E and x0 ∈ (a, b). We saythat f is strongly generalized differential at x0 (Bede-Galdifferential). If there exists an element f

′(x0) ∈ E , such that

(1) for all h > 0 sufficiently small, ∃ f (x0 + h) −h f (x0),∃ f (x0) −h f (x0 − h) and the limits (in the metric D)

limh↘0f (x0+h)−h f (x0)

h = limh↘0f (x0)−h f (x0−h)

h

= f′(x0)

or(2) for all h > 0 sufficiently small, ∃ f (x0) −h f (x0 + h),

∃ f (x0 − h) −h f (x0) and the limits (in the metric D)

limh↘0f (x0)−h f (x0+h)

−h = limh↘0f (x0−h)−h f (x0)

−h

= f′(x0)

or(3) for all h > 0 sufficiently small, ∃ f (x0 + h) −h f (x0),

∃ f (x0 − h) −h f (x0) and the limits (in the metric D)

limh↘0f (x0+h)−h f (x0)

h = limh↘0f (x0−h)−h f (x0)

−h

= f′(x0)

or(4) for all h > 0 sufficiently small, ∃ f (x0) −h f (x0 + h),

∃ f (x0) −h f (x0 − h) and the limits (in the metric D)

limh↘0f (x0)−h f (x0+h)

−h = limh↘0f (x0)−h f (x0−h)

h

= f′(x0)

(the denominators of h and −h denote multiplication by1h and −1

h , respectively).

Remark 2.1 These case (3) and (4) introduced in Bede andGal (2005), in order to ensure a differentiable switch the case(1) and case (2) in Definition (2.4). Of course, as the authorsin Bede and Gal (2005) and in Chalco-Cano and Roman-Flores (2006) have stated, the cases (1) and (2) in Definition(2.4), are more important since case (3) and (4) in Definition(2.4) occur only on a discrete set of points.

As an example supporting these comments, let us con-sider c ∈ E\R be any fuzzy(non-real) constant and letf : [0, a] × E −→ E , f (t) = c � cost, t ∈ [0, a]. Itis natural to expect that f is differentiable everywhere in itsdomain. Let us observe that f is differentiable according toDefinition (2.4) (2), on each sub interval (2kπ, (2k + 1)π)

and differentiable according to Definition (2.4) (1), on eachsub interval of the form ((2k +1)π, 2(k +1)π), k ∈ Z . But,

at the points {kπ}, k ∈ Z , none of the cases (1) and (2) inDefinition (2.4) are fulfilled.

Namely, at these points the H-differences f (kπ + h) −h

f (kπ) and f (kπ) −h f (kπ − h) may not exist simulta-neously. Also, the H-differences f (kπ) −h f (kπ + h) andf (kπ − h) −h f (kπ) cannot exist simultaneously, so f isnot differentiable at kπ in none of the cases (1) and (2) ofdifferentiability in Definition (2.4). Instead, it will be differ-entiable as in the cases (3) and (4) in Definition (2.4). Indeed,above stated theorem does not cover the case when f (t, x)

has not constant monotonicity. In these cases (1) and (2) ofdifferentiability in Definition (2.4), so the cases (3) and (4) inDefinition (2.4) may become important as switch points. Inthe special case when f is a fuzzy-valued function, we havethe following result.

Theorem 2.2 (See e.g. Chalco-Cano and Roman-Flores2006). Let f : R → E be a function and denote f (t) =( f (t, r), f (t, r)), for each r ∈ [0, 1]. Then

(1) If f is (i)-differentiable, then f (t, r) and f (t, r) are dif-ferentiable functions and

f′(t) = ( f

′(t, r), f

′(t, r))

(2) If f is (ii)-differentiable, then f (t, r) and f (t, r) are dif-ferentiable functions and

f′(t) = ( f

′(t, r), f

′(t, r)).

Definition 2.5 (See Allahviranloo et al. 2009a, b; Allahvi-ranloo and Barkhordari 2010) Let f : (a, b) → E andx0 ∈ (a, b). We Define the nth order differential of f asfollow: Let f : (a, b) → E and x0 ∈ (a, b). We say that fis strongly generalized differentiable of the nth order at x0.If f, f ′, . . . , f (s−1) have been strongly generalized differen-tiable and there exists an element f (s)(x0) ∈ E,∀s = 1 . . . n,such that

(1) for all h > 0 sufficiently small, ∃ f (s−1)(x0 + h) −h

f (s−1)(x0),∃ f (s−1)(x0) −h f (s−1)(x0 − h) and the limits (in themetric D)

limh↘0f (s−1)(x0+h)−h f (s−1)(x0)

h

= limh↘0f (s−1)(x0)−h f (s−1)(x0−h)

h = f (s)(x0)

or(2) for all h > 0 sufficiently small, ∃ f (s−1)(x0) −h

f (s−1)(x0 + h),

∃ f (s−1)(x0 − h) −h f (s−1)(x0) and the limits (in themetric D)

limh↘0f (s−1)(x0)−h f (s−1)(x0+h)

−h

= limh↘0f (s−1)(x0−h)−h f (x0)

−h = f (s)(x0)

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2464 M. B. Ahmadi et al.

or(3) for all h > 0 sufficiently small, ∃ f (s−1)(x0 + h) −h

f (s−1)(x0),∃ f (s−1)(x0 − h) −h f (s−1)(x0) and the limits (in themetric D)

limh↘0f (s−1)(x0+h)−h f (s−1)(x0)

h

= limh↘0f (s−1)(x0−h)−h f (s−1)(x0)

−h = f (s)(x0)

or(4) for all h > 0 sufficiently small, ∃ f (s−1)(x0) −h

f (s−1)(x0 + h),∃ f (s−1)(x0) −h f (s−1)(x0 − h) and the limits (in themetric D)

limh↘0f (k−1)(x0)−h f (s−1)(x0+h)

−h

= limh↘0f (s−1)(x0)−h f (s−1)(x0−h)

h = f (s)(x0),

Definition 2.6 (Allahviranloo and Barkhordari 2010) Letf (x) be continuous fuzzy-value function. Suppose thatf (x) � e−px improper fuzzy Rimann integrable on [0,∞),then

∫ ∞0 f (x) � e−px dx is called fuzzy laplace transforms

and is denoted as:

L[ f (x)] =∞∫

0

f (x) � e−px dx (p > 0 and integer)

we have∞∫

0

f (x) � e−px dx

=⎛⎝

∞∫0

f (x, r) � e−px dx,

∞∫0

f (x, r) � e−px dx

⎞⎠

also by using the definition of classical laplace transform:

�[ f (x, r)] =∞∫

0

f (x, r) � e−px dx and �[ f (x, r)]

=∞∫

0

f (x, r) � e−px dx

then, we follow:

L[ f (x)] = (�[ f (x, r)], �[ f (x, r)]).Theorem 2.3 (Allahviranloo and Barkhordari 2010) Letf ′(x) be an integrable fuzzy-valued function, and f (x) isthe primitive of f ′(x) on [0,∞). Then

L[ f ′(x)] = pL[ f (x)] −h f (0)

where f is (i)-differentiable

or

L[ f ′(x)] = (� f (0)) −h (�pL[ f (x)])where f is (ii)-differentiable

Theorem 2.4 (Allahviranloo and Barkhordari 2010) Letf (x), g(x) be continuous-fuzzy -valued functions and c1,c2

are constant. Suppose that f (x)e−px , g(x)e−px are improperfuzzy Rimann-integrable on [0,∞), then

L[(c1 f (x)) + (c2g(x))] = (c1 L[ f (x)]) + (c2 L[g(x)]).

3 Laplace transform on fuzzy Nth-order derivative

In this section, using definition of fuzzy laplace transformand its formula on first order derivative (see in Allahviranlooand Barkhordari 2010), we introduce definition and theoremsabout fuzzy laplace transform on Nth-order derivative.

Definition 3.1 Let u ∈ E and u = (u(r), u(r)),then wedefine −hu and −h(�u), as follows:

(a) −hu = (−u(r),−u(r))

(b) −h(�u) = (u(r), u(r))

(c) −h(�a) −h (�b)u = (ab) � u where a, b ∈ R(d) −h −h (�u) = (−u(r),−u(r))

Theorem 3.1 Let f (t) : (a, b) → E and its derivativesf ′(t), f ′′(t), . . . , f (n−1)(t) be continuous functions for allt ≥ 0 and let f is strongly generalized differentiable ofthe Nth-order such that, there exists elements, f (s)(t0) ∈E, ∀ S = 0, . . . , n then, the laplace transform of f (n)(t) isgiven by

L[ f (n)(x)] =n∏

k=1

s(k) � L[ f (t)] −hn∏

k=2

s(k)sign(1)

� f (t0) −h · · · −h s(n)sign(n − 1)

� f (n−2)(t0) −h sign(n) � f (n−1)(t0)

where

sign(k) ={

1 if f (k) be (i)-derivative−h(�1) if f (k) be (ii)-derivative

and

s(k) ={

p if f (k) be (i)-derivative−h(�p) if f (k) be (ii)-derivative

Proof By induction, we show following steps:Step 1. we first consider the case when n = 1. Then, by

the formula

L[ f ′(t)] = s(1)L[ f ] −h sign(1) f (t0)

where

s(1) ={

p if f (′) be (i)-derivative−h(�p) if f (′) be (i i)-derivative

and it is equivalent to laplace transform formula in Allahvi-ranloo and Barkhordari (2010)

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Laplace transform formula on fuzzy nth-order derivative 2465

Step 2. suppose that n = k holds and by using n = k, weobtain formula for n = k + 1

L[ f (k+1)]= L[( f k)′] = s(k+1) � L[ f (k)]−hsign(k + 1) � f (k)(t0)

= s(k + 1) �(

k∏i=1

s(i) � L[ f (t)] −hk∏

i=2

s(i) � sign(1)

� f (t0) −h · · · −h s(k)sign(k − 1) � f (k−2)(t0)−h

sign(k) � f (k−1)(t0)

)−h sign(k + 1) � f (k)(t0)

=k+1∏i=1

s(i) � L[ f (t)] −hk+1∏i=2

s(i)sign(1)

� f (t0)−h · · · −hs(k+1)s(k)sign(k−1) � f (k−2)(t0)−h

s(k+1)sign(k) � f (k−1)(t0))−hsign(k+1) � f (k)(t0)

so completes the proof. ��Theorem 3.2 Let f ′′(x) be an integrable fuzzy-valued func-tion, and f (x), f ′(x) are the primitive of f ′(x), f ′′(x) on[0,∞). Then

L[ f ′′(x)] = p2 � L[ f (x)] −h p � f (0) −h f ′(0)

where f is (i)-differentiable and f ′ is (i)-differentiable or

L[ f ′′(x)] = p2 � L[ f (x)] −h p � f (0) � f ′(0)

where f is (i i)-differentiable and f ′ is (i i)-differentiable or

L[ f ′′(x)] = −h(�p2) �L[ f (x)](�p) � f (0) � f ′(0)

where f is (i)-differentiable and f ′ is (i i)-differentiable or

L[ f ′′(x)] = −h(�p2) � L[ f (x)](�p) � f (0) −h f ′(0)

where f is (ii)-differentiable and f ′ is (i)-differentiable

Proof Proof follows easily by Theorem 3.1. ��Theorem 3.3 Let f : R → E be a fuzzy valued-functionand denote f (t) = ( f (t, r), f (t, r)), for each r ∈ [0, 1].Then

f (n)(t, r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

( f (n)(t, r), f(n)

(t, r)) if number of(ii)-derivative is even

( f(n)

(t, r), f (n)(t, r)) if number of(ii)-derivative is odd

Proof Since the proof procedure is similar for both of twocases, we consider case (1) without loss of generality. Step1: we first consider n = 1.

(a) if number of (ii)-derivative is 1 f ′(t, r) = ( f′(t, r),

f ′(t, r))

(b) if number of (ii)-derivative is 0 f ′(t, r) = ( f ′(t, r),

f′(t, r))

Step 2: suppose that n = k holds, we obtain for n = k +1using n = k. Two cases occur:

case(I): if number of ii-derivative to (k +1)-order be evenand in (k + 1)th-order is differentiable in the first form (i),then we have:

f (k)(t) = ( f (k)(t, r), f(k)

(t, r))

now, consider g(t) as follow:

g(t) = f (k)(t)

if h > 0 and r ∈ [0, 1], we have

g(t + h) −h g(t)

= (g(t + h, r) − g(t, r), g(t + h, r) − g(t, r))

= ( f (k)(t + h, r) − f (k)(t, r), f(k)

(t + h, r) − f(k)

(t, r))

and, multiplying by 1h , we have:

g(t + h) −h g(t)

h

=(

f (k)(t+h, r)− f (k)(t, r)

h,

f(k)

(t+h, r)− f(k)

(t, r)

h

)

similarly, we have:

f (k)(t) −h f (k)(t − h)

h

=(

f (k)(t, r)− f (k)(t−h, r)

h,

f(k)

(t, r)− f(k)

(t−h, r)

h

)

passing to the limit, we have:

f (k+1)(t) = ( f (k+1)(t, r), f(K+1)

(t, r))

and if number of (ii)-derivative to (k + 1)-order be even andin (k + 1)th-order is differentiable in the second form (ii),then we have:

f (k)(t) = ( f(k)

(t, r), f (k)(t, r))

now, consider g(t) as follow:

g(t) = f (k)(t)

if h < 0 and r ∈ [0, 1], we have

g(t + h) −h g(t)

= (g(t + h, r) − g(t, r), g(t + h, r) − g(t, r))

= ( f(k)

(t + h, r) − f(k)

(t, r), f (k)(t + h, r) − f (k)(t, r))

and, multiplying by 1h , we have:

g(t + h) −h g(t)

h

=(

f (k)(t+h, r)− f (k)(t, r)

h,

f(k)

(t+h, r)− f(k)

(t, r)

h

).

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2466 M. B. Ahmadi et al.

Similarly, we have:

f (k)(t) −h f (k)(t − h)

h

=(

f (k)(t, r)− f (k)(t−h, r)

h,

f(k)

(t, r)− f(k)

(t−h, r)

h

).

Passing to the limit, we have:

f (k+1)(t) = ( f (k+1)(t, r), f(k+1)

(t, r))

case(II) if number of (ii)-derivative to (k +1)th-order be oddin (k + 1)th-order is differentiable in the second form or thefirst form (i), then for h < 0 or h > 0 and r ∈ [0, 1], similarto case(I), we have:

f (k+1)(t, r) = ( f (k+1)(t, r), f(k+1)

(t, r))

so, completes the proof. ��

3.1 2nd-order fuzzy differential equation

we shall now discuss how the laplace transform methodsolves fuzzy differential equations. We begin with an initialvalue problem

y′′ + ay

′ + by = r̃(t) y(t0) = k̃0, y′(t0) = k̃1

with constant a and b.By using laplace transform method, we have:

L[y′′ ] + aL[y

′ ] + bL[y] = L [̃r(t)]Then, we have the following alternatives for solving:

Case I. If we consider y(t) and y′(t) by using (i)-differentiable, then then we have

p2 � L[y(t)] −h p � y(t0) −h y′(t0) ⊕ ap � L[y(t)]−ha � y(t0) ⊕ b � L[y(t)] = L [̃r(t)]

Case II. If we consider y(t) and y′(t) by using (ii)-differentiable, then then we have

p2 � L[y(t)] −h p � y(t0) � y′(t0) ⊕ a(−h(�p)) � L[y(t)]�a � y(t0) ⊕ b � L[y(t)] = L [̃r(t)]

Case III. If we consider y(t) by using (i)-differentiableand y′(t) by using (ii)-differentiable, then then we have

(−h(�p2)) � L[y(t)](�p) � y(t0) � y′(t0) ⊕ ap � L[y(t)]−ha � y(t0) ⊕ b � L[y(t)] = L [̃r(t)]

Case IV. If we consider y(t) by using (ii)-differentiableand y′(t) by using (i)-differentiable, then then we have

(−h(�p2)) � L[y(t)](�p) � y(t0) −h y′(t0)⊕(−h � p)a � L[y(t)](�a) � y(t0) ⊕ b � L[y(t)]

= L [̃r(t)]using this representation for four cases, we have the followingexamples.

4 Examples

Example 4.1 Let us consider the second order fuzzy differ-ential equation equation⎧⎨⎩

y′′ − 3y′ + 2y = 4̃y′(0) = 0y(0) = 1̃

where 1̃ = (0.8+0.2r, 1.5−0.5r) and 4̃ = (3.2+0.8r, 5−r).

by using fuzzy laplace transform method, we have:

L[y′′] � 3L[y′] ⊕ 2L[y] = L [̃4]in (i)-differentiable, then by using case(I), we have

L[y(t, r)] = (3.2 + 0.8r)1

p(p − 1)(p − 2)

+(0.8 + 0.2r)p − 3

(p − 1)(p − 2)

L[y(t, r)] = (5 − r)1

p(p − 1)(p − 2)

+(1.5 − 0.5r)p − 3

(p − 1)(p − 2)

Hence solution is as follows:

y(t, r) = (3.2 + 0.8r)

(1

2− ex + 1

2e2x

)

+ (0.8 + 0.2r)(2ex − e2x )

y(t, r) = (5 − r)

(1

2− ex + 1

2e2x

)

+ (1.5 − 0.5r)(2ex − e2x )

The y(t, r) and y(t, r) at r = 0 and r = 0.5 are presented inFigs. 1 and 2.

Now, if we consider r = 1, then

y(t) = y(t, 1) = y(t, 1) = 2 − 2ex + e2x

which is displayed in Fig. 3.

Example 4.2 consider the initial value problem equation⎧⎨⎩

y′′ + 4y = 4̃xy′(0) = 0y(0) = 1̃

where 1̃ = (0.8+0.2r, 1.5−0.5r) and 4̃ = (3.2+0.8r, 5−r).

in (ii)-differentiable, then by using case(II), we have

L[y(t, r)] − py(0, r) − y′(0, r) + 4L[y(t, r)]= L[(3.2 + 0.8r)x]

L[y(t, r)] − py(0, r) − y′(0, r) + 4L[y(t, r)]= L[(5 − r)x]

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Laplace transform formula on fuzzy nth-order derivative 2467

Fig. 1 y(t, r) and y(t, r) at r = 0 for Example 4.1

Fig. 2 y(t, r) and y(t, r) at r = 0.5 for Example 4.1

Fig. 3 y(t) at r = 1 for Example 4.1

Fig. 4 y(t, r) and y(t, r) at r = 0 for Example 4.2

Fig. 5 y(t, r) and y(t, r) at r = 0.5 for Example 4.2

Hence solution is as follows:

y(t, r) = (0.8 + 0.2r)

(x − 1

2sin2x + cos2x

)

y(t, r) = (5 − r)

(1

4x − 1

8sin2x

)+ (1.5 − 0.5r)cos2x

The y(t, r) and y(t, r) at r = 0 and r = 0.5 are presented inFigs. 4 and 5.

Now, if we consider r = 1, then

y(t, 1) = y(t, 1) = x − 1

2sin2x + cos2x

which is displayed in Fig. 6.

From, Examples 4.1 and 4.2, we see that the solution ofa FDE is dependent of the election of the derivative:in the(i)-differentiable or in the (ii)-differentiable. Thus, as in theabove examples, the solution can be a adequately chosen.

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2468 M. B. Ahmadi et al.

Fig. 6 y(t) at r = 1 for Example 4.1

On the other hand, it is clear that in this new procedure theunicity of the solution is lost, but it is a expected situation inthe fuzzy context.

5 Conclusion

In this paper, the Laplace transform formula provided solu-tions to fuzzy Nth order differential equation which is inter-preted by using the strongly generalized differentiability con-cept. This may confer solutions which have a decreasinglength of their support. The efficiency of method was illus-trated by a numerical example.

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