Numerical Solution of Fuzzy Differential Equations of 2nd-Order by ...
Journal of Mathematical Extension Vol. 7, No. 3, (2013), 47-62
Numerical Solution of Fuzzy Differential Equations of 2nd-Order by Runge-Kutta Method N. Parandin
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Islamic Azad University, Kermanshah Branch
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Abstract. In this paper, solving fuzzy ordinary differential equations
of the n th order by Runge-Kutta method have been done, and the convergence of the proposed method is proved. This method is illustrated by some numerical examples.
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AMS Subject Classification: 65L06 Keywords and Phrases: Fuzzy differential equations; Runge-Kutta method; Convergence of numerical method
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Introduction
The topics of fuzzy differential equations have been rapidly growing in recent years. The theory of fuzzy differential equations was treated by Buckley and Feuring [9], Kaleva [16, 17], Nieto [22], Ouyang and Wu [23], Roman-Flores and Rojas- Medar [26], Seikkala [27], also recently there appeared the papers of Bede [7], Bede and Gal [8], Diamond [10, 11], Georgiou et al., [15] Nieto and Rodriguez-Lopez [21]. In the following, we have mentioned some numerical solution which have proposed by other scientists. Abbasbandy and Allahviranloo have solved fuzzy differential equations by Runge-Kutta and Taylor methods[1, 2]. Also, Allahviranloo et al. solved differential equations by predictor-corrector and transformation methods[4, 5, 6]. Ghazanfari and Shakerami developed Runge-Kutta like formulae of order 4 for solving fuzzy differential
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Received: December 2012; Accepted: July 2013
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N. PARANDIN
equations[14]. Nystrom method has been introduced for solving fuzzy differential equations[18]. Mosleh and Otadi (2012) simulated and evaluate fuzzy differential equations by fuzzy neural network[20]. Pederson and Sambandham (2008) applied Runge-Kutta method for solving hybrid fuzzy differential equations [25]. Runge-Kutta method has been used for solving fuzzy differential equations by Palligkinis et al. (2009)[24]. Also, Kim and Sakthivel could solve hybrid fuzzy differential equations using improved predictor-corrector method[19]. The paper is organized as follows. Section 2 includes preliminaries. In Section 3, we can see the main idea of this paper. In Section 4, the proposed method is illustrated by examples. The conclusion is in Section 5.
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Preliminaries
Definition 2.1. [12] A fuzzy number is a map u : R −→ I = [0, 1] which satisfies
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(i) u is upper semi continuous,
(ii) u(x) = 0 outside some interval [c, d] ⊂ R,
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(iii) There exist real numbers a, b such that c 6 a 6 b 6 d where, 1. u(x) is monotonic increasing on [c, a],
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2. u(x) is monotonic decreasing on [b, d], 3. u(x) = 1, a 6 x 6 b.
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The set of all such fuzzy numbers is represented by E 1 . Definition 2.2. [12] An arbitrary fuzzy number in parametric form is represented by an ordered pair functions (u(r), u ¯(r)), 0 6 r 6 1, which satisfy the following requirements: 1. u(r) is a bounded left-continuous non-decreasing function over [0, 1], 2. u ¯(r) is a bounded left -continuous non -increasing function over [0, 1],
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
¯(r), 3. u(r) 6 u
0 6 r 6 1.
A crisp number α is simply represented by u(r) = u(r) = α, 0 6 r 6 1. For arbitrary u = (u(r), u ¯(r)), v = (v(r), v¯(r)) and k ∈ R, we define equality, addition and multiplication by k as a. u = v if and only if u(r) = v(r) and u(r) = v(r), b. u + v = (u(r) + v(r), u(r) + v(r)), k > 0, (ku, ku), c. ku = (ku, ku), k < 0.
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Definition 2.3. [8] The Hausdorff distance of two fuzzy numbers given by D : R × R −→ R+ ∪ {0}, is defined as follows:
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r r |} = sup {dH ([u]r , [v]r )}, D(u, v) = sup max{|ur− − v− |, |ur+ − v+ r∈[0,1]
r∈[0,1]
where
[u]r
=
[ur− , ur+ ], [v]r
=
r , v r ]. [v− +
We denote k.k = D(., 0).
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Definition 2.4. [27] Let I be a real interval. A mapping x : I → E is called a fuzzy process and its r-level set is denoted by [x(t)]r = [x1 (t; r), x2 (t; r)], t ∈ I, r ∈ (0, 1]
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the derivative x0 (t) of a fuzzy process x is defined by [x0 (t)]r = [x01 (t; r), x02 (t; r)], t ∈ I, r ∈ (0, 1],
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Runge-kutta Mehtods for Solving Fuzzy Differential Equations
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Let us consider the second-order fuzzy ordinary differential equations of the form
d2 x dt2
= f (t, x, dx dt )
x(t0 ) = x0 ,
x0 (t
(1) 0)
=
x00 .
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Equation (1) can be reduced to 2 first-order simultaneous fuzzy differential equations as follows:
dx dt
= y = f1 (t, x, y), t ∈ [t0 , T ]
dy dt
= f2 (t, x, y),
(2)
x(t0 ) = x0 , x0 (t0 ) = y(t0 ) = y0 ,
where x0 and y0 are fuzzy numbers. Assume that Equation (3) and (4) are the exact and approximate solutions of Equation (2) respectively.
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[x(t)]r = [x1 (t; r), x2 (t; r)] [y(t)]r = [y1 (t; r), y2 (t; r)]
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[ˆ x(t)]r = [ˆ x1 (t; r), x ˆ2 (t; r)] [ˆ y (t)]r = [ˆ y1 (t; r), yˆ2 (t; r)]
(3)
(4)
by using the fourth-order Runge-Kutta method for i = 0, 1, ..., N approximate solution is calculated as follows:
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x ˆ1 (ti+1 ; r) = x ˆ1 (ti ; r) + h
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x ˆ2 (ti+1 ; r) = x ˆ2 (ti ; r) + h
yˆ1 (ti+1 ; r) = yˆ1 (ti ; r) + h
yˆ2 (ti+1 ; r) = yˆ2 (ti ; r) + h
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wj kj,1 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1
4 X
wj kj,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1 4 X
(5) wj lj,1 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1 4 X
wj lj,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1
where the wj s are constants. Then, kj,1 , kj,2 , lj,1 and lj,2 for j = 1, 2, 3, 4 are defined as follows:
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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k1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r), v ∈ [ˆ y1 (ti ; r), yˆ2 (ti ; r)]} k1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r), v ∈ [ˆ y1 (ti ; r), yˆ2 (ti ; r)]} k2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti + h2 , u, v)| u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} k2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h2 , u, v)| u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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k3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti + h2 , u, v)| u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} k3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h2 , u, v)| u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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k4,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti + h, u, v)| u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))],
v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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NUMERICAL SOLUTION OF FUZZ
l4,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = m 52 52
u ∈ [p3,1 (ti , x ˆ(ti ; r),
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p3,2 (ti , x ˆ(ti ; r), yˆ(
k4,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h, u, v)| v ∈ [q (t , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 k4,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h, u, v)| 3,1 i u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), l4,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = ma u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], u ∈ [p3,1 (ti , x ˆ(ti ; r), p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} p3,2 (ti , x ˆ(ti ; r), yˆ( v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2
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l1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(tiwhere , u, v)| (6) (7) l1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2 (ti ; r)]} u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2 (ti ; r)]} ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti , u,p1,1 v)|(ti , x l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2 (ti ; r)]} ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + i, x u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2p(t1,2 r)]} i ; (t l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 , u, v)| l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 , u, v)| ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + u ∈ [p1,1 (ti ; x ˆ(ti ; r), yˆ(ti ; r)), p2,1 (ti , x u ∈ [p1,1 (ti ; x ˆ(ti ; r), yˆ(ti ; r)), p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + i, x ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r),pyˆ3,1 (ti(t; r))]} l2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + h2 , u, v)| (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + 3,2v)| l2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + h2 p, u, u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,1 (ti , x p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ,x ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) + ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r),qyˆ1,2 (t(t i ;ir))]} ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} l3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 , u, v)| (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + 2,1v)| l3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 ,qu, u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q (t , x ˆ(ti ; r)) = yˆ2 (ti ; r) + 2,2 i ˆ(ti ; r), y p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + i, x ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), qyˆ3,1 (ti(t ; r))]} ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), y ˆ (t ; r))]} i(t , x ˆ(ti ; r)) = yˆ2 (ti ; r) + 3,2v)| i ˆ(ti ; r), y l3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + h2 ,qu, h l3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + 2 , u, v)| u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ... ... NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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ll4,1 (t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = = min{g(t min{g(ti + + h, h, u, u, v)| v)| l4,1 ˆ(tii ; r), yˆ(tii ; r)) = min{g(tii + h, u, v)| 4,1 (ti , x u ˆˆ(t u∈ ∈ [p [p3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), u ∈ [p3,1 ˆ(tii ; r), yˆ(tii ; r)), 3,1 (ti , x p (t ˆˆ(t p3,2 (tii ,, x x (ti ;; r), r), yyˆ ˆ(t (ti ;; r))], r))], p3,2 ˆ(tii ; r), yˆ(tii ; r))], 3,2 (ti , x vv ∈ ˆˆ(t ˆˆ(t ∈ [q [q3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), qq3,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r))]} r))]} v ∈ [q3,1 ˆ(tii ; r), yˆ(tii ; r)), q3,2 ˆ(tii ; r), yˆ(tii ; r))]} 3,1 (ti , x 3,2 (ti , x ll4,2 (t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = = max{g(t max{g(ti + + h, h, u, u, v)| v)| l4,2 ˆ(tii ; r), yˆ(tii ; r)) = max{g(tii + h, u, v)| 4,2 (ti , x u ˆˆ(t u∈ ∈ [p [p3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), u ∈ [p3,1 ˆ(tii ; r), yˆ(tii ; r)), 3,1 (ti , x p (t ˆˆ(t p3,2 (tii ,, x x (ti ;; r), r), yyˆ ˆ(t (ti ;; r))], r))], p3,2 ˆ(tii ; r), yˆ(tii ; r))], 3,2 (ti , x NUMERICAL SOLUTION OF FUZZY DIFFERENT vv ∈ ˆˆ(t ˆˆ(t ∈ [q [q3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), qq3,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r))]} r))]} v ∈ [q3,1 ˆ(tii ; r), yˆ(tii ; r)), q3,2 ˆ(tii ; r), yˆ(tii ; r))]} 3,1 (ti , x 3,2 (ti , x where (6) (7) l4,1 (ti , x (8) (9) ˆ(ti(8) ; r), yˆ(ti ; r)) = min{g(t where (6) (7) (9) i + h, u, v) where (6) (7) (8) (9) u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), h h ˆˆ(t yyˆˆ(t ;; (t r)) p (t ˆˆ(t ˆˆ11 (t iˆ k1,1 (t (tii ,, x x (ti ;; r), r), (t r)) p1,1 (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = =x x (ti ;; r) r) + + hk ˆ(ti ; r))], 3,2 (t i , ix i ; r), y ˆ(tii ;pr), yˆ(t p1,1 ˆ(tii ; r), yˆ(tii ; r)) = x ˆ1 (tii ; r) + 22 k1,1 1,1 (ti , x i ; r)) 1,1 (ti , x 2 vh (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; h∈k[q3,1 (t ˆˆ(t yyˆˆ(t p ˆˆ(t ˆˆ22 (t 1,2 ii ,, x ii ;; r), ii ;; r)) k (t x (t r), (t r)) p1,2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (tii ;; r) r) + +h 1,2 1,2 (t ˆˆ(t p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + 22 kl1,2 (t i, x (tii;; r), r), yˆyˆ(t (tii;; r)) r)) = max{g(ti + h, u, v 2 4,2 (ti , x h h p (t ˆˆ(t ˆˆ11 (t ˆˆ(t ˆˆ(t p2,1 (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = =x x (ti ;; r) r) + + hk k2,1 (t (tii ,, x x (ti ;; r), r), (tii ;;ir)) r)) [pyyy3,1 ,x ˆ(ti ; r), yˆ(ti ; r)), p2,1 ˆ(tii ; r), yˆ(tii ; r)) = x ˆ1 (tii ; r) + 22 k2,1 ˆ(tuii ;∈r), ˆ(t(t 2,1 (ti , x 2,1 (ti , x i ; r)) 2 h h k (t , x p3,2 (t ˆ(ti ; r), yˆ(ti ; r))], i, x ˆˆ(t p ˆˆ(t ˆˆ22 (t ii ;; r)) k2,2 (tii , x (tii ;; r), r), yyˆˆ(t (t r)) p2,2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (tii ;; r) r) + +h 2,2 (t (t , x ˆ (t ; r), y ˆ (t p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + 22 k2,2 2,2 i i i ; r)) v 2∈ [q (t , x ˆ (t ; r), y ˆ (t ˆ(ti ; r), yˆ(ti ; 3,1 i i i ; r)), q3,2 (ti , x p ˆˆ(t ˆˆ11 (t ˆˆ(t p3,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (tii ;; r) r) + + hk hk3,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) 3,1 (t 3,1 (t p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + hk3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) where (8) p ˆˆ(t ˆˆ22 (t ˆˆ(t ii ;; r) i ;; r), p3,2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (t r) + + hk hk3,2 (tii ,, x x (t(6) r), yyˆˆ(t (ti ;; r)) r)) (7) 3,2 (t 3,2 (t p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + hk3,2 (ti , x ˆ(tii ; r), yˆ(tii ; r)) h h l (t , x qq1,1 ˆˆ(t ˆˆ(t l1,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = = yyˆˆ11 (t (tii ;; r) r) + +h (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) 1,1 (t 1,1 (tii , x h 22, x l q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ;pr) + (t , x ˆ (t ; r), y ˆ (t 1,1 i i ˆ(ti ; r (t ˆ (t ; r), y ˆ (t ; r)) =i ;x ˆr)) 1,1 i2 i i 1 (ti ; r) + k1,1 (ti , x 2 h h qq1,2 (t ˆˆ(t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ2 (t (ti ;; r)) r)) = = yyˆˆ2 (t (ti ;; r) r) + + h ll1,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) h q1,2 ˆ(tii ; r), yˆ22 (tii ; r)) = yˆ22 (tii ; r) + 22 l1,2 ˆ(tii ; r), yˆ(tii ; r)) 1,2 (ti , x 1,2 (ti , x ˆ(ti ; r p1,2 (ti ,2x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + k1,2 (ti , x h 2 h l (t , x qq2,1 (t , x ˆ (t ; r), y ˆ (t ; r)) = y ˆ (t ; r) + ˆ (t ; r), y ˆ (t ; r)) (tii , x ˆ(tii ; r), yˆ(tii ; r)) = yˆ11 (tii ; r) + h l2,1 (tii , x ˆ(tii ; r), yˆ(tii ; r)) h q2,1 ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + 22 l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2,1 (ti , x p2,1 (ti2, x ˆ2,1 (ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + k2,1 (ti , x ˆ(ti ; r h 2 h qq2,2 (t ˆˆ(t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = = yyˆˆ2 (t (ti ;; r) r) + + h ll2,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) h q2,2 ˆ(tii ; r), yˆ(tii ; r)) = yˆ22 (tii ; r) + 22 l2,2 (ti , x ˆ(tii ; r), yˆ(tii ; r)) 2,2 (ti , x ˆ(ti ; r p2,2 (ti2, x ˆ2,2 (ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + k2,2 (ti , x qq3,1 ˆˆ(t ˆˆ(t 2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = = yyˆˆ11 (t (tii ;; r) r) + + hl hl3,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) 3,1 (t 3,1 (t q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + hl3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) pr) (tihl ,x ˆ(t(t yˆ(t;ir), ; r)) ˆr)) ˆ(ti ; r) 3,1+ i ; r), 1 (ti ; r) + hk3,1 (ti , x (t , x ˆ (t ; r), y ˆ (t ; r)) = y ˆ (t ; , x ˆ(t (t yˆˆ(t (t=i ;;x i i i 2 i 3,2 i i ; r), y qq3,2 (t , x ˆ (t ; r), y ˆ (t ; r)) = y ˆ (t ; r) + hl (t , x r)) i i ˆ q3,2 ˆ(tii ; r), yˆ(tii ; r)) = yˆ22 (tii ; r) + hl3,2 ˆ(tii ; r), yˆ(tii ; r)) 3,2 (ti , x 3,2 (ti , x p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + hk3,2 (ti , x ˆ(ti ; r)
D I
S f
o e
v i h
c r
A
q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) +
h
q1,2 (ti , x ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) +
h l1,2 (ti , x ˆ(ti ; r 2 h
l1,1 (ti , x ˆ(ti ; r) www.SID.ir 2
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l4,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h, u, v)|
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N. PARANDIN
u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
(ti the ,x ˆ(tfourthˆ(torder i ; r), y i ; r))],RungeNow, using the initial conditions x0 , y0p3,2 and Kutta formula, we compute v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} h l (ti , x ˆ(t ; r), yˆ(ti ; r)) = max{g(ti + h, u, v)| xˆ1 (ti+1 ; r) = xˆ1 (ti ; r)+ (k1,1 (t4,2 ˆ(ti ; r),i yˆ(ti ; r))+2k ˆ(ti ; r), yˆ(ti ; r))+ i, x 2,1 (ti , x 6 u ∈ [p (t , x ˆ(t ; r), yˆ(t ; r)), 3,1
i
i
i
2k3,1 (ti , xˆ(ti ; r), yˆ(ti ; r)) + k4,1 (ti , xˆ(ti ; r), yˆ(ti ; r))) p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], h xˆ2 (ti+1 ; r) = xˆ2 (ti ; r)+ (k , xˆ(ti ;, r), (t3,2 ˆ(ti ,ix ;ˆr), (ti ;yˆr))+ v1,2 ∈ (t[qi3,1 x ˆ(tyˆi ;(tr), yˆ(ti ; r)), (t (ti ;yˆr), (ti ; r))]} i ; r))+2k 2,2 q i, x 6 2k3,2 (ti ,where xˆSOLUTION (ti ; r), yˆ(ti ; r)) k(6) xˆ(ti ; r), yˆ(t(7) (8) NUMERICAL OF+FUZZY 53 4,2 (ti ,DIFFERENTIAL i ; r))) ...
D I
S f
h yˆ1 (ti+1 ; r) = yˆ1 (ti ; r)+ (l1,1 (ti , xˆ(ti ; r), yˆ(ti ; r))+2l2,1 (ti , xˆ(thi ; r), yˆ(ti ; r))+ l4,1 (ti , x ˆ(t6i ; r), yˆ(ti ; r)) = min{g(ti + h, u, v)| ˆ(ti ; r), yˆ(ti ; r)) p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + k1,1 (ti , x 2 2l3,1 (ti , xˆ(tui ;∈r), y ˆ (t ; r)) + l (t , x ˆ (t ; r), y ˆ (t ; r))) i 4,1 i i i [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), h h ˆ(ti ; r), yˆ(ti ; r)) p1,2 (ti(t ,x ˆi ,(t,x ;(t r), yˆ(tyˆyiˆ(t ;(tr)) =x ˆ2 (ti ;(t r) + 1,2 (t i, x iˆ p(l (t 3,2 i ;i ;r), ii;;r))], yˆ2 (ti+1 ; r) = yˆ2 (ti ; r)+ r), r))+2l ˆ(t2i ;kr), yˆ(t 1,2 (t i ˆx 2,2 i , x i ; r))+ 6 h v ∈2l[q3,1(t(t,ixˆ, (t x ˆ(t; ir), ; r),ˆ(t yˆ(t; r)) (t,i ,xˆx ˆ(t(t;i ;r), r), yˆ(t;ir))) ; r))]} i ; r)), 3,2(t +yˆlq(t p2,1 ,x ˆ(t ˆ1 (tiyˆ;(t r)i + k2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 3,2 i i (tiy i i ; r), 4,2 i = ix i ; r)) 2 l (t , x ˆ (t ; r), y ˆ (t ; r)) = max{g(t + h, u, v)| 4,2 i i following i Then, we illustrated the some lemmasi which express h the conk2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) p (t , x ˆ (t ; r), yˆ(ti ;the r)) = x ˆ (ti ; r) + as 2,2 i i vergence of the approximate u ∈ [p3,1solution (ti , x ˆ(ti ;to r), yˆ(tiexact ; r)),2 solution 2 follows. p3,1 (ti(t ,x ˆ,(tx ˆ(tˆi(t ; r)) =x ˆ1 (ti ; r) + hk3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) i ; r), y p3,2 i ˆ(ti ; r), y i ; r))], lim xˆ1 (t; r) = x1 (t; r) ˆˆ(t (ti ;; r)), r), yˆq(ti ;(t r)) ˆ=(tx ˆ;2r), (ti ;yˆr) hk3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) h→0 3,2;(t i, x v ∈ [q3,1 (ti , x ˆp(t (ti+ ; r))]} i r), y i 3,2 i , x i h lim xˆ2 (t; r) = x2 (t; r) q(6) ˆ(ti ; r), yˆ(t ˆ1 (ti ; r)(8) + l1,1 (ti , x ˆ(ti ;(9) r), yˆ(ti ; r)) h→0 1,1 (t i, x i ; r)) = y where (7) 2 lim yˆ1 (t; r) = y1 (t; r) h q1,2 (th→0 ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) + l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) i, x h 2 lim=yˆ2x r) = y (t; r) k (t , x ˆ (t ; r), y ˆ (t ; r)) p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) ˆ(t; (t ; r) + 2 1,1 i i i 1 i h→0 2 h q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 h k1,2 (t ˆ(tsatisfy yˆ(ti ; r)) p1,2 (t3.1. ˆ(tLet yˆ(tsequence ˆof }Ni , x Lemma i ; r), h i, x i ; r),the i ; r)) = x 2 (tnumbers i ; r) + {w 2 =n yˆn=0 q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) (t ; r) + l2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 i 2 h |wn+1 0 kn2,1 (tNi , x − n p2,1 (ti , x ˆ(ti ; r), yˆ(tqi|; r))A|w =ˆ(t x ˆ|1;+ (tiB ; r) ˆ(t1i ; r), yˆ(ti ; r)) ˆ(t ; r), yˆ(t ; r)) ˆ(t+i ; r)) 3,1 (ti , x i r), y i i 2 = yˆ1 (ti ; r) + hl3,1 (ti , x h (proof for some given positive constants A and [12]). Then (ti ,=x ˆ(t yˆ(ti ;Br)) =(tyˆ2,(t ; r); r), + hl (ti , x ˆ(ti ; r), yˆ(ti ; r)) k2,2 ˆi(t yˆ(t3,2 p2,2 (ti , x ˆ(ti ; r), yˆ(tqi3,2 ; r)) x ˆi2;(tr), i x i i ; r)) i ; r) + 2 An − 1 n =x p3,1 (ti , x ˆ(ti ;|w r),n |yˆ (tiA ; r)) ˆ1B (ti ; r) + hk3,1 (tin, x ˆ(tN ˆ(ti ; r)) i ; r), y |w0 | + 0 A−1 p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + hk3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))
o e
v i h
c r
A
h l1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 www.SID.ir h q1,2 (ti , x ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) + l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 h ˆ(ti ; r), yˆ(ti ; r)) q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + l2,1 (ti , x 2 q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) +
(9
NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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N Lemma 3.2. Let the sequence of numbers {wn }N n=0 , {vn }n=0 (proof [12]) satisfy
|wn+1 | 6 |wn | + A. max{|wn |, |vn |} + B, |vn+1 | 6 |vn | + A. max{|wn |, |vn |} + B, for some given positive constants A and B, and denote |un | = |wn | + |vn |,
06n6N
D I
Then ¯n ¯A −1 |un | 6 A¯n |u0 | + B A¯ − 1 ¯ = 2B. where A¯ = 1 + 2A and B
0 6 n 6 N,
S f
Proof. [12] Generally, if we replace the values of ti s with t and after that replace x ˆ1 (t; r), x ˆ2 (t; r), yˆ1 (t; r) and yˆ2 (t; r) with u ˆ, vˆ, w, ˆ zˆ, then F1 [t, u ˆ, vˆ, w, ˆ zˆ], G1 [t, u ˆ, vˆ, w, ˆ zˆ], F2 [t, u ˆ, vˆ, w, ˆ zˆ], G2 [t, u ˆ, vˆ, w, ˆ zˆ] are defined as follows:
o e
F1 [t, u ˆ, vˆ, w, ˆ zˆ] = k1,1 [t, u ˆ, vˆ, w, ˆ zˆ]
v i h
+2k2,1 [t, u ˆ, vˆ, w, ˆ zˆ] + 2k3,1 [t, u ˆ, vˆ, w, ˆ zˆ] + k4,1 [t, u ˆ, vˆ, w, ˆ zˆ] G1 [t, u ˆ, vˆ, w, ˆ zˆ] = k1,2 [t, u ˆ, vˆ, w, ˆ zˆ] +2k2,2 [t, u ˆ, vˆ, w, ˆ zˆ] + 2k3,2 [t, u ˆ, vˆ, w, ˆ zˆ] + k4,2 [t, u ˆ, vˆ, w, ˆ zˆ]
c r
F2 [t, u ˆ, vˆ, w, ˆ zˆ] = l1,1 [t, u ˆ, vˆ, w, ˆ zˆ]
+2l2,1 [t, u ˆ, vˆ, w, ˆ zˆ] + 2l3,1 [t, u ˆ, vˆ, w, ˆ zˆ] + l4,1 [t, u ˆ, vˆ, w, ˆ zˆ]
A
G2 [t, u ˆ, vˆ, w, ˆ zˆ] = l1,2 [t, u ˆ, vˆ, w, ˆ zˆ]
+2l2,2 [t, u ˆ, vˆ, w, ˆ zˆ] + 2l3,2 [t, u ˆ, vˆ, w, ˆ zˆ] + l4,2 [t, u ˆ, vˆ, w, ˆ zˆ]
The domain where F1 , G1 , F2 , G2 are defined is therefore K = {[t, u ˆ, vˆ, w, ˆ zˆ] | 0 6 t 6 T, − ∞ < u ˆ < ∞, − ∞ < vˆ < ∞, −∞ < w ˆ < ∞, − ∞ < zˆ < ∞}.
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Theorem 3.3. Let F1 [t, u ˆ, vˆ, w, ˆ zˆ], G1 [t, u ˆ, vˆ, w, ˆ zˆ], F2 [t, u ˆ, vˆ, w, ˆ zˆ] and G2 [t, u ˆ, vˆ, w, ˆ zˆ] belongs to C 4 (K) and let the partial derivatives of F1 , G1 , F2 , G2 be bounded over K. Then for arbitrary fixed r : 0 6 r 6 1, the approximate solutions converge uniformly in t to the exact solutions. this theorem is simply proved (see proof theorem 4.1 in [1]).
4.
Examples
D I
Example 4.1. Consider the following fuzzy differential equation with fuzzy initial value problem: 00 x − 2x0 = 0 , t ∈ [0, 0.5]
S f
x(0) = (r, 2 − r), x0 (0) = (3 + r, 4)
where the exact solution is as follow:
o e
1 3 1 x(t) = r − + (3 + r)e2t 2 2 2 x ¯(t) = −r + 2e2t
v i h
In this example, the exact and the approximate solution of the equation and the first differential for t=0.1 have been shown in Tables 1 and 2 respectively. Also, Tables 3, 4 have presented the exact and the approximate solutions of equation and the first differential for t=0.2.
c r
A
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Table 1: Comparison between the exact solution and the approximate solution for t = 0.1 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
xexa. (0.1) 0.3321 0.4432 0.5542 0.6653 0.7764 0.8875 0.9985 1.1096 1.2207 1.3317 1.4428
xapp. (0.1) 0.3315 0.4426 0.5536 0.6647 0.7757 0.8868 0.9978 1.1088 1.2199 1.3310 1.4420
x ¯exa. (0.1) 2.4428 2.3428 2.2428 2.1428 2.0428 1.9428 1.8428 1.7428 1.6428 1.5428 1.4428
x ¯app. (0.1) 2.4420 2.3420 2.2420 2.1420 2.0420 1.9420 1.8420 1.7420 1.6420 1.5420 1.4420
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S f
o e
Table 2: Comparison between the exact solution and the approximate solution of the first differential for t = 0.1 r x0 (0.1) x0 (0.1) x¯0 exa. (0.1) x¯0 app. (0.1)
v i h
exa.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
3.6642 3.7863 3.9085 4.0306 4.1528 4.2749 4.3970 4.5192 4.6413 4.7635 4.8856
A
c r
app.
3.6416 3.7630 3.8844 4.0057 4.1272 4.2485 4.3700 4.4913 4.6128 4.7341 4.8556
4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856
4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554
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Table 3: Comparison between the exact solution and the approximate solution for t = 0.2 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
xexa. (0.2) 0.7377 0.8623 0.9869 1.1115 1.2361 1.3607 1.4853 1.6099 1.7345 1.8591 1.9836
xapp. (0.2) 0.7339 0.8584 0.9829 1.1073 1.2318 1.3563 1.4807 1.6052 1.7296 1.8541 1.9786
x ¯exa. (0.2) 2.9836 2.8836 2.7836 2.6836 2.5836 2.4836 2.3836 2.2836 2.1836 2.0836 1.9836
x ¯app. (0.2) 2.9786 2.8786 2.7786 2.6786 2.5786 2.4786 2.3786 2.2786 2.1786 2.0786 1.9786
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Table 4: Comparison between the exact solution and the approximate solution of the first differential for t = 0.2 r x0 (0.2) x0 (0.2) x¯0 exa. (0.2) x¯0 app. (0.2)
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exa.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.4755 4.6247 4.7738 4.9230 5.0722 5.2214 5.3706 5.5198 5.6689 5.8181 5.9673
A
c r
app.
4.4327 4.5805 4.7283 4.8760 5.0238 5.1716 5.3194 5.4671 5.6149 5.7627 5.9105
5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673
5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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Example 4.2. Consider the following fuzzy differential equation with fuzzy initial value problem: 00 x − 4x0 + 4x = 0 , t ∈ [0, 1] x(0) = (2 + r, 4 − r), x0 (0) = (5 + r, 7 − r)
where the exact solution is as follow: x(t) = (2 + r)e2t + (1 − r)te2t
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x ¯(t) = (4 − r)e2t + (r − 1)te2t
In the following table, the error of the proposed method(PM) is compared with the method introduced by [5]. The results shows that the proposed method is better than the method presented by [5].
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Table 5: The Comparison between the error of the proposed method(PM) and method presented by [5] in t = 0.01 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
5.
error[5] 0.461836e-3 0.476055e-3 0.490273e-3 0.504491e-3 0.518710e-3 0.532928e-3 0.547146e-3 0.561365e-3 0.575584e-3 0.589801e-3 0.604020e-4
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errorP M 4.0369e-4 3.6342e-4 3.2316e-4 2.8289e-4 2.4262e-4 2.0235e-4 1.6209e-4 1.2182e-4 8.1550e-5 4.1283e-5 1.0151e-6
error[5] 0.746204e-3 0.731985e-3 0.717767e-3 0.703549e-3 0.689330e-3 0.675112e-3 0.660894e-3 0.646675e-3 0.632456e-3 0.618239e-3 0.604020e-3
errorP M 4.0166e-4 3.6139e-4 3.2113e-4 2.8086e-4 2.4059e-4 2.0032e-4 1.6006e-4 1.1979e-4 7.9520e-5 3.9253e-5 1.0151e-6
Conclusion
So far, many methods have been proposed for solving the first-order fuzzy differential equations in comparison with the other order of fuzzy differential equations. In this paper the forth-order Runge-Kutta method
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numerically extended to solve the second-order fuzzy ordinary differential equations. In this method, the second-order of fuzzy differential convert to two first-order fuzzy differential equations. Then, by RungeKutta method, the solution of equations is calculated. Also, the proposed method have good and acceptable precise. Acknowledgements: This article has resulted from the research project supported by Islamic Azad University of Kermanshah Branch in Iran.
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References
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[1] S. Abbasbandy and T. Allahviranloo, Numerical solution of fuzzy differential equation by Runge-Kutta method, J. Sci. Teacher Training University, 1(3), 2002. [2] S. Abbasbandy and T. Allahviranloo, Numerical solutions of fuzzy differential equations by Taylor method, Journal of Computational Methods in Applied Mathematics, 2 (2002), 113-124.
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[3] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso, and J. J. Nieto, Numerical methods for fuzzy differential inclusions, Journal of Computer and Mathematics with Applications, 48 (2004), 1633-1641.
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[4] T. Allahviranloo, N. Ahmady, and E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method, Information Sciences, 177 (2007), 1633-1647.
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[5] T. Allahviranloo, E. Ahmady, and N. Ahmady, Nth-order fuzzy linear differential equations, Information Sciences, 178 (2008), 1309-1324. [6] T. Allahviranloo, N. A. Kiani, and N. Motamedi, Solving fuzzy differential equations by differential transformation method, Information Sciences, 179 (2009), 956-966.
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[7] B. Bede, A note on two-point boundary value problems associated with non-linear fuzzy differential equations, Fuzzy Sets and Systems, 157 (2006), 986-989. [8] B. Bede and S. G. Gal, Generalizations of the differntiability of fuzzynumber-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581-599.
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[9] J. J. Buckley and T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 43-54. [10] P. Diamond, Stability and periodicity in fuzzy differential equations, IEEE Trans, Fuzzy Systems, 8 (2000), 583-590. [11] P. Diamond, Brief note on the variation of constants formula for fuzzy differential equations, Fuzzy Sets and Systems, 129 (2002), 65-71. [12] Ming Ma, M. Friedman, and A. Kandel, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105 (1999), 133-138.
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[13] M. Friedman, M. Ming, and A. Kandel, Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets and Systems, 106 (1999), 35-48.
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[14] B. Ghazanfari and A. Shakerami, Numerical solution of fuzzy differential equations by extended Runge-Kutta-like formulae of order 4, Fuzzy Sets and Systems, 189 (2011), 74-91.
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[15] D. N. Georgiou, J. J. Nieto, and R. Rodriguez-Lopez, Initial value problems for higher-order fuzzy differential equations, Nonlinear Anal., 63 (2005), 587-600. [16] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
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[17] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35 (1990), 389-396.
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[18] A. Khastan and K. Ivaz, Numerical solution of fuzzy differential equations by Nystrom method. Chaos, Solitons and Fractals, 41 (2009), 859-868. [19] H. Kim and R. Sakthivel, Numerical solution of hybrid fuzzy differential equations using improved predictor-corrector method, Commun Nonlinear Sci Numer Simulat, 17 (2012), 3788-3794.
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[20] M. Mosleh and M. Otadi, Simulation and evalution of fuzzy differential equations by fuzzy neural network, Applied Soft Computing, 12 (2012), 2817-2827. [21] J. J. Nieto and R. Rodriguez-Lopez, Bounded solutions for fuzzy differential and integral equations, Chaos, Solitons and Fractals, 27 (2006), 1376-1386.
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[22] J. J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems, 102 (1999), 259-262. [23] H. Ouyang and Y. Wu, On fuzzy differential equations, Fuzzy Sets and Systems, 32 (1989), 321-325. [24] S. Ch. Palligkinis, G. Papageorgiou, and I. Th. Famelis, Runge-Kutta methods for fuzzy differential equations, Applied Mathematics and Computation, 209 (2009), 97-105. [25] S. Pederson and M. Sambandham, The Runge-Kutta method for hybrid fuzzy differential equations, Nonlinear Analysis: Hybrid Systems, 2 (2008), 626-634.
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[26] H. Roman-Flores and M. Rojas-Medar, Embedding of level-continuous fuzzy sets on Banach spaces, Information Sciences, 144 (2002), 227-247.
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[27] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.
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Nouredin Parandin Department of Mathematics Assistant Professor of Mathematics Islamic Azad University, Kermanshah Branch Kermanshah, Iran E-mail: n [email protected]
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Numerical Solution of Fuzzy Differential Equations of 2nd-Order by Runge-Kutta Method N. Parandin
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Islamic Azad University, Kermanshah Branch
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Abstract. In this paper, solving fuzzy ordinary differential equations
of the n th order by Runge-Kutta method have been done, and the convergence of the proposed method is proved. This method is illustrated by some numerical examples.
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AMS Subject Classification: 65L06 Keywords and Phrases: Fuzzy differential equations; Runge-Kutta method; Convergence of numerical method
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Introduction
The topics of fuzzy differential equations have been rapidly growing in recent years. The theory of fuzzy differential equations was treated by Buckley and Feuring [9], Kaleva [16, 17], Nieto [22], Ouyang and Wu [23], Roman-Flores and Rojas- Medar [26], Seikkala [27], also recently there appeared the papers of Bede [7], Bede and Gal [8], Diamond [10, 11], Georgiou et al., [15] Nieto and Rodriguez-Lopez [21]. In the following, we have mentioned some numerical solution which have proposed by other scientists. Abbasbandy and Allahviranloo have solved fuzzy differential equations by Runge-Kutta and Taylor methods[1, 2]. Also, Allahviranloo et al. solved differential equations by predictor-corrector and transformation methods[4, 5, 6]. Ghazanfari and Shakerami developed Runge-Kutta like formulae of order 4 for solving fuzzy differential
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Received: December 2012; Accepted: July 2013
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N. PARANDIN
equations[14]. Nystrom method has been introduced for solving fuzzy differential equations[18]. Mosleh and Otadi (2012) simulated and evaluate fuzzy differential equations by fuzzy neural network[20]. Pederson and Sambandham (2008) applied Runge-Kutta method for solving hybrid fuzzy differential equations [25]. Runge-Kutta method has been used for solving fuzzy differential equations by Palligkinis et al. (2009)[24]. Also, Kim and Sakthivel could solve hybrid fuzzy differential equations using improved predictor-corrector method[19]. The paper is organized as follows. Section 2 includes preliminaries. In Section 3, we can see the main idea of this paper. In Section 4, the proposed method is illustrated by examples. The conclusion is in Section 5.
2.
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Preliminaries
Definition 2.1. [12] A fuzzy number is a map u : R −→ I = [0, 1] which satisfies
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(i) u is upper semi continuous,
(ii) u(x) = 0 outside some interval [c, d] ⊂ R,
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(iii) There exist real numbers a, b such that c 6 a 6 b 6 d where, 1. u(x) is monotonic increasing on [c, a],
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2. u(x) is monotonic decreasing on [b, d], 3. u(x) = 1, a 6 x 6 b.
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The set of all such fuzzy numbers is represented by E 1 . Definition 2.2. [12] An arbitrary fuzzy number in parametric form is represented by an ordered pair functions (u(r), u ¯(r)), 0 6 r 6 1, which satisfy the following requirements: 1. u(r) is a bounded left-continuous non-decreasing function over [0, 1], 2. u ¯(r) is a bounded left -continuous non -increasing function over [0, 1],
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
¯(r), 3. u(r) 6 u
0 6 r 6 1.
A crisp number α is simply represented by u(r) = u(r) = α, 0 6 r 6 1. For arbitrary u = (u(r), u ¯(r)), v = (v(r), v¯(r)) and k ∈ R, we define equality, addition and multiplication by k as a. u = v if and only if u(r) = v(r) and u(r) = v(r), b. u + v = (u(r) + v(r), u(r) + v(r)), k > 0, (ku, ku), c. ku = (ku, ku), k < 0.
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Definition 2.3. [8] The Hausdorff distance of two fuzzy numbers given by D : R × R −→ R+ ∪ {0}, is defined as follows:
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r r |} = sup {dH ([u]r , [v]r )}, D(u, v) = sup max{|ur− − v− |, |ur+ − v+ r∈[0,1]
r∈[0,1]
where
[u]r
=
[ur− , ur+ ], [v]r
=
r , v r ]. [v− +
We denote k.k = D(., 0).
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Definition 2.4. [27] Let I be a real interval. A mapping x : I → E is called a fuzzy process and its r-level set is denoted by [x(t)]r = [x1 (t; r), x2 (t; r)], t ∈ I, r ∈ (0, 1]
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the derivative x0 (t) of a fuzzy process x is defined by [x0 (t)]r = [x01 (t; r), x02 (t; r)], t ∈ I, r ∈ (0, 1],
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Runge-kutta Mehtods for Solving Fuzzy Differential Equations
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Let us consider the second-order fuzzy ordinary differential equations of the form
d2 x dt2
= f (t, x, dx dt )
x(t0 ) = x0 ,
x0 (t
(1) 0)
=
x00 .
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Equation (1) can be reduced to 2 first-order simultaneous fuzzy differential equations as follows:
dx dt
= y = f1 (t, x, y), t ∈ [t0 , T ]
dy dt
= f2 (t, x, y),
(2)
x(t0 ) = x0 , x0 (t0 ) = y(t0 ) = y0 ,
where x0 and y0 are fuzzy numbers. Assume that Equation (3) and (4) are the exact and approximate solutions of Equation (2) respectively.
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[x(t)]r = [x1 (t; r), x2 (t; r)] [y(t)]r = [y1 (t; r), y2 (t; r)]
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[ˆ x(t)]r = [ˆ x1 (t; r), x ˆ2 (t; r)] [ˆ y (t)]r = [ˆ y1 (t; r), yˆ2 (t; r)]
(3)
(4)
by using the fourth-order Runge-Kutta method for i = 0, 1, ..., N approximate solution is calculated as follows:
iv
h c
x ˆ1 (ti+1 ; r) = x ˆ1 (ti ; r) + h
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x ˆ2 (ti+1 ; r) = x ˆ2 (ti ; r) + h
yˆ1 (ti+1 ; r) = yˆ1 (ti ; r) + h
yˆ2 (ti+1 ; r) = yˆ2 (ti ; r) + h
4 X
wj kj,1 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1
4 X
wj kj,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1 4 X
(5) wj lj,1 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1 4 X
wj lj,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))
j=1
where the wj s are constants. Then, kj,1 , kj,2 , lj,1 and lj,2 for j = 1, 2, 3, 4 are defined as follows:
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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k1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r), v ∈ [ˆ y1 (ti ; r), yˆ2 (ti ; r)]} k1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r), v ∈ [ˆ y1 (ti ; r), yˆ2 (ti ; r)]} k2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti + h2 , u, v)| u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} k2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h2 , u, v)| u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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k3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti + h2 , u, v)| u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} k3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h2 , u, v)| u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
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p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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k4,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{f (ti + h, u, v)| u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))],
v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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NUMERICAL SOLUTION OF FUZZ
l4,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = m 52 52
u ∈ [p3,1 (ti , x ˆ(ti ; r),
N. PARANDIN N. PARANDIN
p3,2 (ti , x ˆ(ti ; r), yˆ(
k4,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h, u, v)| v ∈ [q (t , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 k4,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{f (ti + h, u, v)| 3,1 i u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), l4,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = ma u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], u ∈ [p3,1 (ti , x ˆ(ti ; r), p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} p3,2 (ti , x ˆ(ti ; r), yˆ( v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2
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l1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(tiwhere , u, v)| (6) (7) l1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2 (ti ; r)]} u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2 (ti ; r)]} ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti , u,p1,1 v)|(ti , x l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti , u, v)| u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2 (ti ; r)]} ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + i, x u ∈ [ˆ x1 (ti ; r), x ˆ2 (ti ; r)], v ∈ [yˆ1 (ti ; r), yˆ2p(t1,2 r)]} i ; (t l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 , u, v)| l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 , u, v)| ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + u ∈ [p1,1 (ti ; x ˆ(ti ; r), yˆ(ti ; r)), p2,1 (ti , x u ∈ [p1,1 (ti ; x ˆ(ti ; r), yˆ(ti ; r)), p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + i, x ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r),pyˆ3,1 (ti(t; r))]} l2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + h2 , u, v)| (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + 3,2v)| l2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + h2 p, u, u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + u ∈ [p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,1 (ti , x p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ,x ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) + ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r),qyˆ1,2 (t(t i ;ir))]} ∈ [q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} l3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 , u, v)| (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + 2,1v)| l3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h2 ,qu, u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q (t , x ˆ(ti ; r)) = yˆ2 (ti ; r) + 2,2 i ˆ(ti ; r), y p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + i, x ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), qyˆ3,1 (ti(t ; r))]} ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), y ˆ (t ; r))]} i(t , x ˆ(ti ; r)) = yˆ2 (ti ; r) + 3,2v)| i ˆ(ti ; r), y l3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + h2 ,qu, h l3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = max{g(ti + 2 , u, v)| u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), u ∈ [p2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} ∈ [q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]}
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v v
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A v v
v v
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ... ... NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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ll4,1 (t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = = min{g(t min{g(ti + + h, h, u, u, v)| v)| l4,1 ˆ(tii ; r), yˆ(tii ; r)) = min{g(tii + h, u, v)| 4,1 (ti , x u ˆˆ(t u∈ ∈ [p [p3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), u ∈ [p3,1 ˆ(tii ; r), yˆ(tii ; r)), 3,1 (ti , x p (t ˆˆ(t p3,2 (tii ,, x x (ti ;; r), r), yyˆ ˆ(t (ti ;; r))], r))], p3,2 ˆ(tii ; r), yˆ(tii ; r))], 3,2 (ti , x vv ∈ ˆˆ(t ˆˆ(t ∈ [q [q3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), qq3,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r))]} r))]} v ∈ [q3,1 ˆ(tii ; r), yˆ(tii ; r)), q3,2 ˆ(tii ; r), yˆ(tii ; r))]} 3,1 (ti , x 3,2 (ti , x ll4,2 (t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = = max{g(t max{g(ti + + h, h, u, u, v)| v)| l4,2 ˆ(tii ; r), yˆ(tii ; r)) = max{g(tii + h, u, v)| 4,2 (ti , x u ˆˆ(t u∈ ∈ [p [p3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), u ∈ [p3,1 ˆ(tii ; r), yˆ(tii ; r)), 3,1 (ti , x p (t ˆˆ(t p3,2 (tii ,, x x (ti ;; r), r), yyˆ ˆ(t (ti ;; r))], r))], p3,2 ˆ(tii ; r), yˆ(tii ; r))], 3,2 (ti , x NUMERICAL SOLUTION OF FUZZY DIFFERENT vv ∈ ˆˆ(t ˆˆ(t ∈ [q [q3,1 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)), r)), qq3,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r))]} r))]} v ∈ [q3,1 ˆ(tii ; r), yˆ(tii ; r)), q3,2 ˆ(tii ; r), yˆ(tii ; r))]} 3,1 (ti , x 3,2 (ti , x where (6) (7) l4,1 (ti , x (8) (9) ˆ(ti(8) ; r), yˆ(ti ; r)) = min{g(t where (6) (7) (9) i + h, u, v) where (6) (7) (8) (9) u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), h h ˆˆ(t yyˆˆ(t ;; (t r)) p (t ˆˆ(t ˆˆ11 (t iˆ k1,1 (t (tii ,, x x (ti ;; r), r), (t r)) p1,1 (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = =x x (ti ;; r) r) + + hk ˆ(ti ; r))], 3,2 (t i , ix i ; r), y ˆ(tii ;pr), yˆ(t p1,1 ˆ(tii ; r), yˆ(tii ; r)) = x ˆ1 (tii ; r) + 22 k1,1 1,1 (ti , x i ; r)) 1,1 (ti , x 2 vh (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; h∈k[q3,1 (t ˆˆ(t yyˆˆ(t p ˆˆ(t ˆˆ22 (t 1,2 ii ,, x ii ;; r), ii ;; r)) k (t x (t r), (t r)) p1,2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (tii ;; r) r) + +h 1,2 1,2 (t ˆˆ(t p1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + 22 kl1,2 (t i, x (tii;; r), r), yˆyˆ(t (tii;; r)) r)) = max{g(ti + h, u, v 2 4,2 (ti , x h h p (t ˆˆ(t ˆˆ11 (t ˆˆ(t ˆˆ(t p2,1 (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = =x x (ti ;; r) r) + + hk k2,1 (t (tii ,, x x (ti ;; r), r), (tii ;;ir)) r)) [pyyy3,1 ,x ˆ(ti ; r), yˆ(ti ; r)), p2,1 ˆ(tii ; r), yˆ(tii ; r)) = x ˆ1 (tii ; r) + 22 k2,1 ˆ(tuii ;∈r), ˆ(t(t 2,1 (ti , x 2,1 (ti , x i ; r)) 2 h h k (t , x p3,2 (t ˆ(ti ; r), yˆ(ti ; r))], i, x ˆˆ(t p ˆˆ(t ˆˆ22 (t ii ;; r)) k2,2 (tii , x (tii ;; r), r), yyˆˆ(t (t r)) p2,2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (tii ;; r) r) + +h 2,2 (t (t , x ˆ (t ; r), y ˆ (t p2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + 22 k2,2 2,2 i i i ; r)) v 2∈ [q (t , x ˆ (t ; r), y ˆ (t ˆ(ti ; r), yˆ(ti ; 3,1 i i i ; r)), q3,2 (ti , x p ˆˆ(t ˆˆ11 (t ˆˆ(t p3,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (tii ;; r) r) + + hk hk3,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) 3,1 (t 3,1 (t p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + hk3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) where (8) p ˆˆ(t ˆˆ22 (t ˆˆ(t ii ;; r) i ;; r), p3,2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = =x x (t r) + + hk hk3,2 (tii ,, x x (t(6) r), yyˆˆ(t (ti ;; r)) r)) (7) 3,2 (t 3,2 (t p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + hk3,2 (ti , x ˆ(tii ; r), yˆ(tii ; r)) h h l (t , x qq1,1 ˆˆ(t ˆˆ(t l1,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = = yyˆˆ11 (t (tii ;; r) r) + +h (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) 1,1 (t 1,1 (tii , x h 22, x l q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ;pr) + (t , x ˆ (t ; r), y ˆ (t 1,1 i i ˆ(ti ; r (t ˆ (t ; r), y ˆ (t ; r)) =i ;x ˆr)) 1,1 i2 i i 1 (ti ; r) + k1,1 (ti , x 2 h h qq1,2 (t ˆˆ(t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ2 (t (ti ;; r)) r)) = = yyˆˆ2 (t (ti ;; r) r) + + h ll1,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) h q1,2 ˆ(tii ; r), yˆ22 (tii ; r)) = yˆ22 (tii ; r) + 22 l1,2 ˆ(tii ; r), yˆ(tii ; r)) 1,2 (ti , x 1,2 (ti , x ˆ(ti ; r p1,2 (ti ,2x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + k1,2 (ti , x h 2 h l (t , x qq2,1 (t , x ˆ (t ; r), y ˆ (t ; r)) = y ˆ (t ; r) + ˆ (t ; r), y ˆ (t ; r)) (tii , x ˆ(tii ; r), yˆ(tii ; r)) = yˆ11 (tii ; r) + h l2,1 (tii , x ˆ(tii ; r), yˆ(tii ; r)) h q2,1 ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + 22 l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2,1 (ti , x p2,1 (ti2, x ˆ2,1 (ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + k2,1 (ti , x ˆ(ti ; r h 2 h qq2,2 (t ˆˆ(t ˆˆ(t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) = = yyˆˆ2 (t (ti ;; r) r) + + h ll2,2 (t (tii ,, x x (ti ;; r), r), yyˆˆ(t (ti ;; r)) r)) h q2,2 ˆ(tii ; r), yˆ(tii ; r)) = yˆ22 (tii ; r) + 22 l2,2 (ti , x ˆ(tii ; r), yˆ(tii ; r)) 2,2 (ti , x ˆ(ti ; r p2,2 (ti2, x ˆ2,2 (ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + k2,2 (ti , x qq3,1 ˆˆ(t ˆˆ(t 2 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) = = yyˆˆ11 (t (tii ;; r) r) + + hl hl3,1 (tii ,, x x (tii ;; r), r), yyˆˆ(t (tii ;; r)) r)) 3,1 (t 3,1 (t q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + hl3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) pr) (tihl ,x ˆ(t(t yˆ(t;ir), ; r)) ˆr)) ˆ(ti ; r) 3,1+ i ; r), 1 (ti ; r) + hk3,1 (ti , x (t , x ˆ (t ; r), y ˆ (t ; r)) = y ˆ (t ; , x ˆ(t (t yˆˆ(t (t=i ;;x i i i 2 i 3,2 i i ; r), y qq3,2 (t , x ˆ (t ; r), y ˆ (t ; r)) = y ˆ (t ; r) + hl (t , x r)) i i ˆ q3,2 ˆ(tii ; r), yˆ(tii ; r)) = yˆ22 (tii ; r) + hl3,2 ˆ(tii ; r), yˆ(tii ; r)) 3,2 (ti , x 3,2 (ti , x p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + hk3,2 (ti , x ˆ(ti ; r)
D I
S f
o e
v i h
c r
A
q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) +
h
q1,2 (ti , x ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) +
h l1,2 (ti , x ˆ(ti ; r 2 h
l1,1 (ti , x ˆ(ti ; r) www.SID.ir 2
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l4,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = min{g(ti + h, u, v)|
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N. PARANDIN
u ∈ [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)),
(ti the ,x ˆ(tfourthˆ(torder i ; r), y i ; r))],RungeNow, using the initial conditions x0 , y0p3,2 and Kutta formula, we compute v ∈ [q3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), q3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))]} h l (ti , x ˆ(t ; r), yˆ(ti ; r)) = max{g(ti + h, u, v)| xˆ1 (ti+1 ; r) = xˆ1 (ti ; r)+ (k1,1 (t4,2 ˆ(ti ; r),i yˆ(ti ; r))+2k ˆ(ti ; r), yˆ(ti ; r))+ i, x 2,1 (ti , x 6 u ∈ [p (t , x ˆ(t ; r), yˆ(t ; r)), 3,1
i
i
i
2k3,1 (ti , xˆ(ti ; r), yˆ(ti ; r)) + k4,1 (ti , xˆ(ti ; r), yˆ(ti ; r))) p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))], h xˆ2 (ti+1 ; r) = xˆ2 (ti ; r)+ (k , xˆ(ti ;, r), (t3,2 ˆ(ti ,ix ;ˆr), (ti ;yˆr))+ v1,2 ∈ (t[qi3,1 x ˆ(tyˆi ;(tr), yˆ(ti ; r)), (t (ti ;yˆr), (ti ; r))]} i ; r))+2k 2,2 q i, x 6 2k3,2 (ti ,where xˆSOLUTION (ti ; r), yˆ(ti ; r)) k(6) xˆ(ti ; r), yˆ(t(7) (8) NUMERICAL OF+FUZZY 53 4,2 (ti ,DIFFERENTIAL i ; r))) ...
D I
S f
h yˆ1 (ti+1 ; r) = yˆ1 (ti ; r)+ (l1,1 (ti , xˆ(ti ; r), yˆ(ti ; r))+2l2,1 (ti , xˆ(thi ; r), yˆ(ti ; r))+ l4,1 (ti , x ˆ(t6i ; r), yˆ(ti ; r)) = min{g(ti + h, u, v)| ˆ(ti ; r), yˆ(ti ; r)) p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ1 (ti ; r) + k1,1 (ti , x 2 2l3,1 (ti , xˆ(tui ;∈r), y ˆ (t ; r)) + l (t , x ˆ (t ; r), y ˆ (t ; r))) i 4,1 i i i [p3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)), h h ˆ(ti ; r), yˆ(ti ; r)) p1,2 (ti(t ,x ˆi ,(t,x ;(t r), yˆ(tyˆyiˆ(t ;(tr)) =x ˆ2 (ti ;(t r) + 1,2 (t i, x iˆ p(l (t 3,2 i ;i ;r), ii;;r))], yˆ2 (ti+1 ; r) = yˆ2 (ti ; r)+ r), r))+2l ˆ(t2i ;kr), yˆ(t 1,2 (t i ˆx 2,2 i , x i ; r))+ 6 h v ∈2l[q3,1(t(t,ixˆ, (t x ˆ(t; ir), ; r),ˆ(t yˆ(t; r)) (t,i ,xˆx ˆ(t(t;i ;r), r), yˆ(t;ir))) ; r))]} i ; r)), 3,2(t +yˆlq(t p2,1 ,x ˆ(t ˆ1 (tiyˆ;(t r)i + k2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 3,2 i i (tiy i i ; r), 4,2 i = ix i ; r)) 2 l (t , x ˆ (t ; r), y ˆ (t ; r)) = max{g(t + h, u, v)| 4,2 i i following i Then, we illustrated the some lemmasi which express h the conk2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) p (t , x ˆ (t ; r), yˆ(ti ;the r)) = x ˆ (ti ; r) + as 2,2 i i vergence of the approximate u ∈ [p3,1solution (ti , x ˆ(ti ;to r), yˆ(tiexact ; r)),2 solution 2 follows. p3,1 (ti(t ,x ˆ,(tx ˆ(tˆi(t ; r)) =x ˆ1 (ti ; r) + hk3,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) i ; r), y p3,2 i ˆ(ti ; r), y i ; r))], lim xˆ1 (t; r) = x1 (t; r) ˆˆ(t (ti ;; r)), r), yˆq(ti ;(t r)) ˆ=(tx ˆ;2r), (ti ;yˆr) hk3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) h→0 3,2;(t i, x v ∈ [q3,1 (ti , x ˆp(t (ti+ ; r))]} i r), y i 3,2 i , x i h lim xˆ2 (t; r) = x2 (t; r) q(6) ˆ(ti ; r), yˆ(t ˆ1 (ti ; r)(8) + l1,1 (ti , x ˆ(ti ;(9) r), yˆ(ti ; r)) h→0 1,1 (t i, x i ; r)) = y where (7) 2 lim yˆ1 (t; r) = y1 (t; r) h q1,2 (th→0 ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) + l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) i, x h 2 lim=yˆ2x r) = y (t; r) k (t , x ˆ (t ; r), y ˆ (t ; r)) p1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) ˆ(t; (t ; r) + 2 1,1 i i i 1 i h→0 2 h q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + l2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 h k1,2 (t ˆ(tsatisfy yˆ(ti ; r)) p1,2 (t3.1. ˆ(tLet yˆ(tsequence ˆof }Ni , x Lemma i ; r), h i, x i ; r),the i ; r)) = x 2 (tnumbers i ; r) + {w 2 =n yˆn=0 q2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) (t ; r) + l2,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 i 2 h |wn+1 0 kn2,1 (tNi , x − n p2,1 (ti , x ˆ(ti ; r), yˆ(tqi|; r))A|w =ˆ(t x ˆ|1;+ (tiB ; r) ˆ(t1i ; r), yˆ(ti ; r)) ˆ(t ; r), yˆ(t ; r)) ˆ(t+i ; r)) 3,1 (ti , x i r), y i i 2 = yˆ1 (ti ; r) + hl3,1 (ti , x h (proof for some given positive constants A and [12]). Then (ti ,=x ˆ(t yˆ(ti ;Br)) =(tyˆ2,(t ; r); r), + hl (ti , x ˆ(ti ; r), yˆ(ti ; r)) k2,2 ˆi(t yˆ(t3,2 p2,2 (ti , x ˆ(ti ; r), yˆ(tqi3,2 ; r)) x ˆi2;(tr), i x i i ; r)) i ; r) + 2 An − 1 n =x p3,1 (ti , x ˆ(ti ;|w r),n |yˆ (tiA ; r)) ˆ1B (ti ; r) + hk3,1 (tin, x ˆ(tN ˆ(ti ; r)) i ; r), y |w0 | + 0 A−1 p3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = x ˆ2 (ti ; r) + hk3,2 (ti , x ˆ(ti ; r), yˆ(ti ; r))
o e
v i h
c r
A
h l1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 www.SID.ir h q1,2 (ti , x ˆ(ti ; r), yˆ2 (ti ; r)) = yˆ2 (ti ; r) + l1,2 (ti , x ˆ(ti ; r), yˆ(ti ; r)) 2 h ˆ(ti ; r), yˆ(ti ; r)) q2,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) + l2,1 (ti , x 2 q1,1 (ti , x ˆ(ti ; r), yˆ(ti ; r)) = yˆ1 (ti ; r) +
(9
NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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N Lemma 3.2. Let the sequence of numbers {wn }N n=0 , {vn }n=0 (proof [12]) satisfy
|wn+1 | 6 |wn | + A. max{|wn |, |vn |} + B, |vn+1 | 6 |vn | + A. max{|wn |, |vn |} + B, for some given positive constants A and B, and denote |un | = |wn | + |vn |,
06n6N
D I
Then ¯n ¯A −1 |un | 6 A¯n |u0 | + B A¯ − 1 ¯ = 2B. where A¯ = 1 + 2A and B
0 6 n 6 N,
S f
Proof. [12] Generally, if we replace the values of ti s with t and after that replace x ˆ1 (t; r), x ˆ2 (t; r), yˆ1 (t; r) and yˆ2 (t; r) with u ˆ, vˆ, w, ˆ zˆ, then F1 [t, u ˆ, vˆ, w, ˆ zˆ], G1 [t, u ˆ, vˆ, w, ˆ zˆ], F2 [t, u ˆ, vˆ, w, ˆ zˆ], G2 [t, u ˆ, vˆ, w, ˆ zˆ] are defined as follows:
o e
F1 [t, u ˆ, vˆ, w, ˆ zˆ] = k1,1 [t, u ˆ, vˆ, w, ˆ zˆ]
v i h
+2k2,1 [t, u ˆ, vˆ, w, ˆ zˆ] + 2k3,1 [t, u ˆ, vˆ, w, ˆ zˆ] + k4,1 [t, u ˆ, vˆ, w, ˆ zˆ] G1 [t, u ˆ, vˆ, w, ˆ zˆ] = k1,2 [t, u ˆ, vˆ, w, ˆ zˆ] +2k2,2 [t, u ˆ, vˆ, w, ˆ zˆ] + 2k3,2 [t, u ˆ, vˆ, w, ˆ zˆ] + k4,2 [t, u ˆ, vˆ, w, ˆ zˆ]
c r
F2 [t, u ˆ, vˆ, w, ˆ zˆ] = l1,1 [t, u ˆ, vˆ, w, ˆ zˆ]
+2l2,1 [t, u ˆ, vˆ, w, ˆ zˆ] + 2l3,1 [t, u ˆ, vˆ, w, ˆ zˆ] + l4,1 [t, u ˆ, vˆ, w, ˆ zˆ]
A
G2 [t, u ˆ, vˆ, w, ˆ zˆ] = l1,2 [t, u ˆ, vˆ, w, ˆ zˆ]
+2l2,2 [t, u ˆ, vˆ, w, ˆ zˆ] + 2l3,2 [t, u ˆ, vˆ, w, ˆ zˆ] + l4,2 [t, u ˆ, vˆ, w, ˆ zˆ]
The domain where F1 , G1 , F2 , G2 are defined is therefore K = {[t, u ˆ, vˆ, w, ˆ zˆ] | 0 6 t 6 T, − ∞ < u ˆ < ∞, − ∞ < vˆ < ∞, −∞ < w ˆ < ∞, − ∞ < zˆ < ∞}.
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N. PARANDIN
Theorem 3.3. Let F1 [t, u ˆ, vˆ, w, ˆ zˆ], G1 [t, u ˆ, vˆ, w, ˆ zˆ], F2 [t, u ˆ, vˆ, w, ˆ zˆ] and G2 [t, u ˆ, vˆ, w, ˆ zˆ] belongs to C 4 (K) and let the partial derivatives of F1 , G1 , F2 , G2 be bounded over K. Then for arbitrary fixed r : 0 6 r 6 1, the approximate solutions converge uniformly in t to the exact solutions. this theorem is simply proved (see proof theorem 4.1 in [1]).
4.
Examples
D I
Example 4.1. Consider the following fuzzy differential equation with fuzzy initial value problem: 00 x − 2x0 = 0 , t ∈ [0, 0.5]
S f
x(0) = (r, 2 − r), x0 (0) = (3 + r, 4)
where the exact solution is as follow:
o e
1 3 1 x(t) = r − + (3 + r)e2t 2 2 2 x ¯(t) = −r + 2e2t
v i h
In this example, the exact and the approximate solution of the equation and the first differential for t=0.1 have been shown in Tables 1 and 2 respectively. Also, Tables 3, 4 have presented the exact and the approximate solutions of equation and the first differential for t=0.2.
c r
A
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Table 1: Comparison between the exact solution and the approximate solution for t = 0.1 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
xexa. (0.1) 0.3321 0.4432 0.5542 0.6653 0.7764 0.8875 0.9985 1.1096 1.2207 1.3317 1.4428
xapp. (0.1) 0.3315 0.4426 0.5536 0.6647 0.7757 0.8868 0.9978 1.1088 1.2199 1.3310 1.4420
x ¯exa. (0.1) 2.4428 2.3428 2.2428 2.1428 2.0428 1.9428 1.8428 1.7428 1.6428 1.5428 1.4428
x ¯app. (0.1) 2.4420 2.3420 2.2420 2.1420 2.0420 1.9420 1.8420 1.7420 1.6420 1.5420 1.4420
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Table 2: Comparison between the exact solution and the approximate solution of the first differential for t = 0.1 r x0 (0.1) x0 (0.1) x¯0 exa. (0.1) x¯0 app. (0.1)
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exa.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
3.6642 3.7863 3.9085 4.0306 4.1528 4.2749 4.3970 4.5192 4.6413 4.7635 4.8856
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3.6416 3.7630 3.8844 4.0057 4.1272 4.2485 4.3700 4.4913 4.6128 4.7341 4.8556
4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856 4.8856
4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554 4.8554
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Table 3: Comparison between the exact solution and the approximate solution for t = 0.2 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
xexa. (0.2) 0.7377 0.8623 0.9869 1.1115 1.2361 1.3607 1.4853 1.6099 1.7345 1.8591 1.9836
xapp. (0.2) 0.7339 0.8584 0.9829 1.1073 1.2318 1.3563 1.4807 1.6052 1.7296 1.8541 1.9786
x ¯exa. (0.2) 2.9836 2.8836 2.7836 2.6836 2.5836 2.4836 2.3836 2.2836 2.1836 2.0836 1.9836
x ¯app. (0.2) 2.9786 2.8786 2.7786 2.6786 2.5786 2.4786 2.3786 2.2786 2.1786 2.0786 1.9786
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Table 4: Comparison between the exact solution and the approximate solution of the first differential for t = 0.2 r x0 (0.2) x0 (0.2) x¯0 exa. (0.2) x¯0 app. (0.2)
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exa.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.4755 4.6247 4.7738 4.9230 5.0722 5.2214 5.3706 5.5198 5.6689 5.8181 5.9673
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c r
app.
4.4327 4.5805 4.7283 4.8760 5.0238 5.1716 5.3194 5.4671 5.6149 5.7627 5.9105
5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673 5.9673
5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103 5.9103
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NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL ...
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Example 4.2. Consider the following fuzzy differential equation with fuzzy initial value problem: 00 x − 4x0 + 4x = 0 , t ∈ [0, 1] x(0) = (2 + r, 4 − r), x0 (0) = (5 + r, 7 − r)
where the exact solution is as follow: x(t) = (2 + r)e2t + (1 − r)te2t
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x ¯(t) = (4 − r)e2t + (r − 1)te2t
In the following table, the error of the proposed method(PM) is compared with the method introduced by [5]. The results shows that the proposed method is better than the method presented by [5].
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Table 5: The Comparison between the error of the proposed method(PM) and method presented by [5] in t = 0.01 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
5.
error[5] 0.461836e-3 0.476055e-3 0.490273e-3 0.504491e-3 0.518710e-3 0.532928e-3 0.547146e-3 0.561365e-3 0.575584e-3 0.589801e-3 0.604020e-4
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errorP M 4.0369e-4 3.6342e-4 3.2316e-4 2.8289e-4 2.4262e-4 2.0235e-4 1.6209e-4 1.2182e-4 8.1550e-5 4.1283e-5 1.0151e-6
error[5] 0.746204e-3 0.731985e-3 0.717767e-3 0.703549e-3 0.689330e-3 0.675112e-3 0.660894e-3 0.646675e-3 0.632456e-3 0.618239e-3 0.604020e-3
errorP M 4.0166e-4 3.6139e-4 3.2113e-4 2.8086e-4 2.4059e-4 2.0032e-4 1.6006e-4 1.1979e-4 7.9520e-5 3.9253e-5 1.0151e-6
Conclusion
So far, many methods have been proposed for solving the first-order fuzzy differential equations in comparison with the other order of fuzzy differential equations. In this paper the forth-order Runge-Kutta method
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numerically extended to solve the second-order fuzzy ordinary differential equations. In this method, the second-order of fuzzy differential convert to two first-order fuzzy differential equations. Then, by RungeKutta method, the solution of equations is calculated. Also, the proposed method have good and acceptable precise. Acknowledgements: This article has resulted from the research project supported by Islamic Azad University of Kermanshah Branch in Iran.
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References
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[22] J. J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems, 102 (1999), 259-262. [23] H. Ouyang and Y. Wu, On fuzzy differential equations, Fuzzy Sets and Systems, 32 (1989), 321-325. [24] S. Ch. Palligkinis, G. Papageorgiou, and I. Th. Famelis, Runge-Kutta methods for fuzzy differential equations, Applied Mathematics and Computation, 209 (2009), 97-105. [25] S. Pederson and M. Sambandham, The Runge-Kutta method for hybrid fuzzy differential equations, Nonlinear Analysis: Hybrid Systems, 2 (2008), 626-634.
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Nouredin Parandin Department of Mathematics Assistant Professor of Mathematics Islamic Azad University, Kermanshah Branch Kermanshah, Iran E-mail: n [email protected]
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