Abstract- In this paper, we use the fuzzy Laplace transform method to calculate the exact solutions of a system of tow fuzzy linear differential equations, under generalized Hukuhara differentiability. Then, we transform a second order fuzzy linear differential equation into a fuzzy differential system, which we solve by the fuzzy Laplace algorithm.
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Keywords: fuzzy differential equation; fuzzy differential system; fuzzy second order differential equation; fuzzy laplace transform. I.
Introduction
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Fuzzy differential equations (FDEs) and systems (FDSs) arise in a variety of domains of biological, physical, and engineering applications. In, the recent years, several authors have attracted much attention to the study of FDEs. We refer to [7, 8, 11, 12, 14, 15], where the reader will find the theoretical results necessary to deal with this kind of equations. In the other hand, some other researchers have proceeded to the ”fuzzification” of numerous approaches that are commonly used in the crisp case , and they have developed some fuzzy version of numerical algorithms and methods: fuzzy Laplace transform and fuzzy differential transform method... (see [1, 3–5, 10, 13, 17, 18] and the references therein) . Allahviranloo et al. proposed in [3] a novel method for solving fuzzy linear differential equations which its construction based on the equivalent integral forms of original problems under the assumption of strongly generalized differentiability. In 2015, we developed in [13] an operator method for solving some first order fuzzy linear differential equations, with variable coefficients and we gave the general formula’s solution with necessary proofs. Before in 2010 , T. Allahviranloo and M. B. Ahmadi introduced in [2] fuzzy Laplace transform, which they used under the strongly generalized differentiability, in an analytic solution method for some first order fuzzy differential equations (FDEs). In 2013, S. Salahshour et al. gave in [17] some applications of fuzzy Laplace transform method, and studied sufficient conditions ensuring the existence of the fuzzy
Author α σ ρ : Laboratoire de Mathematiques Appliquees & Calcul Scientifique, Sultan Moulay Slimane University, Beni Mellal, Morocco. e-mail:
[email protected]
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Elhassan ElJaoui α, Said Melliani σ & L. Saadia Chadli ρ
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
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7. B. Bede and S. G. Gal, Generalization of the Differentiability of Fuzzy-NumberValued Functions with Applications to Fuzzy Differentiel Equations, Fuzzy Sets and Systems, Vol. 151 (2005), pp. 581{599.
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
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Preliminaries
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II.
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By PK (R) we denote the family of all nonempty compact convex subsets of R and define the addition and scalar multiplication in PK (R) as usual. Denote n o E = u : R −→ [0, 1] | u satisfies (i) – (iv) below where (i) u is normal, i.e. ∃x0 ∈ R for which u(x0 ) = 1, (ii) u is fuzzy convex, i.e.
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u(λx + (1 − λy)) ≥ min(u(x), u(y))
for any x, y ∈ R, and λ ∈ [0, 1],
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(iii) u is upper semi-continuous, (iv) suppu = {x ∈ R|u(x) > 0} is the support of the u, and its closure cl (supp u) is compact. For 0 < α ≤ 1, denote α
[u] = {x ∈ R | u(x) ≥ α} α
Then, from (i)-(iv), it follows that the α-level set [u] ∈ PK (R) for all 0 ≤ α ≤ 1. 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 α
α
α
[u + v] = [u] + [v] ,
α
α
[ku] = k [u] .
Let D : E × E −→ [0, ∞) be a function which is defined by the equation D(u, v) = sup d [u]α , [v]α 0≤α≤1
where d is the Hausdorff metric defined in PK (R). Then, it is easy to see that D is a metric in E and has the following properties [16]: (1) (E, D) is a complete metric space; (2) D(u + w, v + w) = D(u, v) for all u, v, w ∈ E; (3) D(k u, k v) = |k| D(u, v) for all u, v ∈ E and k ∈ R; © 2015
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10. E. ElJaoui, S. Melliani and L.S. Chadli, Solving second-order fuzzy differential equations by the fuzzy Laplace transform method, Advances in Difference
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Ref Equations, Vol. 66 (2015), pp. 1{14.
Laplace transform for fuzzy-valued functions. In 2015, we extend and use this method, in [10] to solve second-order fuzzy linear differential equations under generalized Hukuhara differentiability. The present work can be viewed as a continuation of the recent results on this issue. Here,we develop the fuzzy Laplace algorithm to solve a system of two first order fuzzy linear differential equations, under generalized differentiability. Then, we apply this algorithm to solve second order fuzzy linear differential equation, after transforming it into a fuzzy differential system. The remainder of this paper is organized as follows: Section 2 is reserved for some preliminaries. In section 3, fuzzy Laplace transform is introduced, its basic properties are studied. Then in section 4, the procedure for solving a system of two first order fuzzy linear differential equations by fuzzy Laplace transform is proposed. Section 5 deals with some numerical examples. In section 6, a second-order fuzzy linear differential equation is transformed into a fuzzy differential system, which is solved by fuzzy Laplace transform algorithm. In the last section, we present conclusion and a further research topic.
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
(4) D(u + w, v + t) ≤ D(u, v) + D(w, t) for all u, v, w, t ∈ E.
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A crisp number k is simply represented by u(r) = u(r) = k, 0 ≤ r ≤ 1. Let T = [c, d] ⊂ R be a compact interval.
Z
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Let F : T → E, then the integral of F over T denoted by is defined by the equation
α
Z
F (t)dt
α
F (t)dt
α ∈]0, 1]
Fα (t)dt;
Z
f (t)dt/f : T → R is a mesurable selection for Fα
=
T
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T
F (t)dt, c
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Z
T
Z
=
T
i.e
d
Z
F (t)dt or
Also, a strongly mesurable Z and integrably bounded mapping F : T → E is said to F (t)dt ∈ E. be integrable over T if T
Proposition 2.4 (Aumann [6]). If F : T → E is strongly measurable and integrably bounded, then F is integrable.
For more measurability, integrability properties for fuzzy set-valued mappings see [2, 7, 11]. Theorem 2.5 (see [20, 21]) Let f (x) be a fuzzy value function on [a, ∞[ and it is represented by (f (x, r), f (x, r)). For any fixed r ∈ [0, 1], assume f (x, r), f (x, r) are Riemann-integrable on [a, b] for every b ≥ a, and assume there are two positive Rb Rb constants M (r) and M (r) such that a |f (x, r)|dx ≤ M (r) and a |f (x, r)|dx ≤ M (r) for every b ≥ a. Then f (x) is improper fuzzy Riemann-integrable on [a, ∞[ and the improper fuzzy Riemann-integral is a fuzzy number. Further more, we have: ∞
Z
Z f (x)dx =
a
∞
∞
Z f (x, r)dx,
a
f (x, r)dx .
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Definition 2.3
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Definition 2.2 A mapping F : T → E is strongly mesurable if for all α ∈ [0, 1] the set-valued function Fα : T → PK (R) defined by Fα (t) = [F (t)]α is Lebesgue mesurable. A mapping F : T → E is called integrably bounded if there exists an integrable function k such that kxk ≤ k(t) for all x ∈ F0 (t).
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2. T. Allahviranloo , M. B. Ahmadi, Fuzzy Laplace transforms, Soft Computing 14 (2010), pp. 235{243.
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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 the following requirements: (1) u(r) is a bounded non-decreasing left continuous function in (0, 1], and right continuous at 0; (2) u(r) is a bounded non-increasing left continuous function in (0, 1], and right continuous at 0; (3) u(r) ≤ u(r) for all 0 ≤ r ≤ 1.
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Proposition 2.6 (see [20]) If each of f (x) and g(x) is a fuzzy valued function and fuzzy Riemann integrable on [a, ∞[ then f (x) + g(x) is fuzzy Riemann-integrable on [a, ∞[. Moreover, we have ∞
Z
∞
Z (f (x) + g(x))dx =
a
∞
Z f (x)dx +
a
g(x)dx. a
For u, v ∈ E, if there exists w ∈ E such that u = v + w, then w is the Hukuhara difference of u and v denoted by u v.
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lim+
h→0
f (x0 + h) f (x0 ) = h
lim
h→0+
f (x0 ) f (x0 − h) = f 0 (x0 ) h
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or (ii) for all h > 0 sufficiently small, there exist f (x0 ) f (x0 + h); f (x0 − h) f (x0 ) and the limits
h→0
f (x0 ) f (x0 + h) f (x0 − h) f (x0 ) = lim = f 0 (x0 ) + (−h) (−h) h→0
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or (iii) for all h > 0 sufficiently small, there exist f (x0 + h) f (x0 ); f (x0 − h) f (x0 ) and the limits lim
h→0+
f (x0 + h) f (x0 ) f (x0 − h) f (x0 ) = lim = f 0 (x0 ) h (−h) h→0+
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or (iv) for all h > 0 sufficiently small, there exist f (x0 ) f (x0 + h); f (x0 ) f (x0 − h) and the limits lim
h→0+
f (x0 ) f (x0 + h) f (x0 ) f (x0 − h) = lim = f 0 (x0 ) (−h) h h→0+
The following theorem (see [7]) allows us to consider case (i) or (ii) of the previous definition almost everywhere in the domain of the functions under discussion. Theorem 2.8 Let f : (a, b) → E be strongly generalized differentiable on each point x ∈ (a, b) in the sense of Definition 2.3, (iii) or (iv). Then f 0 (x) ∈ R for all x ∈ (a, b). Theorem 2.9 (see e.g. [9]) 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 differentiable functions and
0
f 0 (t) = (f 0 (t, r), f (t, r)).
(2) If f is (ii)-differentiable, then f (t, r) and f (t, r) are differentiable functions and © 2015
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f 0 (t) = (f (t, r), f 0 (t, r)).
Systems, Vol. 151 (2005), pp. 581{599.
(i) for all h > 0 sufficiently small, there exist f (x0 + h) f (x0 ); f (x0 ) f (x0 − h) and the limits
7. B. Bede and S. G. Gal, Generalization of the Differentiability of Fuzzy-NumberValued Functions with Applications to Fuzzy Differentiel Equations, Fuzzy Sets and
Definition 2.7 We say that a mapping f : (a, b) → E is strongly generalized differentiable at x0 ∈ (a, b); if there exists an element f 0 (x0 ) ∈ E; such that
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Definition 2.10 We say that a mapping f : (a, b) → E is strongly generalized differentiable of the second-order at x0 ∈ (a, b); if there exists an element f ”(x0 ) ∈ E; such that (i) for all h > 0 sufficiently small, there exist f 0 (x0 +h) f 0 (x0 ); f 0 (x0 ) f 0 (x0 −h) and the limits f 0 (x0 ) f 0 (x0 − h) = f ”(x0 ) h
lim+
h→0
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or (ii) for all h > 0 sufficiently small, there exist f 0 (x0 ) f 0 (x0 +h); f 0 (x0 −h) f 0 (x0 ) and the limits f 0 (x0 ) f 0 (x0 + h) f 0 (x0 − h) f 0 (x0 ) = lim = f ”(x0 ) (−h) (−h) h→0+
h→0
ew
or (iii) for all h > 0 sufficiently small, there exist f 0 (x0 +h) f 0 (x0 ); f 0 (x0 −h) f 0 (x0 ) and the limits lim+
f 0 (x0 + h) f 0 (x0 ) f 0 (x0 − h) f 0 (x0 ) = lim = f ”(x0 ) h (−h) h→0+
f 0 (x0 ) f 0 (x0 + h) f 0 (x0 ) f 0 (x0 − h) = lim = f ”(x0 ) + (−h) h h→0
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h→0+
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or (iv) for all h > 0 sufficiently small, there exist f 0 (x0 ) f 0 (x0 +h); f 0 (x0 ) f 0 (x0 −h) and the limits lim
All the limits are taken in the metric space (E , D) at the end points of (a, b) we consider only one-sided derivatives.
Fuzzy Laplace Transform
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Definition 3.1 (see [2].) Let f (x) be continuous fuzzy-valued function. Suppose R∞ that e−px f (x) is improper fuzzy Riemann integrable on [0, ∞[ then 0 e−px f (x)dx is called fuzzy Laplace transform of f and is denoted as Z ∞ e−px f (x)dx, p > 0. L[f (x)] = 0
Denote L(g(x)) the classical Laplace transform of a crisp function g(x). R∞ R∞ R∞ Since 0 e−px f (x)dx = ( 0 e−px f (x, r)dx, 0 e−px f (x, r)dx), then L[f (x)] = L(f (x, r)), L(f (x, r))
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h→0+
Theorem 3.2 (see [2].) Let f 0 (x) be an integrable fuzzy-valued function, and f (x) is the primitive of f 0 (x) on [0, ∞[. Then, if f is (i)-differentiable: L[f 0 (x)] = pL[f (x)] f (0)
(1)
or if f is (ii)-differentiable: L[f 0 (x)] = (−f (0)) (−p)L[f (x)].
(2)
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f 0 (x0 + h) f 0 (x0 ) = h
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2. T. Allahviranloo , M. B. Ahmadi, Fuzzy Laplace transforms, Soft Computing 14 (2010), pp. 235{243.
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lim+
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Theorem 3.3 (see [2].) Let f (x), g(x) be continuous fuzzy-valued functions and c1 , c2 two real constants, then L[c1 f (x) + c2 g(x)] = c1 L[f (x)] + c2 L[g(x)].
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Theorem 3.4 (see [10].) Let f (x) be a continuous fuzzy-valued function such that e−px f (x), e−px f 0 (x) and e−px f ”(x) exist, are continuous and integrable on [0, ∞[. We distinguish the following cases (a) If f (x) and f 0 (x) are (i)-differentiable, then
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L[f ”(x)] = {p2 L[f (x)] pf (0)} f 0 (0)
(3)
(b) If f (x) is (i)-differentiable and f 0 (x) is (ii)-differentiable, then L[f ”(x)] = (−f 0 (0)) {−p2 L[f (x)] (−pf (0))}
(4)
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(c) If f (x) is (ii)-differentiable and f 0 (x) is (i)-differentiable, then L[f ”(x)] = {(−pf (0)) −p2 L[f (x)]} f 0 (0)
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(d) If f (x) is (ii)-differentiable and f 0 (x) is (ii)-differentiable, then L[f ”(x)] = (−f 0 (0)) {pf (0) p2 L[f (x)]}
(6)
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Our aim now is to solve the following fuzzy linear differential system using fuzzy Laplace transform method under strongly generalized differentiability: x0 (t) = f (t, x(t), y(t)) y 0 (t) = g(t, x(t), y(t)) x(0) = x0 = (x0 , x0 ) ∈ E y(0) = y0 = (y 0 , y 0 ) ∈ E
(7)
where x(t) = (x(t, α), x(t, α)), y(t) = (y(t, α), y(t, α)) are fuzzy functions of t ≥ 0, f (t, x(t), y(t)) and g(t, x(t), y(t)) are fuzzy-valued functions, which are linear with respect to (x(t), y(t)). By using fuzzy Laplace transform, we have (
L[x0 (t, α)] = L[f (t, x(t), y(t), α)] L[y 0 (t, α)] = L[g(t, x(t), y(t), α)]
Then, we have the following alternatives for solving (8): (a) Case I: If x and y are (i)-differentiable: i.e x0 (t, α) = (x0 (t, α), x0 (t, α)) and y 0 (t, α) = (y 0 (t, α), y 0 (t, α)). Then by theorem (3.2), we have
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
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(b) Case II: If x is (i)-differentiable and y is (ii)-differentiable: i.e x0 (t) = (x0 (t, α), x0 (t, α)) and y 0 (t) = (y 0 (t, α), y 0 (t, α)). Then by theorem (3.2), we have (
L[f (t, x(t), y(t), α)] = pL[x(t, α)] x(0, α) L[g(t, x(t), y(t), α)] = −y(0, α) (−pL[y(t, α)])
Hence pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x0 (α) pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x (α) 0 pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y 0 (α) pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y 0 (α)
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where (H1 (p, α), K1 (p, α), N1 (p, α), M1 (p, α)) is solution of the system (10). By using the inverse Laplace transform we get x(t, α) = L−1 [H1 (p, α)] x(t, α) = L−1 [K (p, α)] 1 y(t, α) = L−1 [N1 (p, α)] y(t, α) = L−1 [M1 (p, α)]
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where F (t, x(t), y(t), α) = min{F (t, u, v)/u ∈ (x(t, α), x(t, α)); v ∈ (y(t, α), y(t, α))}, F (t, x(t), y(t), α) = max{F (t, u, v)/u ∈ (x(t, α), x(t, α)); v ∈ (y(t, α), y(t, α))}, for F = f or F = g. Assume that this leads to L[x(t, α)] = H1 (p, α) L[x(t, α)] = K (p, α) 1 L[y(t, α)] = N1 (p, α) L[y(t, α)] = M1 (p, α)
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Hence pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x0 (α) pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x (α) 0 pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y (α) 0 pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y 0 (α)
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L[f (t, x(t), y(t), α)] = pL[x(t, α)] x(0, α) L[g(t, x(t), y(t), α)] = pL[y(t), α] y(0, α)
(11)
(12)
Assume that this leads to L[x(t, α)] = H2 (p, α) L[x(t, α)] = K (p, α) 2 L[y(t, α)] = N2 (p, α) L[y(t, α)] = M2 (p, α) where (H2 (p, α), K2 (p, α), N2 (p, α), M2 (p, α)) is solution of the system (12). By using the inverse Laplace transform we get © 2015
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
x(t, α) = L−1 [H2 (p, α)] x(t, α) = L−1 [K (p, α)] 2 −1 y(t, α) = L [N 2 (p, α)] −1 y(t, α) = L [M2 (p, α)] (c) Case III: If x is (ii)-differentiable and y is (i)-differentiable: i.e
x0 (t) = (x0 (t, α), x0 (t, α)) and y 0 (t) = (y 0 (t, α), y 0 (t, α)).
Then by theorem (3.2), we have
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Notes
L[f (t, x(t), y(t), α)] = −x(0, α) (−pL[x(t, α)]) L[g(t, x(t), y(t), α)] = pL[y(t, α)] y(0, α)
(13)
Hence
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Assume that this leads to L[x(t, α)] = H3 (p, α) L[x(t, α)] = K (p, α) 3 L[y(t, α)] = N3 (p, α) L[y(t, α)] = M3 (p, α)
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pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x0 (α) pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x (α) 0 pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y (α) 0 pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y 0 (α)
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where (H3 (p, α), K3 (p, α), N3 (p, α), M3 (p, α)) is solution of the system (??). By using the inverse Laplace transform we get x(t, α) = L−1 [H3 (p, α)] x(t, α) = L−1 [K (p, α)] 3 −1 y(t, α) = L [N 3 (p, α)] −1 y(t, α) = L [M3 (p, α)]
(d) Case IV: If x and y are (ii)-differentiable: i.e x0 (t) = (x0 (t, α), x0 (t, α)) and y 0 (t) = (y 0 (t, α), y 0 (t, α)). Then by theorem (3.2), we have ( L[f (t, x(t), y(t), α)] = −x(0, α) (−pL[x(t, α)]) L[g(t, x(t), y(t), α)] = −y(0, α) (−pL[y(t, α)]) Hence pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x0 (α) pL[x(t, α)] = L[f (t, x(t), y(t), α)] + x (α) 0 pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y 0 (α) pL[y(t, α)] = L[g(t, x(t), y(t), α)] + y 0 (α) Assume that this leads to L[x(t, α)] = H4 (p, α) L[x(t, α)] = K4 (p, α) L[y(t, α)] = N4 (p, α) L[y(t, α)] = M4 (p, α) © 2015
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
where (H4 (p, α), K4 (p, α), N4 (p, α), M4 (p, α)) is solution of the system (??). By using the inverse Laplace transform we get
X
sgn(σ)
Y
ai,σ(i) = ±
σ∈S4
1≤i≤4
Y
1≤i≤4
(p−bi )±
X σ∈S4 \\{id}
Y
sgn(σ)
ai,σ(i)
1≤i≤4
where sgn(σ) = ±1 is the signature of the permutation σ. Clearly, the function ϕ : p → det(A(p, α)) is polynomial of degree 4, with leading coefficient equals ±1. Hence, ϕ admit four real roots at most and there exists p0 > 0 such that ϕ(p) 6= 0, for all p ≥ p0 . So, the matrix A(p, α) is invertible and the system (10) has a unique solution (H1 (p, α), K1 (p, α), N1 (p, α), M1 (p, α)) ( which can be calculated by Cramer’s formula), for all p ≥ p0 . For instance, if x(t), y(t) are (i)-differentiable and a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0, the system (10) become (p − a)L[x(t, α)] − bL[y(t, α)] = L[h(t, α)] + x0 (p − a)L[x(t, α)] − bL[y(t, α)] = L[h(t, α)] + x 0 −cL[x(t, α)] + (p − d)L[y(t, α)] = L[k(t, α)] + y 0 −cL[x(t, α)] + (p − d)L[y(t, α)] = L[k(t, α)] + y 0
Then, ϕ(p) = (p2 − (a + d)p + ad − bc)2 is clearly polynomial of degree 4. Similarly, Hj (p, α), Kj (p, α), Nj (p, α), Mj (p, α) exist and their expressions can be easily computed (using Matlab or Maple), for j ∈ {2, 3, 4}. In the next section, we investigate some various case, where a, b, c, d or some of them are negative. © 2015
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det(A(p, α)) = ±
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where ai,j is independent of p for i 6= j and ai,i = p ± bi , with bi ∈ {0, ±a, ±b, ±c, ±d}. If S4 denotes the symmetric group of the bijections from {1, . . . , 4} into itself, we have
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det(A(p, α)) = ± det ((ai,j )1≤i,j≤4 )
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Remark 4.1 To show that H1 (p, α), K1 (p, α), N1 (p, α) and M1 (p, α) exist and can be calculated, we suppose that the fuzzy linear functions f and g are given by f (t, x(t), y(t)) = ax(t) + by(t) + h(t), g(t, x(t), y(t)) = cx(t) + dy(t) + k(t), where a, b, c, d are real constants and h(t), k(t) are fuzzy continuous mappings. We have to discuss exactly 16 cases: it depends on the respective signs of the real numbers a, b, c, d. One can verify that in each case, the matrix A(p, α) of the linear system (10) is invertible for adequate values of p. Indeed: Let a fixed α ∈ [0, 1]. By performing permutations on the rows of the system (10), one gets
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x(t, α) = L−1 [H4 (p, α)] x(t, α) = L−1 [K (p, α)] 4 y(t, α) = L−1 [N4 (p, α)] y(t, α) = L−1 [M4 (p, α)]
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
V.
Numerical Examples
Example 1 x0 (t) = 6x(t) + 3y(t) y 0 (t) = (−4).x(t) + (−1).y(t) x(0, α) = e 1 = (α, 2 − α) f y(0, α) = −1 = (α − 2, −α)
(17)
Year
2015
• Case I: If x(t) and y(t) are (i)-differentiable, then 0 x (t, α) = 6x(t, α) + 3y(t, α) x0 (t, α) = 6x(t, α) + 3y(t, α) y 0 (t, α) = −4x(t, α) − y(t, α) 0 y (t, α) = −4x(t, α) − y(t, α)
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Using theorem (3.2), we get (p − 6)L[x(t, α)] − 3L[y(t, α)] = α (p − 6)L[x(t, α)] − 3L[y(t, α)] = 2 − α 4L[x(t, α)] + pL[y(t, α)] + L[y(t, α)] = α − 2 4L[x(t, α)] + L[y(t, α)] + pL[y(t, α)] = −α Therefore
Global Journal of Science Frontier Research
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2 −(α+6)p−6α L[x(t, α)] = pαp 3 −10p2 +15p+18 2 L[x(t, α)] = (2−α)p +(α−8)p+6α−12 p3 −10p2 +15p+18 (α−2)p2 +(12−5α)p+6α L[y(t, α)] = p3 −10p2 +15p+18 −αp2 +(5α+2)p+12−6α L[y(t, α)] = p3 −10p2 +15p+18
By the inverse Laplace transform we deduce
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√ √ √ 7t 7t 73t 73t x(t, α) = e3t + (α − 1)e 2 cosh + 117373 (α − 1)e 2 sinh 2 2 √ √ √ 7t 73t 73t 11 73 x(t, α) = e3t + (1 − α)e 7t2 cosh + 73 (1 − α)e 2 sinh 2 √ √2 √ 7t 7t 73t 73t 3 73 3t 2 2 y(t, α) = −e + (α − 1)e cosh + 73 (α − 1)e sinh √2 √2 √ 7t 7t 73t 73t y(t, α) = −e3t + (1 − α)e 2 cosh + 3 7373 (1 − α)e 2 sinh 2 2
In this case, the solution is valid all over [0, ∞[. • Case II: If x(t) is (i)-differentiable and y(t) is (ii)-differentiable, then 0 x (t, α) = 6x(t, α) + 3y(t, α) 0 x (t, α) = 6x(t, α) + 3y(t, α) 0 y (t, α) = −4x(t, α) − y(t, α) 0 y (t, α) = −4x(t, α) − y(t, α) Using theorem (3.2), we get (p − 6)L[x(t, α)] − 3L[y(t, α)] = α (p − 6)L[x(t, α)] − 3L[y(t, α)] = 2 − α 4L[x(t, α)] + (p + 1)L[y(t, α)] = −α 4L[x(t, α)] + (p + 1)L[y(t, α)] = α − 2 © 2015
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Notes
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Therefore L[x(t, α)] = αp+4α−6 p2 −5p+6 L[x(t, α)] = (2−α)p+2−4α p2 −5p+6 (α−2)p+12−10α L[y(t, α)] = p2 −5p+6 L[y(t, α)] = −αp+10α−8 p2 −5p+6
Year
2015
x(t, α) = 6(1 − α)e2t + (7α − 6)e3t x(t, α) = 6(α − 1)e2t + (8 − 7α)e3t y(t, α) = 8(α − 1)e2t + (6 − 7α)e3t y(t, α) = 8(1 − α)e2t + (7α − 8)e3t
One can verify that there is a unique real number γ ∈]0.133; 0.134[ such that len(y(t, α)) ≥ 0, only for t ∈ [0, γ]. We also have ∀t ∈ [0, γ],
Vi
len(y 0 (t, α)) = y 0 (t, α) − y 0 (t, α) = 2(1 − α)e2t 21et − 16 ≥ 0;
Global Journal of Science Frontier Research
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which means that y(t) is (ii)-differentiable on [0; γ]. Therefore, the solution (x(t), y(t)) is valid only on the small interval [0, γ]. • Case III: If x(t) is (ii)-differentiable and y(t) is (i)-differentiable, then 0 x (t, α) = 6x(t, α) + 3y(t, α) x0 (t, α) = 6x(t, α) + 3y(t, α) y 0 (t, α) = −4x(t, α) − y(t, α) 0 y (t, α) = −4x(t, α) − y(t, α) Using theorem (3.2), we get
−6L[x(t, α)] + L[x(t, α)] − 3L[y(t, α)] = 2 − α pL[x(t, α)] − 6L[x(t, α)] − 3L[y(t, α)] = α 4L[x(t, α)] + pL[y(t, α)] + L[y(t, α)] = α − 2 4L[x(t, α)] + L[y(t, α)] + pL[y(t, α)] = −α
Therefore 2 L[x(t, α)] = αp +(12−7α)p+12α−6 p3 +2p2 −9p−18 L[x(t, α)] = (2−α)p2 +(7α−2)p+18−12α p3 +2p2 −9p−18 (α−2)p2 +(7α−12)p+24−30α L[y(t, α)] = p3 +2p2 −9p−18 −αp2 +(2−7α)p+30α−36 L[y(t, α)] = p3 +2p2 −9p−18 By the inverse Laplace transform we deduce x(t, α) = e3t + 6(1 − α)e−2t + 7(α − 1)e−3t x(t, α) = e3t + 6(α − 1)e−2t + 7(1 − α)e−3t y(t, α) = −e3t + 8(α − 1)e−2t + 7(1 − α)e−3t y(t, α) = −e3t + 8(1 − α)e−2t + 7(α − 1)e−3t © 2015
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The mapping x(t, α) is (i)-differentiable on [0, ∞[. And the length of y(t, α) is given by len(y(t, α)) = y(t, α) − y(t, α) = 2(1 − α)e2t 8 − 7et
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Notes
By the inverse Laplace transform we deduce
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
The length of x(t, α) is given by len(x(t, α)) = x(t, α) − x(t, α) = 2(1 − α)e−3t 7 − 6et
One can verify that there is a unique real number β ∈]0.154; 0.155[ such that len(x(t, α)) ≥ 0, only for t ∈ [0, β]. We also have len(x0 (t, α)) = x0 (t, α) − x0 (t, α) = 6(1 − α)e−3t 7 − 4et
Year
2015
And there is a unique real number λ ∈]0.559; 0.560[ such that len(x0 (t, α)) ≥ 0, only for t ∈ [0, λ]. which means that x(t) is (ii)-differentiable on [0; β], since β < λ. In the other hand, the length of y(t, α) is given by len(y(t, α)) = y(t, α) − y(t, α) = 2(1 − α)e−3t 8et − 7 ≥ 0, ∀t ≥ 0
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We also have
ew
len(y 0 (t, α)) = y 0 (t, α) − y 0 (t, α) = 2(1 − α)e−3t 21 − 16et
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Vi
One can verify that there is a unique real number ξ ∈]0.271; 0.272[ such that len(y 0 (t, α)) ≥ 0, only for t ∈ [0, ξ]. which means that y(t) is (ii)-differentiable on [0; ξ]. Therefore, the solution (x(t), y(t)) is valid only on the small interval [0, β], since β < ξ < λ. • Case IV: If x(t) and y(t) are (ii)-differentiable , then 0 x (t, α) = 6x(t, α) + 3y(t, α) x0 (t, α) = 6x(t, α) + 3y(t, α) y 0 (t, α) = −4x(t, α) − y(t, α) 0 y (t, α) = −4x(t, α) − y(t, α) Using theorem (3.2), we get −6L[x(t, α)] + pL[x(t, α)] − 3L[y(t, α)] = 2 − α pL[x(t, α)] − 6L[x(t, α)] − 3L[y(t, α)] = α 4L[x(t, α)] + (p + 1)L[y(t, α)] = −α 4L[x(t, α)] + (p + 1)L[y(t, α)] = α − 2 Therefore 2 L[x(t, α)] = αp p+(12−5α)p+6α−12 3 +4p2 −27p+18 L[x(t, α)] = (2−α)p2 +(5α+2)p−6α p3 +4p2 −27p+18 (α−2)p2 −(α+6)p+12−6α L[y(t, α)] = p3 +4p2 −27p+18 2 +(α−8)p+6α L[y(t, α)] = −αp p3 +4p2 −27p+18 By the inverse Laplace transform we deduce √ √ √ −7t −7t 73t 11 73 73t 3t 2 cosh 2 sinh α) = e + (α − 1)e + (1 − α)e x(t, 2 73 2 √ √ √ −7t 73t 11 73 73t x(t, α) = e3t + (1 − α)e −7t 2 cosh 2 sinh + (α − 1)e 2 73 √ √2 √ −7t −7t 73t 73t 3 73 3t 2 2 cosh sinh 2 + 73 (1 − α)e 2 y(t, α) = −e + (α − 1)e √ √ √ −7t 73t 3 73 73t y(t, α) = −e3t + (1 − α)e −7t 2 cosh 2 sinh + (α − 1)e 2 73 2 © 2015
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Notes
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Ref
Similarly, we verify that, x is (ii)-differentiable on [0, λ1 ], where λ1 ∈ ]0.242; 0.243[, and y is (ii)-differentiable only on [0, γ1 ], where γ1 ∈]0.356; 0.357[. So, this solution is valid only on the small interval [0, λ1 ], since λ1 < γ1 . Please, note that in each case the classical solution of the corresponding crisp problem is given by ( x(t, 1) = x(t, 1) = e3t y(t, 1) = y(t, 1) = −e3t
x(0, α) = (α − 1, 1 − α) y(0, α) = (α, 2 − α)
2015
Year (18)
93
F ) Volume XV Issue IV Version I
x0 (t) + y 0 (t) = (−1).x(t) + (−1).y(t) + (α, 2 − α) y 0 (t) = 2x(t) + y(t)
Vi
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• Case I: If x(t) and y(t) are (i)-differentiable, then x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + α x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + 2 − α y 0 (t, α) = 2x(t, α) + y(t, α) y 0 (t, α) = 2x(t, α) + y(t, α) Using theorem (3.2), we get
rly
pL[x(t, α)] + L[x(t, α)] + pL[y(t, α)] + L[y(t, α)] = 2α − 1 + αp L[x(t, α)] + pL[x(t, α)] + L[y(t, α)] + pL[y(t, α)] = 3 − 2α + 2−α p 2L[x(t, α)] + (1 − p)L[y(t, α)] = −α 2L[x(t, α)] + (1 − p)L[y(t, α)] = α − 2
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Therefore
Ea
19. M. Thongmoona, S. Pusjuso, The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations, Nonlinear Analysis: Hybrid Systems, Vol. 4 (2010), pp.425{431.
Example 2 This second example was studied in [19] using differential transform method ( D.T.M) in the crisp case.
(α−1)p+α−2 p(p+1) L[x(t, α)] = (1−α)p−α p(p+1) 2 L[y(t, α)] = αp +(3α−2)p+2α−4 p(p2 −1) (2−α)p2 +(4−3α)p−2α L[y(t, α)] = p(p2 −1)
L[x(t, α)] =
By the inverse Laplace transform we deduce x(t, α) = e−t + α − 2 x(t, α) = e−t − α y(t, α) = −e−t + 3(α − 1)et + 4 − 2α y(t, α) = −e−t + 3(1 − α)et + 2α We can verify easily that, in this case, the solution is valid all over [0, ∞[. • Case II: If x(t) is (i)-differentiable and y(t) is (ii)-differentiable, then x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + α x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + 2 − α y 0 (t, α) = 2x(t, α) + y(t, α) y 0 (t, α) = 2x(t, α) + y(t, α) © 2015
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Using theorem (3.2), we get pL[x(t, α)] + L[x(t, α)] + (p + 1)L[y(t, α)] = 1 + αp L[x(t, α)] + pL[x(t, α)] + (p + 1)L[y(t, α)] = 1 + 2−α p α)] + L[y(t, α)] − pL[y(t, α)] = α − 2 2L[x(t, 2L[x(t, α)] − pL[y(t, α)] + L[y(t, α)] = −α Therefore
2015
L[x(t, α)] = (α−1)p+α−2 p(p+1) L[x(t, α)] = (1−α)p−α p(p+1) αp+4−2α L[y(t, α)] = p(p+1) L[y(t, α)] = (2−α)p+2α
Year
p(p+1)
By the inverse Laplace transform we deduce x(t, α) = e−t + α − 2 x(t, α) = e−t − α y(t, α) = (3α − 4)e−t + 4 − 2α y(t, α) = (2 − 3α)e−t + 2α
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Obviously, x(t, α) is well defined and is (i)-differentiable on [0, ∞[. The length of y(t, α) is given by len(y(t, α)) = 2(1 − α) 3e−t − 2
One can verify that len(y(t, α)) ≥ 0, for all t ∈ [0, log(3/2)]. We also have ∀t ∈ [0, log(3/2)],
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len(y 0 (t, α)) = 6(1 − α)e−t ≥ 0;
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Global Journal of Science Frontier Research
which means that y(t) is (ii)-differentiable on [0, log(3/2)]. Therefore the solution (x(t), y(t)) is valid only on the small interval [0, log(3/2)]. • Case III: If x(t) is (ii)-differentiable and y(t) is (i)-differentiable, then x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + α x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + 2 − α y 0 (t, α) = 2x(t, α) + y(t, α) y 0 (t, α) = 2x(t, α) + y(t, α)
Using theorem (3.2), we get (p + 1)L[x(t, α)] + pL[y(t, α)] + L[y(t, α)] = 1 + αp (p + 1)L[x(t, α)] + L[y(t, α)] + pL[y(t, α)] = 1 + 2−α p 2L[x(t, α)] + (1 − p)L[y(t, α)] = −α 2L[x(t, α)] + (1 − p)L[y(t, α)] = α − 2 Solving this system and by the inverse Laplace transform one obtains
x(t, α) = e−t + α − 2 x(t, α) = e−t − α y(t, α) = −e−t + 3(α − 1)et + 4 − 2α y(t, α) = −e−t + 3(1 − α)et + 2α © 2015
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
We get here the same solution (x(t), y(t)) as in case I, which is valid all over [0, ∞[. • Case IV: If x(t) and y(t) are (ii)-differentiable , then x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + α x0 (t, α) + y 0 (t, α) = −x(t, α) − y(t, α) + 2 − α y 0 (t, α) = 2x(t, α) + y(t, α) y 0 (t, α) = 2x(t, α) + y(t, α)
Notes
2015
Using theorem (3.2), we get
Year
(p + 1)L[x(t, α)] + (p + 1)L[y(t, α)] = 3 − 2α + αp (p + 1)L[x(t, α)] + (p + 1)L[y(t, α)] = 2α − 1 + 2−α p 2L[x(t, α)] + L[y(t, α)] − pL[y(t, α)] = α − 2 2L[x(t, α)] − pL[y(t, α)] + L[y(t, α)] = −α
F ) Volume XV Issue IV Version I
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Solving this system and using the inverse Laplace transform we have
Vi
x(t, α) = e−t + α − 2 x(t, α) = e−t − α y(t, α) = (3α − 4)e−t + 4 − 2α y(t, α) = (2 − 3α)e−t + 2α
We get here the same solution as in case II, which is valid only on the small interval [0, log(3/2)]. Since x(t, α) = x(t, α) = −e−t , then x is (i) and (ii)-differentiable on [0, ∞[.
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VI.
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Because of that, cases I and III (respectively II and IV) are analogous. Please, note that in each case the classical solution of the corresponding crisp problem is given by ( x(t, 1) = x(t, 1) = e−t − 1 y(t, 1) = y(t, 1) = 2 − e−t
Application to Solve Second order Fuzzy Differential Equations
We consider the following fuzzy differential equation of second order
x”(t) = g(t, x(t), x0 (t)) x(0, α) = x0 (α) ; x0 ∈ E 0 x (0, α) = y0 (α) ; y0 ∈ E
(19)
where the function g verifies the same assumption as in section 4. Let y = x0 , then the equation (19) is equivalent to following F.D.S x0 (t) = y(t) y 0 (t) = g(t, x(t), y(t)) (20) x(0, α) = x0 (α) y(0, α) = y0 (α) which represents a particular case for the F.D.S (7), with f (t, x(t), y(t)) = y(t), so we can use the fuzzy Laplace transform method to solve it. © 2015
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Example 3 We consider the following fuzzy differential equation of second order y”(t) = y 0 (t) + 2y(t) (21) 3 = (α + 2, 4 − α) y(0, α) = e 0 e y (0, α) = 0 = (α − 1, 1 − α)
Year
2015
Let x = y 0 , then (21) is identically equivalent to x0 (t) = x(t) + 2y(t) y 0 (t) = x(t) x(0, α) = (α − 1, 1 − α) y(0, α) = (α + 2, 4 − α)
(22)
• Case I: If y and y 0 are (i)-differentiable i.e x and y are (i)-differentiable, then (p − 1)L[x(t, α)] − 2L[y(t, α)] = α − 1 (p − 1)L[x(t, α)] − 2L[y(t, α)] = 1 − α −L[x(t, α)] + pL[y(t, α)] = α + 2 −L[x(t, α)] + pL[y(t, α)] = 4 − α
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Solving this linear system and using the inverse Laplace transform we deduce
Vi
x(t, α) = 13 (−α − 5)e−t + 23 (2α + 1)e2t x(t, α) = 1 (α − 7)e−t + 2 (5 − 2α)e2t 3 3 y(t, α) = 13 (α + 5)e−t + 13 (2α + 1)e2t y(t, α) = 13 (7 − α)e−t + 13 (5 − 2α)e2t
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One can verify that, in this case, the solution is valid all over [0, ∞[. • Case II: If y is (i)-differentiable and y 0 is (ii)-differentiable i.e y is (i)differentiable and x is (ii)-differentiable, then −L[x(t, α)] + pL[x(t, α)] − 2L[y(t, α)] = 1 − α pL[x(t, α)] − L[x(t, α)] − 2L[y(t, α)] = α − 1 −L[x(t, α)] + pL[y(t, α)] = α + 2 −L[x(t, α)] + pL[y(t, α)] = 4 − α Solving this linear system and using the inverse Laplace transform we get x(t, α) = 2e2t − 2e−t + x(t, α) = 2e2t − 2e−t + y(t, α) = 2e2t − 2e−t + y(t, α) = 2e2t − 2e−t +
√ −t 5 7 2 7 (1 − α)e √ −t 5 7 2 7 (α − 1)e √ −t 3 7 2 7 (α − 1)e √ −t 3 7 2 7 (1 − α)e
√
−t 7t + (α − 1)e 2 √2 −t sin 27t + (1 − α)e 2 √ −t sin 27t + (α − 1)e 2 √ −t sin 27t + (1 − α)e 2
sin
cos cos cos cos
√ 7t
√2 7t
√2 7t
√2 7t 2
One can verify that this solution is valid only on the small interval [0, β], where β ∈]0.367; 0.368[. • Case III: If y is (ii)-differentiable and y 0 is (i)-differentiable i.e y is (ii)differentiable and x is (i)-differentiable, then (p − 1)L[x(t, α)] − 2L[y(t, α)] = α − 1 (p − 1)L[x(t, α)] − 2L[y(t, α)] = 1 − α −L[x(t, α)] + pL[y(t, α)] = 4 − α −L[x(t, α)] + pL[y(t, α)] = α + 2 © 2015
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Notes
Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
Solving this linear system and using the inverse Laplace transform we obtain √ √ √ t t 5 7 7t 7t 2t −t 2 sin 2 cos + (α − 1)e α) = 2e − 2e + (α − 1)e x(t, 7 2 2 √ √ √ t 7t 7t x(t, α) = 2e2t − 2e−t + 5 7 (1 − α)e 2t sin 2 7 2 + (1 − α)e cos 2 √ √ √ t t y(t, α) = e2t + 2e−t + 3 7 7 (1 − α)e 2 sin 27t + (α − 1)e 2 cos 27t √ √ √ t 7t y(t, α) = e2t + 2e−t + 3 7 (α − 1)e 2t sin + (1 − α)e 2 cos 27t 7 2
2015
One can verify that this solution is valid only on the small interval [0, γ], where γ ∈]1.09; 1.1[.
Year
• Case IV: If y and y 0 are (ii)-differentiable i.e x and y are (ii)-differentiable, then −L[x(t, α)] + pL[x(t, α)] − 2L[y(t, α)] = 1 − α pL[x(t, α)] − L[x(t, α)] − 2L[y(t, α)] = α − 1 −L[x(t, α)] + pL[y(t, α)] = 4 − α −L[x(t, α)] + pL[y(t, α)] = α + 2
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Solving this linear system and using the inverse Laplace transform we obtain
Vi
x(t, α) = 2e2t − 2e−t + 13 (1 − α)et + 43 (α − 1)e−2t x(t, α) = 2e2t − 2e−t + 1 (α − 1)et + 4 (1 − α)e−2t 3 3 y(t, α) = e2t + 2e−t + 13 (α − 1)et + 23 (α − 1)e−2t y(t, α) = e2t + 2e−t + 13 (1 − α)et + 32 (1 − α)e−2t
y(t, 1) = y(t, 1) = e2t + 2e−t
VII.
Conclusion
Using Laplace transform method, the solutions for some systems of fuzzy linear first order differential equations (FDSs) are given. Then, a second order fuzzy linear differential equation is converting to a fuzzy differential system and both are solved by Laplace transform algorithm. For future research, one can apply this method to solve FDSs of high order.
VIII.
Acknowledgements
The authors are grateful to anonymous referees for numerous helpful and constructive suggestions which have improved the presentation of results. References Références Referencias 1. S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso and J.J.Nieto, Numerical methods for fuzzy differential inclusions, Comput.Math.Appl., Vol. 48 (2004), pp. 1633{1641. 2. T. Allahviranloo , M. B. Ahmadi, Fuzzy Laplace transforms, Soft Computing, Vol. 14 (2010), pp. 235{243. 3. T. Allahviranloo , S. Abbasbandy, S. Salahshour and A. Hakimzadeh, A new method for solving fuzzy linear differential equations, Computing, Vol. 92 (2011), pp. 181{197. © 2015
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One can verify that this solution is valid only on the small interval [0, 13 log(4)]. Please, note that in each case the classical solution of the corresponding crisp problem is given by
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Year
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Solving a System of tow Fuzzy Linear Differential Equations by Fuzzy Laplace Transform Method
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