Hindawi Advances in Fuzzy Systems Volume 2018, Article ID 9730502, 10 pages https://doi.org/10.1155/2018/9730502
Research Article Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method Elhassan Eljaoui , Said Melliani, and L. Saadia Chadli Department of Mathematics, University of Sultan Moulay Slimane, Laboratory of Applied Mathematics & Scientific Calculus, P.O. Box 523, Beni Mellal, Morocco Correspondence should be addressed to Elhassan Eljaoui;
[email protected] Received 14 August 2017; Accepted 27 November 2017; Published 1 January 2018 Academic Editor: Erich Peter Klement Copyright Β© 2018 Elhassan Eljaoui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.
1. Introduction Integrals of set-valued functions have been studied in connection with statistical problems and have arisen in connection with economic problems. The basic theory of such integrals was developed by Aumann [1]. Ralescu and Adams defined in [2] the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure, and studied some properties of this integral. Dubois and Prade [3] generalized the Riemann integral over a closed interval to fuzzy mappings. Their approach was more directly related to the works by Aumann [1] and Debreu [4] on multifunctions integration. Puri and Ralescu [5] generalized the integral of a setvalued function to define the concepts of fuzzy random variable and its expectation. Wu proposed in [6] two types of the fuzzy Riemann integral; the first one was based on the crisp compact interval and the second one was considered on the fuzzy interval, provided a numerical method to approximate this integral by invoking the Simpsonβs rule, and transformed its membership function into nonlinear programming problem. In [7], Allahviranloo et al. proposed an integral method for solving fuzzy linear differential equations, under the assumption of strongly generalized differentiability, but they omitted the proofs of their main results. Extending their method, we developed in [8] a more general integral operator
method for solving some first-order fuzzy linear differential equations with variable coefficients, and we gave the general formulaβs solution with necessary proofs. The notions of the fuzzy improper Riemann integral, the fuzzy random variable, and its expectation were also investigated and studied by Wu in [9] using a different approach. This concept of improper fuzzy Riemann integral was later exploited by Allahviranloo and Ahmadi in [10] to introduce the fuzzy Laplace transform, which they used to solve some first-order fuzzy differential equations (FDEs). Salahshour and Allahviranloo gave in [11] some applications of fuzzy Laplace transform and studied sufficient conditions ensuring its existence. Recently in [12], we extended and used the fuzzy Laplace transform method to solve second-order fuzzy linear differential equations under strongly generalized Hukuhara differentiability. Then we established in [13] some important results about continuity and strongly generalized Hukuhara differentiability of functions defined via improper fuzzy Riemann integrals, and we proved some properties of fuzzy Laplace transforms for two variables functions, which we applied to solve fuzzy linear partial differential equations of first order. In the same context, Salahshour et al. developed in [14] the fuzzy Laplace transform method to solve fuzzy convolution Volterra integral equation (FCVIE) of the second kind.
2
Advances in Fuzzy Systems
But the proof proposed for their main result, Theorem 4.1 was invalid and the arguments presented in this demonstration were incorrect. One can remark that it was literally identical to the corresponding proof in the classical case, without taking into consideration the fuzzy nature of the data. First let us recall and enounce Theorem 4.1 in [14]; then we will show the invalid arguments presented by the authors, to prove the fuzzy convolution formula. Theorem 1 (convolution theorem: see Theorem 4.1 in [14]). If π and π are piecewise continuous fuzzy-valued functions on [0, β[ and of exponential order π, then πΏ (π β π) (π‘) = πΏ {π (π‘)} β
πΏ {π (π‘)} = πΉ (π ) β
πΊ (π ) ,
(1)
π > π.
First notice that π and π are fuzzy-valued functions, β so both of the improper integrals β«0 πβπ π π(π)ππ and β
β«0 πβπ π’ π(π’)ππ’ are fuzzy numbers. Then, we cannot justify the following passage by a simple integral linearity argument: πΏ {π (π‘)} β
πΏ {π (π‘)} β
β
0
0
= (β« πβπ π π (π) ππ) (β« πβπ π’ π (π’) ππ’) β
β
0
0
(2)
= β« (β« πβπ (π+π’) π (π) π (π’) ππ’) ππ, β
without proving that for each fuzzy number π β πΈ : β«0 π β
β
π(π‘)ππ‘ = π β«0 π(π‘)ππ‘. Moreover, the authors claimed that due to the hypothesis on π and π, the fuzzy Laplace integrals of π and π converge β β absolutely and hence β«0 β«0 |πβπ π‘ π(π)π(π‘βπ)ππ‘|ππ converges. It was the most important key of their proof as in the crisp case, since it allows us to reverse the order of the double integrals, but unfortunately it is also incorrect, because the notion of the absolute value of a fuzzy number is not defined at least in [14]. Furthermore, the concept of the absolute convergence of a fuzzy improper integral does not make sense in the fuzzy literature. To overcome all of these obstacles, we propose in the actual paper the convolution product of a crisp mapping and a fuzzy function in Section 4, and we intend to investigate rigorously the case of two fuzzy functions in a future work. The theory of fuzzy integro-differential equations has many applications and have been studied extensively in the fuzzy literature; for the reader, we refer to [15β17] and the references therein. Concerning the classical integro-differential equations, one can consult [18β20]. The aim of this work is to define the convolution product and to prove a fuzzy Laplace convolution formula, in the purpose of solving the following fuzzy integro-differential equations (FIDEs) with kernel of convolution type: π₯
π¦σΈ (π₯) = π (π₯) + β« π (π₯ β π‘) π¦ (π‘) ππ‘, 0
π¦ (0) = π¦0 = (π¦ , π¦0 ) β πΈ, 0
π₯
π¦σΈ (π₯) = π (π₯, π¦ (π₯)) + β« π (π₯ β π‘) π¦ (π‘) ππ‘ 0
π¦ (0) = π¦0 = (π¦ , π¦0 ) β πΈ 0
(3) provided that π : [0, β[β πΈ, π : [0, β[ΓπΈ β πΈ are continuous fuzzy-valued functions and π : [0, β[β R is a crisp continuous function verifying some assumptions to be mentioned later. Then we give some examples to illustrate the efficiency of our method for solving FIDEs. To achieve this goal, we first introduce the Aumann fuzzy improper integral concept, which we utilize instead of the Riemann fuzzy improper integral used in [10, 12β14]. This new definition of fuzzy generalized (improper) integral is essentially based on the notion of fuzzy integral and the expectation of a fuzzy random variable, introduced by Puri and Ralescu in [5]. The remainder of this paper is organized as follows. Section 2 is reserved for some preliminaries. And Section 3 is devoted to the definition of the Aumann fuzzy improper integral. In Section 4, fuzzy Laplace transform is introduced, its basic properties are studied, and a particular case of Laplace convolution is investigated. Then in Section 5, the main result about Laplace convolution is enounced and proved. The procedure for solving fuzzy integro-differential equations by fuzzy Laplace transform is proposed and some numerical examples are given in Section 6. In the last section, we present conclusion and a further research topic.
2. Preliminaries Denote by ππΎ (R) the family of all nonempty compact convex subsets of R and define the addition and scalar multiplication in ππΎ (R) as usual. The distance between two nonempty bounded subsets π΄ and π΅ of R is defined by the Hausdorff metric π (π΄, π΅) = max {sup inf |π β π| , sup inf |π β π|} . πβπ΄ πβπ΅
πβπ΅ πβπ΄
(4)
Define πΈ = {π’ : R σ³¨β [0, 1] | π’ satisfies (i)β(iv) below} ,
(5)
where (i) π’ is normal, that is, βπ₯0 β R for which π’(π₯0 ) = 1, (ii) π’ is fuzzy convex, (iii) π’ is upper semicontinuous, (iv) supp π’ = {π₯ β R | π’(π₯) > 0} is the support of π’, and its closure cl (supp π’) is compact. For 0 < πΌ β€ 1, the πΌ-cut (or level) of π’ is denoted [π’]πΌ = {π₯ β R | π’ (π₯) β₯ πΌ} .
(6)
Advances in Fuzzy Systems
3
Then, from (i) to (iv), it follows that the πΌ-level set [π’]πΌ β ππΎ (R) for all 0 β€ πΌ β€ 1. It is well known that [π’ + V]πΌ = [π’]πΌ + [V]πΌ , πΌ
πΌ
[ππ’] = π [π’] .
(7)
Let π· : πΈ Γ πΈ β [0, β) be a function which is defined by the equation π· (π’, V) = sup π ([π’]πΌ , [V]πΌ ) , 0β€πΌβ€1
(8)
where π is the Hausdorff metric defined in ππΎ (R). Then, the following properties hold true (see [5, 21]): (1) (πΈ, π·) is a complete metric space. (2) π·(π’ + π€, V + π€) = π·(π’, V) for π’, V, π€ β πΈ. (3) π·(ππ’, πV) = |π| π·(π’, V) for all π’, V β πΈ and π β R. (4) π·(π’ + π€, V + π‘) β€ π·(π’, V) + π·(π€, π‘) for all π’, V, π€, π‘ β πΈ. Definition 2. A fuzzy number π’ in parametric form is a pair (π’, π’) of functions π’(π), π’(π), 0 β€ π β€ 1, which satisfy the following requirements: (1) π’(π) is a bounded nondecreasing left continuous function in (0, 1] and right continuous at 0. (2) π’(π) is a bounded nonincreasing left continuous function in (0, 1] and right continuous at 0. (3) π’(π) β€ π’(π) for all 0 β€ π β€ 1. A crisp number π is simply represented by π’(π) = π’(π) = π, 0 β€ π β€ 1. The following general definition and properties were developed by Puri and Ralescu in [5], for the fuzzy Aumann integral theory in πΈπ . Here, we restrict their theory to πΈ = πΈ1 instead of πΈπ . Let (Ξ©, A, π) be a probability space where the probability measure π is assumed to be nonatomic. Definition 3 (Puri and Ralescu [5]). A mapping πΉ : Ξ© β πΈ is strongly measurable if for all πΌ β [0, 1] the set-valued function πΉπΌ : Ξ© β PπΎ (R) defined by πΉπΌ (π‘) = [πΉ(π‘)]πΌ is Lebesgue measurable. A mapping πΉ : Ξ© β πΈ is called integrably bounded if there exists an integrable function π such that |π₯| β€ π(π‘) for all π₯ β πΉ0 (π‘). Definition 4 (Puri and Ralescu [5]). Let (Ξ©, A, π) be a probability space where the probability measure π is assumed to be nonatomic. A set-valued function is a function πΉ : Ξ© β P(R) such that πΉ(π) =ΜΈ 0 for every π β Ξ©. By πΏ1 (π) we denote the space of π-integrable functions π : Ξ© β R. We denote by π(πΉ) the set of all πΏ1 (π) selections of πΉ; that is, π (πΉ) = {π β πΏ1 (π) | π (π) β πΉ (π) a.e} .
(9)
The Aumann integral of πΉ, denoted by β«Ξ© πΉ ππ or β« πΉ for short, is defined by β« πΉ = {β« π ππ | π β π (πΉ)} . Ξ©
Ξ©
(10)
Definition 5. A strongly measurable and integrably bounded mapping πΉ : Ξ© β πΈ is said to be integrable over Ξ© if β«Ξ© πΉ β πΈ. Lemma 6 (Puri and Ralescu [5]). If πΉ : Ξ© β P(R) is measurable and integrably bounded, then πΉ is integrable over Ξ©. Theorem 7 (Puri and Ralescu [5]). If πΉπ : Ξ© β P(R) are measurable and if there exists β β πΏ1 (π, R) such that supπβ₯1 βππ (π)β β€ β(π) for every ππ β π(πΉπ ) and if πΉπ (π) β πΉ(π) (in the sense of Kuratowski), then β«Ξ© πΉπ β β«Ξ© πΉ. Remark 8 (Puri and Ralescu [5]). It is important to observe that Theorem 7 can be stated in a different form by replacing convergence in the sense of Kuratowski by convergence in the Hausdorff metric. The statement of the theorem remains unchanged provided that we assume that all functions take values in π(R), the set of all nonempty, compact subsets of R. Now, we define the Hukuhara difference and the strongly generalized differentiability. For π’, V β πΈ, if there exists π€ β πΈ such that π’ = V+π€, then π€ is the Hukuhara difference of π’ and V denoted by π’ β V. Definition 9. We say that a fuzzy mapping π : (π, π) β πΈ is strongly generalized differentiable at π₯0 β (π, π), if there exists an element πσΈ (π₯0 ) β πΈ such that (i) for all β > 0 sufficiently small, there exist π(π₯0 + β) β π(π₯0 ); π(π₯0 ) β π(π₯0 β β) and lim+
ββ0
π (π₯0 + β) β π (π₯0 ) π (π₯0 ) β π (π₯0 β β) = lim+ ββ0 β β
(11)
σΈ
= π (π₯0 ) or (ii) for all β > 0 sufficiently small, there exist π(π₯0 ) β π(π₯0 + β); π(π₯0 β β) β π(π₯0 ) and lim+
ββ0
π (π₯0 ) β π (π₯0 + β) π (π₯0 β β) β π (π₯0 ) = lim+ ββ0 (ββ) (ββ)
(12)
σΈ
= π (π₯0 ) or (iii) for all β > 0 sufficiently small, there exist π(π₯0 + β) β π(π₯0 ); π(π₯0 β β) β π(π₯0 ) and lim
ββ0+
π (π₯0 + β) β π (π₯0 ) π (π₯0 β β) β π (π₯0 ) = lim+ ββ0 β (ββ) σΈ
= π (π₯0 ) or
(13)
4
Advances in Fuzzy Systems (iv) for all β > 0 sufficiently small, there exist π(π₯0 ) β π(π₯0 + β); π(π₯0 ) β π(π₯0 β β) and lim+
ββ0
π (π₯0 ) β π (π₯0 + β) π (π₯0 ) β π (π₯0 β β) = lim+ ββ0 β (ββ)
(14)
Taking πΉπ,πΌ (π₯) = π[π,π] (π₯) β
πΉπΌ (π₯) in Theorem 7 and Remark 8 implies the following result. Theorem 13. If πΉ : [π, β[β πΈ is measurable and integrably bounded, then for all 0 β€ πΌ β€ 1
= πσΈ (π₯0 ) .
π
β
π
π
β« πΉπΌ (π₯) ππ₯ σ³¨β β« πΉπΌ (π₯) ππ₯
All the limits are taken in the metric space (πΈ, π·). At the end points of (π, π), we consider only one-sided derivatives. The following theorem (see [22]) allows us to consider case (i) or (ii) of the previous definition almost everywhere in the domain of the functions under discussion. Theorem 10. Let π : (π, π) β πΈ be strongly generalized differentiable on each point π₯ β (π, π) in the sense of Definition 9, (iii) or (iv). Then πσΈ (π₯) β R for all π₯ β (π, π).
σΈ
σΈ
σΈ
ferentiable functions and π (π‘) = (π (π‘, π), π (π‘, π)). (2) If π is (ii)-differentiable, then π(π‘, π) and π(π‘, π) are difσΈ
ferentiable functions and πσΈ (π‘) = (π (π‘, π), πσΈ (π‘, π)).
3. Aumann Fuzzy Improper Integral
Lemma 14. If πΉ, πΊ : [π, β[β πΈ are (fuzzy) integrable over [π, β[, then for all real π the mappings πΉ + πΊ and ππΉ are integrable over [π, β[ and we have β
π
β
β
πΌ
β
[β« πΉ (π₯) ππ₯] = β« πΉπΌ (π₯) ππ₯ π
π
(15)
β
= {β« π (π₯) ππ₯ | π β π (πΉπΌ )} . π
Definition 12. A strongly measurable and integrably bounded mapping πΉ : [π, β[β πΈ is said to be integrable over [π, β[ if β β«π πΉ(π₯)ππ₯ β πΈ. Using Lemma 6, we deduce that if πΉ : [π, β[β πΈ is measurable and integrably bounded, then it is integrable β over [π, β[ and β«π πΉπΌ (π₯)ππ₯ is a real interval, since it is a nonempty, convex, and compact subset of R; that is, β
β
β
β« πΉπΌ (π₯) ππ₯ = [β« πΉ (π₯, πΌ) ππ₯, β« πΉ (π₯, πΌ) ππ₯] . π
π
π
(16)
In the parametric form, the fuzzy improper integral β β«π πΉ(π₯)ππ₯ can be written β
β
β
β« πΉ (π₯) ππ₯ = (β« πΉ (π₯, πΌ) ππ₯, β« πΉ (π₯, πΌ) ππ₯) . π
π
π
π
π
β
β
π
π
(19)
Remark 15. Analogously, we define the integrability and the π Aumann fuzzy improper integral β«ββ πΉ(π₯)ππ₯ of a fuzzy function πΉ :] β β, π] β πΈ. Then, we said that a fuzzy mapping πΉ : R β πΈ is integrable over R, if it is integrable over ] β β, π] and over [π, β[, for each real π. In this case, we define β
πΉ (π₯) ππ₯ = β«
π
ββ
β
πΉ (π₯) ππ₯ + β« πΉ (π₯) ππ₯. π
(20)
For more details concerning Aumann fuzzy improper integral, one can see [5]. Remark 16. The concepts of the fuzzy improper integral, the fuzzy random variable, and its expectation were defined and studied in a different way by Wu in [9]. His proposal of the improper fuzzy Riemann integral was an appropriate attempt for finding the expectations of fuzzy random variables numerically. He stated that the developments in [5] were in measuretheoretic sense; thus, it was difficult to provide a numerical method in applications. However, this statement seems to be false because of the approach developed in our present article and precisely by the identities (16) and (17); the Aumann fuzzy improper integral (and the integral over a compact subset of R) has the same properties and qualities as well as the improper fuzzy Riemann integral.
4. Fuzzy Laplace Convolution Definition 17 (see [10]). Let π(π₯) be continuous fuzzy-valued function. Suppose that πβπ0 π₯ π(π₯) is integrable on [0, β[, for some π0 > 0, then for all π β₯ π0 the improper integral β β«0 πβππ₯ π(π₯)ππ₯, which is well defined, is called fuzzy Laplace transform of π and is denoted as β
(17)
β
β« ππΉ (π₯) ππ₯ = π β« πΉ (π₯) ππ₯.
ββ
as follows: [β«π πΉ(π₯)ππ₯]πΌ = β«Ξ© πΉπΌ (π₯) exp(π₯)ππ; that is,
β
β« (πΉ (π₯) + πΊ (π₯)) ππ₯ = β« πΉ (π₯) ππ₯ + β« πΊ (π₯) ππ₯,
β«
Considering the positive measure related to the exponential law on the positive real line Ξ© = [π, β[, defined by ππ = exp(βπ₯)ππ₯, where ππ₯ refers to the Lebesgue measure. We define the Aumann fuzzy improper integral β β«π πΉ(π₯)ππ₯ of a fuzzy function πΉ : [π, β[β πΈ, by its πΌ-levels
(18)
Since the Aumann integral over [π, π] is linear (see [24]), then from Theorem 13, we deduce the linearity of the Aumann improper fuzzy integral over [π, β[.
Theorem 11 (see, e.g., [23]). Let π : R β πΈ be a function and denote π(π‘) = (π(π‘, π), π(π‘, π)), for each π β [0, 1]. (1) If π is (i)-differentiable, then π(π‘, π) and π(π‘, π) are dif-
as π σ³¨β β.
L [π (π₯)] = β« πβππ₯ π (π₯) ππ₯, 0
π β₯ π0 .
(21)
Advances in Fuzzy Systems
5
If L(π(π₯)) denotes the classical Laplace transform of a crisp function π(π₯), then since β
L [(π β π) (π₯)]
β« πβππ₯ π (π₯) ππ₯ 0
β
βππ₯
= (β« π 0
Then using (29)-(30) and the fact that L[π(π₯)] β₯ 0, we get
β
(22) βππ₯
π (π₯, π) ππ₯, β« π 0
π (π₯, π) ππ₯) ,
= (L [π (π₯)] β
L [π (π₯)] , L [π (π₯)] β
L [π (π₯)])
we have L [π (π₯)] = (L (π (π₯, π)) , L (π (π₯, π))) .
(23)
Theorem 18. Let π be a differentiable fuzzy-valued function such that πβππ₯ π(π₯) and πβππ₯ πσΈ (π₯) are integrable on [0, β[. (a) If π is (i)-differentiable, then L [π (π₯)] = πL [π (π₯)] β π (0) .
(24)
(b) If π is (ii)-differentiable, then L [πσΈ (π₯)] = (βπ (0)) β (βπ) L [π (π₯)] .
(25)
Proof. To prove Theorem 18, one can adopt the proof in [10] using Aumann fuzzy improper integral instead of Riemann fuzzy improper integral. Theorem 19. Let π(π₯), π(π₯) be continuous fuzzy-valued functions such that πβππ₯ π(π₯) and πβππ₯ π(π₯) are integrable on [0, β[ and π1 , π2 two real constants; then L [π1 π (π₯) + π2 π (π₯)] = π1 L [π (π₯)] + π2 L [π (π₯)] .
(26)
Theorem 19 is an obvious consequence of linearity of the Aumann fuzzy improper integral. Definition 20. Let π : [0, β[β R be a crisp continuous function and π : [0, β[β πΈ a fuzzy-valued continuous mapping. We define the convolution product of π and π on [0, β[ as follows: (π β π) (π₯) = β« π (π₯ β π‘) π (π‘) ππ‘, 0
(31)
= L [π (π₯)] β
(L [π (π₯)] , L [π (π₯)]) = L [π (π₯)] β
L [π (π₯)] . (b) If the function π is nonpositive on [0, β[, then (π β π) (π₯)
σΈ
π₯
= (L [(π β π) (π₯)] , L [(π β π) (π₯)])
π₯ β₯ 0.
(27)
Remark 21. Suppose that πβππ₯ π(π₯) and πβππ₯ π(π₯) are integrable on [0, β[. We examine the two following alternatives: (a) If the function π is nonnegative on [0, β[, then
π₯
π₯
0
0
= (β« π (π₯ β π‘) π (π‘) ππ‘, β« π (π₯ β π‘) π (π‘) ππ‘) . Therefore, (π β π) (π₯) = ((π β π) (π₯) , (π β π) (π₯)) .
(33)
Then from (30)-(33) and since L[π(π₯)] β€ 0, we deduce L [(π β π) (π₯)] = (L [π (π₯)] β
L [π (π₯)] , L [π (π₯)] β
L [π (π₯)]) = L [π (π₯)] β
(L [π (π₯)] , L [π (π₯)])
(34)
= L [π (π₯)] β
L [π (π₯)] . In both cases, we have L [(π β π) (π₯)] = L [π (π₯)] β
L [π (π₯)] .
(35)
Remark 22. Now let us recall the error in [25] Example 1. The authors studied the following fuzzy integro-differential equation using fuzzy differential transform method (DTM): π₯
π’σΈ (π₯) = (1 + π₯) (π + 1, π β 2) + β« π’ (π‘) ππ‘, 0
(36)
π’ (0) = (0, 0) ,
(π β π) (π₯)
(32)
π’σΈ (0) = (π + 1, π β 2) .
π₯
π₯
0
0
= (β« π (π₯ β π‘) π (π‘) ππ‘, β« π (π₯ β π‘) π (π‘) ππ‘) .
(28)
But π = (π + 1, π β 2) is not a fuzzy number in the parametric form, since the function π = π β 2 is not decreasing. Note that the second initial data π’σΈ (0) can be obviously deduced by taking π₯ = 0 in the equation.
(29)
Example 23. We correct the previous fuzzy Volterra integrodifferential equation as follows:
Therefore, (π β π) (π₯) = ((π β π) (π₯) , (π β π) (π₯)) .
If π1 and π2 are two crisp functions defined from [0, β[ into R, then, we recall the well-known classical convolution Laplace formula: L [(π1 β π2 ) (π₯)] = L [π1 (π₯)] β
L [π2 (π₯)] .
(30)
π₯
π¦σΈ (π₯) = (1 + π₯) π + β« π¦ (π‘) ππ‘, 0
π¦ (0, πΌ) = (0, 0) ,
(37)
6
Advances in Fuzzy Systems
where π(π₯) = (1 + π₯)π, π = (πΌ β 1, 1 β πΌ) and π(π₯) = 1 is nonnegative.
In this case, the solution is acceptable since π¦(π₯) is (i)differentiable.
Case 1. If π¦(π₯) is (i)-differentiable, then from (35) we have
Case 2. If π¦(π₯) is (ii)-differentiable, then from (35) we get
L [π¦ (π₯, πΌ)] =
πΌβ1 , π (π β 1)
1βπΌ L [π¦ (π₯, πΌ)] = . π (π β 1)
L [π¦ (π₯, πΌ)] = (1 β πΌ) (38)
π+1 L [π¦ (π₯, πΌ)] = (πΌ β 1) . π (π2 + 1)
By the inverse Laplace transform, we get the lower and upper functions of solution of (37) for π₯ β₯ 0 π¦ (π₯, πΌ) = (πΌ β 1) (exp (π₯) β 1) , π¦ (π₯, πΌ) = (1 β πΌ) (exp (π₯) β 1) .
(39)
In this case, since π¦(π₯) is (i)-differentiable, the solution is valid. Case 2. If π¦(π₯) is (ii)-differentiable, then from (35) we obtain L [π¦ (π₯, πΌ)] = (1 β πΌ)
π+1 , π (π2 + 1)
π+1 L [π¦ (π₯, πΌ)] = (πΌ β 1) . π (π2 + 1)
(40)
π¦ (π₯, πΌ) = (πΌ β 1) (cos (π₯) β sin (π₯) β 1) , (41)
In this case, π¦(π₯) is (ii)-differentiable only for π₯ β [7π/4, 2π] and the solution is acceptable only over this interval. Example 24. We consider the following fuzzy Volterra integro-differential equation: 0
(42)
π¦ (0, πΌ) = (0, 0) ,
Case 1. If π¦(π₯) is (i)-differentiable, then from (35) we have
L [π¦ (π₯, πΌ)] =
π¦ (π₯, πΌ) = (2 β πΌ) sin (π₯) , π¦ (π₯, πΌ) = πΌ sin (π₯) .
(46)
In this case, π¦(π₯) is (ii)-differentiable only for π₯ β [3π/2, 2π], so the solution is valid only over this interval.
To overcome all the obstacles and to avoid the error in [14], we propose in this paper the convolution product of crisp and fuzzy functions, and we intend to investigate rigorously the case of two fuzzy functions in a future work. Now, we enounce our main result giving the convolution Laplace formula generalizing the result in Section 4. Theorem 25. Let πΉ : [0, β[β πΈ be a fuzzy-valued continuous mapping and let π : [0, β[β R be a crisp continuous function. Assume that the mappings πβππ₯ π(π₯), πβππ₯ πΉ(π₯), and πβππ₯ (π β πΉ)(π₯) are integrable over [0, β[ for all π > 0; then L [(π β πΉ) (π₯)] = L [π (π₯)] β
L [πΉ (π₯)] .
(47)
Proof. Let π₯ β₯ 0 and π > 0. It is obvious that [(π β πΉ)(π₯)]πΌ = (π β πΉπΌ )(π₯).
(π β πΉπΌ ) (π₯) = {(π β π) (π₯) | π β π (πΉπΌ )} .
(48)
π₯
where π(π₯) = (πΌ, 2 β πΌ) and π(π₯) = β1 is nonpositive.
L [π¦ (π₯, πΌ)] =
Using the inverse Laplace transform, we obtain the solution of (42) for π₯ β [π, 2π]:
Step 1. We claim that
π₯
π¦σΈ (π₯) = (πΌ, 2 β πΌ) + β« (β1) π¦ (π‘) ππ‘,
(45)
5. Main Result
Then by the inverse Laplace transform the lower and upper functions of solution of (37) are given for π₯ β [3π/2, 2π] as follows:
π¦ (π₯, πΌ) = (1 β πΌ) (cos (π₯) β sin (π₯) β 1) .
π+1 , π (π2 + 1)
πΌπ2 + πΌ β 2 , π4 β 1 (2 β πΌ) π2 β πΌ . π4 β 1
(43)
By the inverse Laplace transform we get the lower and upper functions of solution of (42) for π₯ β₯ 0 π¦ (π₯, πΌ) = (πΌ β 1) sinh (π₯) + sin (π₯) , π¦ (π₯, πΌ) = (1 β πΌ) sinh (π₯) + sin (π₯) .
(44)
Let π¦ β (π β πΉπΌ )(π₯) = β«0 π(π₯ β π‘)πΉπΌ (π‘)ππ‘. So, there exists a measurable selection π of π‘ σ³¨β π(π₯ β π‘)πΉπΌ (π‘) such that π¦ = π₯ β«0 π(π‘)ππ‘. It is clear that the function π defined by π (π‘) { π (π‘) = { π (π₯ β π‘) {πΉπΌ (π‘)
if π (π₯ β π‘) =ΜΈ 0 if π (π₯ β π‘) = 0
(49)
is a measurable selection of πΉπΌ verifying π(π‘) = π(π₯ β π‘)π(π‘). Hence, π¦ = (π β π)(π₯), which implies that (π β πΉπΌ )(π₯) β {(π β π)(π₯) | π β π(πΉπΌ )}. Let π be a measurable selection of πΉπΌ . It is clear that π‘ σ³¨β π(π₯ β π‘)π(π‘) is a measurable selection of π‘ σ³¨β π(π₯ β π‘)πΉπΌ (π‘) and (π β π) (π₯) β (π β πΉπΌ ) (π₯) ,
(50)
Advances in Fuzzy Systems
7
because π₯
π₯
0
0
β« π (π₯ β π‘) π (π‘) ππ‘ β β« π (π₯ β π‘) πΉπΌ (π‘) ππ‘.
(51)
Therefore, (48) is proved.
Please notice that Theorem 10 allows us to use only (i) or (ii) type of strongly generalized differentiability. Assume in a first time that L[π(π₯)] β₯ 0. By using the fuzzy Laplace transform and Theorem 25, we have L [π¦σΈ (π₯)] = L [π (π₯)] + L [π (π₯)] β
L [π¦ (π₯)] .
Step 2. Now we show that L [(π β πΉπΌ ) (π₯)] = L [π (π₯)] β
L [πΉπΌ (π₯)] . If we denote π1 (π‘) = exp(βππ‘)π(π‘) and π»πΌ (π‘) exp(βππ‘)πΉπΌ (π‘), then using (48) we can write
(52) =
Then, we have the following alternatives for solving (56). Case 1. If π¦ is (i)-differentiable, then π¦σΈ (π₯) = (π¦σΈ (π₯, πΌ) , π¦σΈ (π₯, πΌ)) ,
β
L [π¦σΈ (π₯)] = πL [π¦ (π₯)] β π¦ (0) .
L [(π β πΉπΌ ) (π₯)] = β« exp (βππ₯) (π β πΉπΌ ) (π₯) ππ₯ 0
(57)
Then from (56), it follows that
β
= β« (π1 β π»πΌ ) (π₯) ππ₯
πL [π¦ (π₯)] = π¦ (0) + L [π (π₯)] + L [π (π₯)] β
L [π¦ (π₯)] . (58)
0
β
Using L[π(π₯)] β₯ 0, we deduce
0
πL [π¦ (π₯, πΌ)] = π¦ (πΌ) + L [π (π₯, πΌ)] + L [π (π₯)]
= {β« (π1 β β) (π₯) ππ₯ | β β π (π»πΌ )}
0
β
= {β« (π1 β β) (π₯) ππ₯ | β : π‘
(53)
β
L [π¦ (π₯, πΌ)] ,
0
β
βππ₯
= {β« π
β
L [π¦ (π₯, πΌ)] .
(π β π) (π₯) ππ₯ | π β π (πΉπΌ )}
Therefore, L [π¦ (π₯, πΌ)] =
= {L [(π β π) (π₯)] | π β π (πΉπΌ )} . Since L[π(π₯)] is a real number, then from (30) it follows that L [(π β πΉπΌ ) (π₯)] = {L [π (π₯)] β
L [π (π₯)] | π β π (πΉπΌ )} = L [π (π₯)] β
{L [π (π₯)] | π β π (πΉπΌ )}
(59)
πL [π¦ (π₯, πΌ)] = π¦0 (πΌ) + L [π (π₯, πΌ)] + L [π (π₯)]
σ³¨β πβππ‘ π (π‘) ; with π β π (πΉπΌ )}
0
(56)
L [π¦ (π₯, πΌ)] =
π¦ (πΌ) + L [π (π₯, πΌ)] 0
π β L [π (π₯)]
π¦0 (πΌ) + L [π (π₯, πΌ)] π β L [π (π₯)]
= π»1 (π, πΌ) , (60) = πΎ1 (π, πΌ) .
By using the inverse Laplace transform, we get π¦ (π₯, πΌ) = Lβ1 [π»1 (π, πΌ)] ,
(54)
π¦ (π₯, πΌ) = Lβ1 [πΎ1 (π, πΌ)] .
= L [π (π₯)] β
πΏ [πΉπΌ (π₯)]
(61)
Case 2. If π¦ is (ii)-differentiable, then π¦σΈ (π₯) = (π¦σΈ (π₯, πΌ) , π¦σΈ (π₯, πΌ)) ,
6. Fuzzy Laplace Transform Algorithm for Solving Fuzzy Integro-Differential Equations
L [π¦σΈ (π₯)] = βπ¦ (0) β (βπL [π¦ (π₯)]) .
Our aim now is to solve the following fuzzy integro-differential equation using fuzzy Laplace transform method under strongly generalized differentiability: π₯
π¦σΈ (π₯) = π (π₯) + β« π (π₯ β π‘) π¦ (π‘) ππ‘, 0
(62)
Then from (56), it follows that β π¦ (0) β (βπL [π¦ (π₯)]) = L [π (π₯)] + L [π (π₯)] β
L [π¦ (π₯)] .
(63)
Using L[π(π₯)] β₯ 0, we deduce (55)
π¦ (0) = π¦0 = (π¦ , π¦0 ) β πΈ, 0
where the unknown function π¦(π₯) = (π¦(π₯, πΌ), π¦(π₯, πΌ)) is a fuzzy function of π₯ β₯ 0, provided that π : [0, β[β πΈ is a continuous fuzzy-valued function and π : [0, β[β R is a crisp continuous function.
β π¦0 (πΌ) + πL [π¦ (π₯, πΌ)] = L [π (π₯, πΌ)] + L [π (π₯)] β
L [π¦ (π₯, πΌ)] , β π¦ (πΌ) + πL [π¦ (π₯, πΌ)] 0
= L [π (π₯, πΌ)] + L [π (π₯)] β
L [π¦ (π₯, πΌ)] .
(64)
8
Advances in Fuzzy Systems That is,
By using the inverse Laplace transform, we obtain π¦ (π₯, πΌ) = Lβ1 [π»4 (π, πΌ)] ,
L [π (π₯)] β
L [π¦ (π₯, πΌ)] β πL [π¦ (π₯, πΌ)] = π΄ (π, πΌ) , β πL [π¦ (π₯, πΌ)] + L [π (π₯)] β
L [π¦ (π₯, πΌ)] = π΅ (π, πΌ) ,
(65)
where π΄(π, πΌ) = βπ¦0 (πΌ)βL[π(π₯, πΌ)] and π΅(π, πΌ) = βπ¦ (πΌ)β L[π(π₯, πΌ)]. Then by solving the linear system (65), we have L [π¦ (π₯, πΌ)] =
Example 27. We consider the following fuzzy integro-differential equation: π₯
π¦σΈ (π₯) + π¦ (π₯) = β« sin (π₯ β π‘) π¦ (π‘) ππ‘, 0
L [π (π₯)] β
π΄ (π, πΌ) + ππ΅ (π, πΌ)
L [π (π₯)] β
π΅ (π, πΌ) + ππ΄ (π, πΌ)
(66)
Case 1. If π¦(π₯) is (i)-differentiable, then from Theorems 18 and 25 we have
(L [π (π₯)])2 β π2
L [π¦ (π₯, πΌ)] = (πΌ β 1)
= πΎ2 (π, πΌ) .
L [π¦ (π₯, πΌ)] = (1 β πΌ)
By using the inverse Laplace transform, we get π¦ (π₯, πΌ) = Lβ1 [π»2 (π, πΌ)] , π¦ (π₯, πΌ) = Lβ1 [πΎ2 (π, πΌ)] .
(67)
Remark 26. Similarly, if we assume that L[π(π₯)] < 0, we obtain the following results.
π2 β (L [π (π₯)])2
L [π¦ (π₯, πΌ)] =
L [π (π₯)] β
πΆ (π, πΌ) + ππ· (π, πΌ)
(68)
where πΆ(π, πΌ) = π¦ (πΌ) + L[π(π₯, πΌ)] and π·(π, πΌ) =
= (πΌ β 1) [1 β
In this case, the solution is invalid over [0, β[, since π¦(π₯) is not (i)-differentiable.
πL [π¦ (π₯, πΌ)] + (π2 + 1) πL [π¦ (π₯, πΌ)]
By using the inverse Laplace transform, we get
(69)
(2) If π¦ is (ii)-differentiable, then
L [π¦ (π₯, πΌ)] =
π β L [π (π₯)]
π¦0 (πΌ) + L [π (π₯, πΌ)] π β L [π (π₯)]
π2 + 1 , π
(π2 + 1) L [π¦ (π₯, πΌ)] + πL [π¦ (π₯, πΌ)]
β1
0
(74)
β3π₯ 2β 3 π₯ exp (β ) sin ( )] . 3 2 2
= (1 β πΌ)
π¦ (πΌ) + L [π (π₯, πΌ)]
.
β3π₯ 2β3 π₯ exp (β ) sin ( )] , 3 2 2
0
π¦0 (πΌ) + L[π(π₯, πΌ)].
L [π¦ (π₯, πΌ)] =
π3 + π2 + π
(73)
Case 2. If π¦(π₯) is (ii)-differentiable, then Theorems 18 and 25 yield
= πΎ3 (π, πΌ) ,
π¦ (π₯, πΌ) = Lβ1 [πΎ3 (π, πΌ)] .
(π2 + 1)
,
π¦ (π₯, πΌ)
= (1 β πΌ) [1 β
π2 β (L [π (π₯)])2
π¦ (π₯, πΌ) = L [π»3 (π, πΌ)] ,
π3 + π2 + π
π¦ (π₯, πΌ)
L [π (π₯)] β
π· (π, πΌ) + ππΆ (π, πΌ)
= π»3 (π, πΌ) ,
(π2 + 1)
By the inverse Laplace transform we get the lower and upper functions of solution of (72) for π₯ β₯ 0
(1) If π¦ is (i)-differentiable, then L [π¦ (π₯, πΌ)] =
(72)
π¦ (0, πΌ) = (πΌ β 1, 1 β πΌ) .
(L [π (π₯)])2 β π2
= π»2 (π, πΌ) , L [π¦ (π₯, πΌ)] =
0
(71)
π¦ (π₯, πΌ) = Lβ1 [πΎ4 (π, πΌ)] .
= (πΌ β 1)
(75)
π2 + 1 . π
By solving the linear system (75) and using the inverse Laplace transform, we get = π»4 (π, πΌ) ,
π¦ (π₯, πΌ) = (πΌ β 1) [1 + (70)
= πΎ4 (π, πΌ) .
β3π₯ 2β3 π₯ exp ( ) sin ( )] , 3 2 2
β3π₯ 2β 3 π₯ π¦ (π₯, πΌ) = (1 β πΌ) [1 + exp ( ) sin ( )] . 3 2 2
(76)
Advances in Fuzzy Systems
9
One can verify that in this case the solution is acceptable over a closed interval [π, π] such that [2.45, 3.75] β [π, π] β [2.4, 3.8]. Remark 28. Analogously, we can solve the following generalized fuzzy integro-differential equation, with kernel of convolution type via Laplace transform method: π¦σΈ (π₯) = π (π₯, π¦ (π₯)) + β« π (π₯ β π‘) π¦ (π‘) ππ‘, (77)
π¦ (0) = π¦0 = (π¦ , π¦0 ) β πΈ 0
provided that π : [0, β[ΓπΈ β πΈ is a continuous fuzzy-valued function, which is linear with respect to its second argument, and π : [0, β[β R is a crisp continuous function over [0, β[. Example 29. We consider the following known fuzzy integrodifferential equation: π₯
π¦σΈ (π₯) + 3π¦ (π₯) + β« πβ(π₯βπ‘) π¦ (π‘) ππ‘ 0
(78)
= (1 + π₯) (πΌ β 1, 1 β πΌ) ,
Its corresponding crisp problem, studied in [20], is as follows:
0
(79)
π¦ (0) = 1. Case 1. If π¦(π₯) is (i)-differentiable, then from Theorems 18 and 25 we have
L [π¦ (π₯, πΌ)] = (1 β πΌ)
(π + 1) π2
(π + 2)
(π + 1) π2
2
+
2
(π + 2)
2
+
π+1 (π + 2)
2
(80)
π+1 (π + 2)
,
2
.
By the inverse Laplace transform we get the lower and upper functions of solution of (78) for π₯ β₯ 0 π¦ (π₯, πΌ) =
(πΌ β 1) [π₯ cosh (π₯) + sinh (π₯)] πβπ₯ 2 + (1 β π₯) πβ2π₯ ,
π¦ (π₯, πΌ) =
π+1 + π + 1, π2
= (πΌ β 1)
(82)
π+1 + π + 1. π2
By solving the linear system (82) and using the inverse Laplace transform, we get the lower and upper functions of solution of (78) for π₯ β₯ 0 as follows: π¦ (π₯, πΌ) = (1 β π₯) exp (β2π₯) + (πΌ β 1)
β2π₯ β 3 8
3 7β5 + (πΌ β 1) ππ₯ ( cosh (β5π₯) + sinh (β5π₯)) , 8 40 β2π₯ β 3 π¦ (π₯, πΌ) = (1 β π₯) exp (β2π₯) + (1 β πΌ) 8
(83)
Notice that the length of π¦σΈ (π₯, πΌ) len (π¦σΈ (π₯, πΌ)) = π¦σΈ (π₯, πΌ) β π¦σΈ (π₯, πΌ)
π₯
π¦σΈ (π₯) = β3π¦ (π₯) β β« exp (β (π₯ β π‘)) π¦ (π‘) ππ‘,
L [π¦ (π₯, πΌ)] = (πΌ β 1)
= (1 β πΌ)
3 7β5 + (1 β πΌ) ππ₯ ( cosh (β5π₯) + sinh (β5π₯)) . 8 40
π¦ (0, πΌ) = (1, 1) .
2
π (π + 1) L [π¦ (π₯, πΌ)] + (3π + 4) L [π¦ (π₯, πΌ)]
(3π + 4) L [π¦ (π₯, πΌ)] + π (π + 1) L [π¦ (π₯, πΌ)]
π₯
0
Case 2. If π¦(π₯) is (ii)-differentiable, then Theorems 18 and 25 yield
(1 β πΌ) [π₯ cosh (π₯) + sinh (π₯)] πβπ₯ 2
(81)
+ (1 β π₯) πβ2π₯ . In this case, the solution is valid over [0, β[, since π¦(π₯) is (i)-differentiable.
=β
5 (πΌ β 1) + (πΌ β 1) ππ₯ cosh (β5π₯) (84) 2 2
+
11β5 (πΌ β 1) ππ₯ sinh (β5π₯) 10
is a nonnegative increasing function over [0, β[; then π¦(π₯) is (ii)-differentiable. So, in this case the solution is acceptable for all π₯ β₯ 0. Taking πΌ = 1 in formulas (81) and (83) yields the crisp solution, π¦(π₯) = (1 β π₯) exp(β2π₯), of the classic problem (79) (see [20] page 8 Example 1.2.1).
7. Conclusion In this paper, we have introduced the Aumann fuzzy improper integral, and also we have applied Laplace transform method for solving FIDEs, with kernel of convolution type, under the assumption of strongly generalized differentiability. Clearly, the suggested formula allows us to solve more difficult FIDEs by Laplace method compared to the previously reported works. Indeed, in the most fuzzy examples studied before, the considered kernels π(π₯) were real and nonnegative constants. But in this paper, we treated various cases for this kernel π(π₯): positive or negative in the first and second examples, respectively; π(π₯) = sin(π₯) and π(π₯) = exp(βπ₯) were nonconstant functions of π₯ in the third and fourth ones.
10 For future research, we will apply Laplace transform method to solve FIDEs with a fuzzy kernel.
Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.
References [1] R. J. Aumann, βIntegrals of set-valued functions,β Journal of Mathematical Analysis and Applications, vol. 12, pp. 1β12, 1965. [2] D. Ralescu and G. Adams, βThe fuzzy integral,β Journal of Mathematical Analysis and Applications, vol. 75, no. 2, pp. 562β570, 1980. [3] D. Dubois and H. Prade, βTowards fuzzy differential calculus. I. INTegration of fuzzy mappings,β Fuzzy Sets and Systems, vol. 8, no. 1, pp. 1β17, 1982. [4] G. Debreu, βIntegration of correspondences,β in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Ed., pp. 351β372. [5] M. L. Puri and D. A. Ralescu, βFuzzy random variables,β Journal of Mathematical Analysis and Applications, vol. 114, no. 2, pp. 409β422, 1986. [6] H.-C. Wu, βThe fuzzy Riemann integral and its numerical integration,β Fuzzy Sets and Systems, vol. 110, no. 1, pp. 1β25, 2000. [7] T. Allahviranloo, S. Abbasbandy, S. Salahshour, and A. Hakimzadeh, βA new method for solving fuzzy linear differential equations,β Computing (Vienna/New York), vol. 92, no. 2, pp. 181β197, 2011. [8] S. Melliani, E. Eljaoui, and L. S. Chadli, βSolving fuzzy linear differential equations by a new method,β Annals of Fuzzy Mathematics and Informatics, vol. 9, no. 2, pp. 307β323, February 2015. [9] H.-C. Wu, βThe improper fuzzy Riemann integral and its numerical integration,β Information Sciences, vol. 111, no. 1-4, pp. 109β137, 1998. [10] T. Allahviranloo and M. B. Ahmadi, βFuzzy Laplace transforms,β Soft Computing, vol. 14, no. 3, pp. 235β243, 2010. [11] S. Salahshour and T. Allahviranloo, βApplications of fuzzy laplace transforms,β Soft Computing, vol. 17, no. 1, pp. 145β158, 2013. [12] E. ElJaoui, S. Melliani, and L. S. Chadli, βSolving second-order fuzzy differential equations by the fuzzy Laplace transform method,β Advances in Difference Equations, vol. 2015, no. 1, 2015. [13] E. Eljaoui and S. Melliani, βOn Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations,β International Journal of Differential Equations, vol. 2016, Article ID 7246027, 2016. [14] S. Salahshour, M. Khezerloo, S. Hajighasemi, and M. Khorasany, βSolving fuzzy integral equations of the second kind by fuzzy laplace transform method,β International Journal of Industrial Mathematics, vol. 4, no. 1, pp. 21β29, 2012. [15] P. Balasubramaniam and S. Muralisankar, βExistence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations,β Applied Mathematics Letters, vol. 14, no. 4, pp. 455β462, 2001. [16] J. Y. Park and J. U. Jeong, βOn existence and uniqueness of solutions of fuzzy integrodifferential equations,β Indian Journal of Pure and Applied Mathematics, vol. 34, no. 10, pp. 1503β1512, 2003.
Advances in Fuzzy Systems [17] N. Phuong, N. Phung, and H. Vu, βExistence and Uniqueness of Fuzzy Control Integro-Differential Equation with Perturbed,β Bulletin of Mathematical Sciences and Applications, vol. 3, pp. 37β44, 2013. [18] V. Volterra, Theory of Functionals And of Integral And IntegroDifferential Equations, Dover Publications, Inc., New York, NY, USA, 1959. [19] V. Lakshmikantham and M. Rama Mohana Rao, Theory of integro-differential equations, vol. 1 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Lausanne, Switzerland, 1995. [20] T. A. Burton, Volterra Integral and Differential Equations, vol. 202 of Mathematics in Science and Engineering, Elsevier, Amsterdam, The Netherlands, 2005. [21] E. P. Klement, M. L. Puri, and D. A. Ralescu, βLimit theorems for fuzzy random variables,β Proceedings of the Royal Society A Mathematical, Physical and Engineering Sciences, vol. 407, no. 1832, pp. 171β182, 1986. [22] B. Bede, I. J. Rudas, and A. L. Bencsik, βFirst order linear fuzzy differential equations under generalized differentiability,β Information Sciences, vol. 177, no. 7, pp. 1648β1662, 2007. [23] Y. Chalco-Cano and H. RomΒ΄an-Flores, βOn new solutions of fuzzy differential equations,β Chaos, Solitons & Fractals, vol. 38, no. 1, pp. 112β119, 2008. [24] O. Kaleva, βFuzzy differential equations,β Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301β317, 1987. [25] N. Mikaeilvand, S. Khakrangin, and T. Allahviranloo, βSolving fuzzy volterra integro-differential equation by fuzzy differential transform method,β in Proceedings of the Joint 7th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2011 and 17th French Days on Fuzzy Logic and Applications, LFA 2011, pp. 891β896, July 2011.
Journal of
Advances in
Industrial Engineering
Multimedia
Hindawi Publishing Corporation http://www.hindawi.com
The Scientific World Journal Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Applied Computational Intelligence and Soft Computing
International Journal of
Distributed Sensor Networks Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 201
Advances in
Fuzzy Systems Modelling & Simulation in Engineering Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at https://www.hindawi.com
-RXUQDORI
&RPSXWHU1HWZRUNV DQG&RPPXQLFDWLRQV
βAdvancesβinβ
Artificial Intelligence +LQGDZL3XEOLVKLQJ&RUSRUDWLRQ KWWSZZZKLQGDZLFRP
HindawiβPublishingβCorporation http://www.hindawi.com
9ROXPH
International Journal of
Biomedical Imaging
Volumeβ2014
Advances in
$UWLΓFLDO 1HXUDO6\VWHPV
International Journal of
Computer Engineering
Computer Games Technology
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Advances in
Volume 2014
Advances in
Software Engineering Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 201
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Reconfigurable Computing
Robotics Hindawi Publishing Corporation http://www.hindawi.com
Computational Intelligence and Neuroscience
Advances in
Human-Computer Interaction
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Journal of
Electrical and Computer Engineering Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014