solving second-order fuzzy differential equations by the fuzzy

14
ElJaoui et al. Advances in Difference Equations (2015) 2015:66 DOI 10.1186/s13662-015-0414-x RESEARCH Open Access Solving second-order fuzzy differential equations by the fuzzy Laplace transform method Elhassan ElJaoui * , Said Melliani and L Saadia Chadli * Correspondence: [email protected] Laboratoire de Mathématiques Appliquées & Calcul Scientifique, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal, 23000, Morocco Abstract In this paper, we study the fuzzy Laplace transforms introduced by the authors in (Allahviranloo and Ahmadi in Soft Comput. 14:235-243, 2010) to solve only first-order fuzzy linear differential equations. We extend and use this method to solve second-order fuzzy linear differential equations under generalized Hukuhara differentiability. Keywords: fuzzy differential equation; fuzzy second-order differential equation; fuzzy Laplace transform; generalized Hukuhara differentiability 1 Introduction In recent years, the theory of FDEs has attracted widespread attention and has been rapidly growing. It was massively studied by several authors (see [–]). Concerning the solutions of this kind of equations, some numerical methods and algorithms have been developed by other researchers (see [–]). Allahviranloo et al. proposed in [] a novel method for solving fuzzy linear differential equations, which its construction based on the equivalent integral forms of original prob- lems under the assumption of strongly generalized differentiability. But their method was limited to solving only fuzzy linear differential equations with crisp constant coefficients, and its main result was formal and lacks proof. Motivated by their work, we have developed and extended in (see []) this op- erator method to solve some first-order fuzzy linear differential equations, with variable coefficients. Moreover, we gave the general formula’s solution with necessary proofs. Before, in , Allahviranloo and Ahmadi introduced in [] the fuzzy Laplace trans- form, which they used under the strongly generalized differentiability, in an analytic solu- tion method for some first-order fuzzy differential equations (FDEs). In their main result the authors established the relation between the fuzzy Laplace trans- forms of a fuzzy function and its first derivative. They gave two numerical examples to illustrate the efficiency of the method, but these two examples are all first-order FDEs. The aim of this work is to develop their method and to extend their main result by estab- lishing the relationship between the fuzzy Laplace transforms of a function and its second © 2015 ElJaoui et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons At- tribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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Page 1: Solving second-order fuzzy differential equations by the fuzzy

ElJaoui et al. Advances in Difference Equations (2015) 2015:66 DOI 10.1186/s13662-015-0414-x

R E S E A R C H Open Access

Solving second-order fuzzy differentialequations by the fuzzy Laplace transformmethodElhassan ElJaoui*, Said Melliani and L Saadia Chadli

*Correspondence:[email protected] de MathématiquesAppliquées & Calcul Scientifique,Sultan Moulay Slimane University,P.O. Box 523, Beni Mellal, 23000,Morocco

AbstractIn this paper, we study the fuzzy Laplace transforms introduced by the authors in(Allahviranloo and Ahmadi in Soft Comput. 14:235-243, 2010) to solve only first-orderfuzzy linear differential equations. We extend and use this method to solvesecond-order fuzzy linear differential equations under generalized Hukuharadifferentiability.

Keywords: fuzzy differential equation; fuzzy second-order differential equation;fuzzy Laplace transform; generalized Hukuhara differentiability

1 IntroductionIn recent years, the theory of FDEs has attracted widespread attention and has been rapidlygrowing. It was massively studied by several authors (see [–]). Concerning the solutionsof this kind of equations, some numerical methods and algorithms have been developedby other researchers (see [–]).

Allahviranloo et al. proposed in [] a novel method for solving fuzzy linear differentialequations, which its construction based on the equivalent integral forms of original prob-lems under the assumption of strongly generalized differentiability. But their method waslimited to solving only fuzzy linear differential equations with crisp constant coefficients,and its main result was formal and lacks proof.

Motivated by their work, we have developed and extended in (see []) this op-erator method to solve some first-order fuzzy linear differential equations, with variablecoefficients. Moreover, we gave the general formula’s solution with necessary proofs.

Before, in , Allahviranloo and Ahmadi introduced in [] the fuzzy Laplace trans-form, which they used under the strongly generalized differentiability, in an analytic solu-tion method for some first-order fuzzy differential equations (FDEs).

In their main result the authors established the relation between the fuzzy Laplace trans-forms of a fuzzy function and its first derivative.

They gave two numerical examples to illustrate the efficiency of the method, but thesetwo examples are all first-order FDEs.

The aim of this work is to develop their method and to extend their main result by estab-lishing the relationship between the fuzzy Laplace transforms of a function and its second

© 2015 ElJaoui et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons At-tribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly credited.

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ElJaoui et al. Advances in Difference Equations (2015) 2015:66 Page 2 of 14

derivative, with the purpose of solving second-order fuzzy linear differential equationsunder strongly generalized differentiability.

The remainder of this paper is organized as follows:In Section , which is reserved for some preliminaries, we collect some useful results

on fuzzy derivation and integration. In Section , we first introduce the fuzzy Laplacetransform, we recall its basic properties. Then we announce and prove our main result. InSection , we propose the procedure for solving second-order FDEs by the fuzzy Laplacetransform. For illustration, we give some numerical examples in Section . In the last sec-tion, we present our conclusion and a further research topic.

2 PreliminariesBy PK (R) we denote the family of all nonempty, compact, and convex subsets of R anddefine the addition and scalar multiplication in PK (R) as usual. Denote

E ={

u : R → [, ]|u satisfies (i)-(iv) below}

,

where(i) u is normal, i.e. ∃x ∈R for which u(x) = ,

(ii) u is fuzzy convex, i.e.

u(λx + ( – λy)

) ≥ min(u(x), u(y)

)for any x, y ∈R, and λ ∈ [, ],

(iii) u is upper semi-continuous,(iv) supp u = {x ∈R|u(x) > } is the support of the u, and its closure cl (supp u) is

compact.For < α ≤ , denote

[u]α ={

x ∈R|u(x) ≥ α}

.

Then, from (i)-(iv), it follows that the α-level set [u]α ∈ PK (R) for all ≤ α ≤ .According to Zadeh’s extension principle, we have addition and scalar multiplication in

fuzzy-number space E as usual.It is well known that the following properties are true for all levels:

[u + v]α = [u]α + [v]α , [ku]α = k[u]α .

Let D : E × E → [,∞) be a function which is defined by the equation

D(u, v) = sup≤α≤

d([u]α , [v]α

),

where d is the Hausdorff metric defined in PK (R). Then it is easy to see that D is a metricin E and has the following properties []:

() (E, D) is a complete metric space;() D(u + w, v + w) = D(u, v) for all u, v, w ∈ E;() D(ku, kv) = |k|D(u, v) for all u, v ∈ E and k ∈R;() D(u + w, v + t) ≤ D(u, v) + D(w, t) for all u, v, w, t ∈ E.

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Definition . A fuzzy number u in parametric form is a pair (u, u) of functions u(r), u(r), ≤ r ≤ , which satisfy the following requirements:

() u(r) is a bounded non-decreasing left continuous function in (, ], and rightcontinuous at ;

() u(r) is a bounded non-increasing left continuous function in (, ], and rightcontinuous at ;

() u(r) ≤ u(r) for all ≤ r ≤ .

A crisp number k is simply represented by u(r) = u(r) = k, ≤ r ≤ .Let T = [c, d] ⊂R be a compact interval.

Definition . A mapping F : T → E is strongly measurable if for all α ∈ [, ] the set-valued function Fα : T →PK (R) defined by Fα(t) = [F(t)]α is Lebesgue measurable.

A mapping F : T → E is called integrably bounded if there exists an integrable functionk such that ‖x‖ ≤ k(t) for all x ∈ F(t).

Definition . Let F : T → E, then the integral of F over T , denoted by∫

T F(t) dt or∫ d

c F(t) dt, is defined by the equation

[∫

TF(t) dt

=∫

TFα(t) dt; α ∈ ], ]

i.e.

[∫

TF(t) dt

={∫

Tf (t) dt|f : T →R is a measurable selection for Fα

}.

Also, a strongly measurable and integrably bounded mapping F : T → E is said to be inte-grable over T if

∫T F(t) dt ∈ E.

Proposition . (Aumann []) If F : T → E is strongly measurable and integrablybounded, then F is integrable.

For more measurability, integrability properties for fuzzy set-valued mappings see [, ,].

Theorem . (see [, ]) Let f (x) be a fuzzy valued-function on [a,∞[ which is repre-sented by (f (x, r), f (x, r)). For any fixed r ∈ [, ], assume f (x, r), f (x, r) are Riemann inte-grable on [a, b] for every b ≥ a, and assume there are two positive constants M(r) and M(r)such that

∫ ba |f (x, r)|dx ≤ M(r) and

∫ ba |f (x, r)|dx ≤ M(r) for every b ≥ a. Then f (x) is im-

proper fuzzy Riemann integrable on [a,∞[ and the improper fuzzy Riemann integral is afuzzy number. Furthermore, we have

∫ ∞

af (x) dx =

(∫ ∞

af (x, r) dx,

∫ ∞

af (x, r) dx

).

Proposition . (see []) If each of f (x) and g(x) is a fuzzy valued function and fuzzy Rie-mann integrable on [a,∞[ then f (x) + g(x) is fuzzy Riemann integrable on [a,∞[. Moreover,

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we have∫ ∞

a

(f (x) + g(x)

)dx =

∫ ∞

af (x) dx +

∫ ∞

ag(x) dx.

For u, v ∈ E, if there exists w ∈ E such that u = v + w, then w is the Hukuhara differenceof u and v denoted by u v.

Definition . We say that a mapping f : (a, b) → E is strongly generalized differentiableat x ∈ (a, b) if there exists an element f ′(x) ∈ E such that

(i) for all h > sufficiently small, there exist f (x + h) f (x), f (x) f (x – h), and thelimits

limh→+

f (x + h) f (x)h

= limh→+

f (x) f (x – h)h

= f ′(x)

or(ii) for all h > sufficiently small, there exist f (x) f (x + h), f (x – h) f (x), and the

limits

limh→+

f (x) f (x + h)(–h)

= limh→+

f (x – h) f (x)(–h)

= f ′(x)

or(iii) for all h > sufficiently small, there exist f (x + h) f (x), f (x – h) f (x), and the

limits

limh→+

f (x + h) f (x)h

= limh→+

f (x – h) f (x)(–h)

= f ′(x)

or(iv) for all h > sufficiently small, there exist f (x) f (x + h), f (x) f (x – h), and the

limits

limh→+

f (x) f (x + h)(–h)

= limh→+

f (x) f (x – h)h

= f ′(x).

The following theorem (see []) allows us to consider case (i) or (ii) of the previousdefinition almost everywhere in the domain of the functions under discussion.

Theorem . Let f : (a, b) → E be strongly generalized differentiable on each point x ∈(a, b) in the sense of Definition ., (iii) or (iv). Then f ′(x) ∈ R for all x ∈ (a, b).

Theorem . (see e.g. []) Let f : R → E be a function and denote f (t) = (f (t, r), f (t, r)),for each r ∈ [, ]. Then

() If f is (i)-differentiable, then f (t, r) and f (t, r) are differentiable functions andf ′(t) = (f ′(t, r), f ′(t, r)).

() If f is (ii)-differentiable, then f (t, r) and f (t, r) are differentiable functions andf ′(t) = (f ′(t, r), f ′(t, r)).

Definition . We say that a mapping f : (a, b) → E is strongly generalized differentiableof the second-order at x ∈ (a, b); if there exists an element f ′′(x) ∈ E such that

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(i) for all h > sufficiently small, there exist f ′(x + h) f ′(x), f ′(x) f ′(x – h), andthe limits

limh→+

f ′(x + h) f ′(x)h

= limh→+

f ′(x) f ′(x – h)h

= f ′′(x)

or(ii) for all h > sufficiently small, there exist f ′(x) f ′(x + h), f ′(x – h) f ′(x), and

the limits

limh→+

f ′(x) f ′(x + h)(–h)

= limh→+

f ′(x – h) f ′(x)(–h)

= f ′′(x)

or(iii) for all h > sufficiently small, there exist f ′(x + h) f ′(x), f ′(x – h) f ′(x), and

the limits

limh→+

f ′(x + h) f ′(x)h

= limh→+

f ′(x – h) f ′(x)(–h)

= f ′′(x)

or(iv) for all h > sufficiently small, there exist f ′(x) f ′(x + h), f ′(x) f ′(x – h), and

the limits

limh→+

f ′(x) f ′(x + h)(–h)

= limh→+

f ′(x) f ′(x – h)h

= f ′′(x).

All the limits are taken in the metric space (E, D), and at the end points of (a, b) and weconsider only one-sided derivatives.

3 Fuzzy Laplace transformDefinition . (see []) Let f (x) be a continuous fuzzy-valued function. Suppose thate–pxf (x) is improper fuzzy Riemann integrable on [,∞[, then

∫ ∞ e–pxf (x) dx is called the

fuzzy Laplace transform of f and is denoted

L[f (x)

]=

∫ ∞

e–pxf (x) dx, p > .

Denote by L(g(x)) the classical Laplace transform of a crisp function g(x).Since

∫ ∞ e–pxf (x) dx = (

∫ ∞ e–pxf (x, r) dx,

∫ ∞ e–pxf (x, r) dx), then

L[f (x)

]=

(L

(f (x, r)

),L

(f (x, r)

)).

Theorem . (see []) Let f ′(x) be an integrable fuzzy-valued function and f (x) the prim-itive of f ′(x) on [,∞[. Then

(a) if f is (i)-differentiable:

L[f ′(x)

]= pL

[f (x)

] f () ()

(b) or if f is (ii)-differentiable:

L[f ′(x)

]=

(–f ()

) (–p)L[f (x)

]. ()

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Theorem . (see []) Let f (x), g(x) be continuous fuzzy-valued functions and c, c tworeal constants, then

L[cf (x) + cg(x)

]= cL

[f (x)

]+ cL

[g(x)

].

Now, we present our main result about the relation between the Laplace transforms ofa function and its second derivative.

Theorem . Let f (x) be a continuous fuzzy-valued function such that e–pxf (x), e–pxf ′(x),and e–pxf ′′(x) exist, are continuous, and Riemann integrable on [,∞[. We distinguish thefollowing cases:

(a) If f (x) and f ′(x) are (i)-differentiable, then

L[f ′′(x)

]=

{pL

[f (x)

] pf ()} f ′(). ()

(b) If f (x) is (i)-differentiable and f ′(x) is (ii)-differentiable, then

L[f ′′(x)

]=

(–f ′()

) {–pL

[f (x)

] (–pf ()

)}. ()

(c) If f (x) is (ii)-differentiable and f ′(x) is (i)-differentiable, then

L[f ′′(x)

]=

{(–pf ()

) (–pL

[f (x)

])} f ′(). ()

(d) If f (x) is (ii)-differentiable and f ′(x) is (ii)-differentiable, then

L[f ′′(x)

]=

(–f ′()

) {pf () pL

[f (x)

]}. ()

Proof(a) Assume that f (x) and f ′(x) are (i)-differentiable, then applying () to f (x) and f ′(x),

respectively, we get

L[f ′(x)

]= pL

[f (x)

] f () and L[f ′′(x)

]= pL

[f ′(x)

] f ′().

Combining these identities yields

L[f ′′(x)

]= p

{pL

[f (x)

] f ()} f ′()

={

pL[f (x)

] pf ()} f ′().

(b) Assume that f (x) is (i)-differentiable and f ′(x) is (ii)-differentiable, then applying ()and () to f (x) and f ′(x), respectively, we get

L[f ′(x)

]= pL

[f (x)

] f () and L[f ′′(x)

]=

(–f ′()

) (–p)L[f ′(x)

].

By combination of these identities we get

L[f ′′(x)

]=

(–f ′()

) (–p){

pL[f (x)

] f ()}

=(–f ′()

) {–pL

[f (x)

] (–pf ()

)}.

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(c) If f (x) is (ii)-differentiable and f ′(x) is (i)-differentiable, then

L[f ′(x)

]=

(–f ()

) (–p)L[f (x)

]and L

[f ′′(x)

]= pL

[f ′(x)

] f ′().

By combination of these identities we get

L[f ′′(x)

]= p

{(–f ()

) (–p)L[f (x)

]} f ′()

={(

–pf ()) (

–p)L[f (x)

]} f ′().

(d) Assume that f (x) and f ′(x) are (ii)-differentiable, then

L[f ′(x)

]=

(–f ()

) (–p)L[f (x)

]and L

[f ′′(x)

]=

(–f ′()

) (–p)L[f ′(x)

].

Combining these identities yields

L[f ′′(x)

]=

(–f ′()

) (–p){(

–f ()) (–p)L

[f (x)

]}

=(–f ′()

) {pf () pL

[f (x)

]}. �

4 Fuzzy Laplace transform algorithm for second-order fuzzy differentialequations

Our aim now is to solve the following second-order fuzzy differential equation, using thefuzzy Laplace transform method under strongly generalized differentiability:

⎧⎪⎪⎨

⎪⎪⎩

y′′(t) = f (t, y(t), y′(t)),

y() = y = (y, y) ∈ E,

y′() = z = (z, z) ∈ E,

()

where y(t) = (y(t,α), y(t,α)) is a fuzzy function of t ≥ and f (t, y(t), y′(t)) is a fuzzy-valuedfunction, which is linear with respect to (y(t), y′(t)).

By using the fuzzy Laplace transform, we obtain

L[y′′(t)

]= L

[f(t, y(t), y′(t)

)]. ()

Then we have the following alternatives for solving ():(a) Case I: If y and y′ are (i)-differentiable: y′(t) = (y′(t,α), y′(t,α)) and

y′′(t) = (y′′(t,α), y′′(t,α)) and

L[y′′(t)

]=

{pL

[y(t)

] py()} y′().

Therefore

L[f(t, y(t), y′(t)

)]=

{pL

[y(t)

] py} z.

Hence⎧⎨

⎩L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py

(α) – z(α),

L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py(α) – z(α),()

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where f (t, y(t), y′(t),α) = min{f (t, u, v)/u ∈ (y(t,α), y(t,α)); v ∈ (y′(t,α), y′(t,α))} andf (t, y(t), y′(t),α) = max{f (t, u, v)/u ∈ (y(t,α), y(t,α)); v ∈ (y′(t,α), y′(t,α))}. Assumethat this leads to

⎧⎨

⎩L[y(t,α)] = H(p,α),

L[y(t,α)] = K(p,α),

where the couple (H(p,α), K(p,α)) is a solution of the system ().By using the inverse Laplace transform we get

⎧⎨

⎩y(t,α) = L–[H(p,α)],

y(t,α) = L–[K(p,α)].

(b) Case II: If y is (i)-differentiable and y′ is (ii)-differentiable: y′(t) = (y′(t,α), y′(t,α)) andy′′(t) = (y′′(t,α), y′′(t,α)) and

L[y′′(t)

]=

(–y′()

) {–pL

[y(t)

] (–py()

)}.

Therefore

L[f(t, y(t), y′(t)

)]= (–z) {

–pL[y(t)

] (–py)}

.

Hence⎧⎨

⎩L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py

(α) – z(α),

L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py(α) – z(α).()

Assume that this implies

⎧⎨

⎩L[y(t,α)] = H(p,α),

L[y(t,α)] = K(p,α),

where (H(p,α), K(p,α)) is a solution of the system ().By using the inverse Laplace transform we get

⎧⎨

⎩y(t,α) = L–[H(p,α)],

y(t,α) = L–[K(p,α)].

(c) Case III: If y is (ii)-differentiable and y′ is (i)-differentiable: y′(t) = (y′(t,α), y′(t,α))and y′′(t) = (y′′(t,α), y′′(t,α)) and

L[y′′(t)

]=

{(–py()

) (–pL

[y(t)

])} y′().

Therefore

L[f(t, y(t), y′(t)

)]=

{(–py()

) (–pL

[y(t)

])} y′().

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Hence⎧⎨

⎩L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py

(α) – z(α),

L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py(α) – z(α).()

Assume that this leads to⎧⎨

⎩L[y(t,α)] = H(p,α),

L[y(t,α)] = K(p,α),

where (H(p,α), K(p,α)) is a solution of the system ().By using the inverse Laplace transform we get

⎧⎨

⎩y(t,α) = L–[H(p,α)],

y(t,α) = L–[K(p,α)].

(d) Case IV: If y and y′ are (ii)-differentiable: y′(t) = (y′(t,α), y′(t,α)) andy′′(t) = (y′′(t,α), y′′(t,α)) and

L[y′′(t)

]=

(–y′()

) {py() pL

[y(t)

]}.

Therefore

L[f(t, y(t), y′(t)

)]=

(–y′()

) {py() pL

[y(t)

]}.

Hence⎧⎨

⎩L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py

(α) – z(α),

L[f (t, y(t), y′(t),α)] = pL[y(t,α)] – py(α) – z(α).()

Assume that this implies

⎧⎨

⎩L[y(t,α)] = H(p,α),

L[y(t,α)] = K(p,α),

where (H(p,α), K(p,α)) is a solution of the system ().By using the inverse Laplace transform we get

⎧⎨

⎩y(t,α) = L–[H(p,α)],

y(t,α) = L–[K(p,α)].

Remark . To show that H(p,α) and K(p,α) can easily be calculated, we suppose thatthe fuzzy linear function f is given by f (t, y(t), y′(t)) = ay(t) + by′(t) + c(t), where a, b arereal constants and c(t) is a crisp mapping.

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We have to discuss the following cases:() Case (I.): If a ≥ and b ≥ , then the system () is equivalent to

⎧⎨

⎩pL[y(t,α)] – py

(α) – z(α) = (a + bp)L[y(t,α)] – by

(α) + L[c(t)],

pL[y(t,α)] – py(α) – z(α) = (a + bp)L[y(t,α)] – by(α) + L[c(t)].

By consequence⎧⎨

⎩H(p,α) = L[y(t,α)] =

(p–b)y(α)+z(α)+L[c(t)]p–bp–a ,

K(p,α) = L[y(t,α)] = (p–b)y(α)+z(α)+L[c(t)]p–bp–a .

() Case (I.): If a ≥ and b < , then () is equivalent to the following system:⎧⎨

⎩(p – a)L[y(t,α)] – bpL[y(t,α)] = py

(α) + z(α) – by(α) + L[c(t)],

bpL[y(t,α)] + (p – a)L[y(t,α)] = py(α) + z(α) – by(α) + L[c(t)].

Denote⎧⎨

⎩B(p,α) = py

(α) + z(α) – by(α) + L[c(t)],

C(p,α) = py(α) + z(α) – by(α) + L[c(t)].

()

Hence⎧⎨

⎩H(p,α) = L[y(t,α)] = (p–a)B(p,α)+bpC(p,α)

(p–a)+(bp) ,

K(p,α) = L[y(t,α)] = (p–a)C(p,α)–bpB(p,α)(p–a)+(bp) .

() Case (I.): If a < and b ≥ , then () is equivalent to the following system:⎧⎨

⎩(p – bp)L[y(t,α)] – aL[y(t,α)] = py

(α) + z(α) – by(α) + L[c(t)],

–aL[y(t,α)] + (p – bp)L[y(t,α)] = py(α) + z(α) – by(α) + L[c(t)].

Therefore⎧⎨

⎩H(p,α) = L[y(t,α)] = (p–bp)B(p,α)+aC(p,α)

(p–bp)+a ,

K(p,α) = L[y(t,α)] = (p–a)C(p,α)+aB(p,α)(p–bp)+a .

() Case (I.): If a < and b < , then () is equivalent to the following system:⎧⎨

⎩pL[y(t,α)] – (a + bp)L[y(t,α)] = py

(α) + z(α) – by(α) + L[c(t)],

–(a + bp)L[y(t,α)] + pL[y(t,α)] = py(α) + z(α) – by(α) + L[c(t)].

Therefore⎧⎨

⎩H(p,α) = L[y(t,α)] = pB(p,α)+(a+bp)C(p,α)

p+(a+bp) ,

K(p,α) = L[y(t,α)] = pC(p,α)+(a+bp)B(p,α)p+(a+bp) .

Here B(p,α) and C(p,α) are given by ().

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Similarly, the respective expressions of H(p,α), K(p,α), H(p,α), K(p,α), H(p,α),K(p,α) can be computed.

5 Numerical examplesThe following first example was studied in [] using the fuzzy double integral method:

Example

⎧⎪⎪⎨

⎪⎪⎩

y′′(x) + y(x) = σ, σ = (α, – α),

y(,α) = (α – , – α),

y′(,α) = (α – , – α).

()

• Case I: If y(x) and y′(x) are (i)-differentiable, then

⎧⎨

⎩y′′(x,α) + y(x,α) = α,

y′′(x,α) + y(x,α) = – α.

Therefore

⎧⎨

⎩L[y′′(x,α)] + L[y(x,α)] = α

p ,

L[y′′(x,α)] + L[y(x,α)] = –αp .

Using Theorem ., we get

⎧⎨

⎩L[y(x,α)] = (α – ) p+

p+ + α( p – p

p+ ),

L[y(x,α)] = ( – α) p+p+ + ( – α)(

p – pp+ ).

By the inverse Laplace transform we deduce

⎧⎨

⎩y(x,α) = α( + sin(x)) – sin(x) – cos(x),

y(x,α) = ( – α)( + sin(x)) – sin(x) – cos(x).

In this case, no solution exists, since y′(x) is not an (i)-differentiable fuzzy-valuedfunction (see []).

• Case II: If y(x) is (i)-differentiable and y′(x) is (ii)-differentiable, then

⎧⎨

⎩L[y′′(x,α)] + L[y(x,α)] = α

p ,

L[y′′(x,α)] + L[y(x,α)] = –αp .

Using Theorem ., we get

⎧⎨

⎩pL[y(x,α)] + L[y(x,α)] = ( – α)(p + ) + α

p ,

L[y(x,α)] + pL[y(x,α)] = (α – )(p + ) + –αp .

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Thus⎧⎨

⎩L[y(x,α)] = α(

(p–) – (p+) +

p ) + (p+) –

(p–) – pp+ ,

L[y(x,α)] = α( (p+) –

(p–) – p ) +

p + (p–) –

(p+) – pp+ .

By the inverse Laplace transform we deduce

⎧⎨

⎩y(x,α) = α( + sinh(x)) – sinh(x) – cos(x),

y(x,α) = ( – α)( + sinh(x)) – sinh(x) – cos(x).

As in case I, no solution exists (see []).• Case III: If y(x) is (ii)-differentiable and y′(x) is (i)-differentiable, then

⎧⎨

⎩L[y(x,α)] = α(

(p+) – (p–) +

p ) + (p–) –

(p+) – pp+ ,

L[y(x,α)] = α( (p–) –

(p+) – p ) +

p + (p+) –

(p–) – pp+ .

By the inverse Laplace transform we deduce

⎧⎨

⎩y(x,α) = α( – sinh(x)) + sinh(x) – cos(x),

y(x,α) = ( – α)( – sinh(x)) + sinh(x) – cos(x).

In this case, since y(x) is (ii)-differentiable and y′(x) is (i)-differentiable, the solution isacceptable for x ∈ (, ln( +

√)) (see []).

• Case IV: If y(x) and y′(x) are (ii)-differentiable, then

⎧⎨

⎩L[y(x,α)] = (α – )( p

p+ – p+ ) + α(

p – pp+ ),

L[y(x,α)] = ( – α)( pp+ –

p+ ) + ( – α)( p – p

p+ ).

By the inverse Laplace transform we deduce

⎧⎨

⎩y(x,α) = α( – sin(x)) + sin(x) – cos(x),

y(x,α) = ( – α)( – sin(x)) + sin(x) – cos(x).

In this case, the solution is acceptable for x ∈ (, π ) (see []).

Example We consider the following fuzzy differential equation:

⎧⎪⎪⎨

⎪⎪⎩

y′′(x) = y′(x) + x + ,

y(,α) = (α – , – α),

y′(,α) = (α – , – α).

()

• Case I: If y(x) and y′(x) are (i)-differentiable, then

⎧⎨

⎩L[y′′(x,α)] = L[y′(x)] + L[t + ],

L[y′′(x,α)] = L[y′(x)] + L[t + ].

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Using Theorems . and ., we get

⎧⎨

⎩L[y(x,α)] = α–

p– + p+p(p–) ,

L[y(x,α)] = –αp– + p+

p(p–) .

By the inverse Laplace transform we deduce

⎧⎨

⎩y(x,α) = αex – x

– x – ,

y(x,α) = ex( – α) – x

– x – .

In this case, the solution is valid for all x ∈R.• Case II: If y(x) is (i)-differentiable and y′(x) is (ii)-differentiable, then by Theorems .

and . we get

⎧⎨

⎩–pL[y(x,α)] + pL[y(x,α)] = ( – α)p + – α + p+

p ,

pL[y(x,α)] – pL[y(x,α)] = (α – )p + α – + p+p .

Therefore⎧⎨

⎩L[y(x,α)] = α–

p– + –α

p(p–) + p+p(p–) ,

L[y(x,α)] = –αp– + α–

p(p–) + p+p(p–) .

Using the inverse Laplace transform we deduce

⎧⎨

⎩y(x,α) = αex + ( – α) cosh(x) – x

– x – + α,

y(x,α) = ( – α)ex + (α – ) cosh(x) – x

– x + – α.

This solution is not acceptable since y′′(x,α) = y′(x,α) + x + .• Case III: If y(x) is (ii)-differentiable and y′(x) is (i)-differentiable, then by Theorems .

and . we get

⎧⎨

⎩L[y(x,α)] = –α

p– + α–p(p–) + p+

p(p–) ,

L[y(x,α)] = α–p– + –α

p(p–) + p+p(p–) .

Using the inverse Laplace transform we deduce

⎧⎨

⎩y(x,α) = ( – α)ex – x

– x – + α,

y(x,α) = αex – x

– x + – α.

In this case, the solution is valid for all x ∈R.• Case IV: If y(x) and y′(x) are (ii)-differentiable, then by Theorems . and . we get

⎧⎨

⎩pL[y(x,α)] – pL[y(x,α)] = (α – )p + p+

p ,

–pL[y(x,α)] + pL[y(x,α)] = ( – α)p + p+p .

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By solving this linear system and using the inverse Laplace transform, we get

⎧⎨

⎩y(x,α) = (α – ) cosh(x) + ( – α) sinh(x) + ex – x

– x – ,

y(x,α) = ( – α) cosh(x) + (α – ) sinh(x) + ex – x

– x – .

As in case II, no solution exists.

6 ConclusionIn the present work, the relation between the fuzzy Laplace transforms of a fuzzy func-tion and its second derivative is established and proved. The main purpose of the paperis to solve fuzzy linear second-order differential equations (FDEs) using the fuzzy Laplacetransform method, under generalized differentiability. The efficiency of the proposed al-gorithm is illustrated by giving two numerical examples.

For future research, we will investigate the relationship between the fuzzy Laplace trans-form of a fuzzy function and that of its kth derivative, with k ≥ , then we will apply theLaplace method to solve a large class of FDEs.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

AcknowledgementsThe authors express their sincere thanks to the anonymous referees for numerous helpful and constructive suggestionswhich have improved the manuscript.The first author would like to thank his teacher professor Said Melliani for valuable supervision.

Received: 16 November 2014 Accepted: 11 February 2015

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