- Email: [email protected]
Tuning of reachable set in one dimensional fuzzy differential inclusions
Chaos, Solitons and Fractals 26 (2005) 1337–1341 www.elsevier.com/locate/chaos
Tuning of reachable set in one dimensional fuzzy differential inclusions S. Abbasbandy
a,*
, Juan J. Nieto b, M. Alavi
c
a
Department of Mathematics, Imam Khomeini International University, P.O. Box 34194-288, Qazvin, Iran Departamento de Ana´lisis Matema´tico, Facultad de Matema´ticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Department of Mathematics, Science and Research Branch, Islamic Azad University, P.O. Box 14515-397, Tehran, Iran b
c
Accepted 29 March 2005
Abstract This paper presents a new numerical method for solving one dimensional fuzzy differential inclusions. An efficient algorithm to solve them in MAPLE has been devised, which is easy to implement. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy models are now used to study a variety of problems ranging from fuzzy topological spaces [11], fuzzy metric spaces [25], population models [18], the golden mean [12] to control chaotic systems [16,21], radiotherapy treatment [4], to fuzzy differential equations [3] and particle physics [30–33]. Knowledge about differential equations is often incomplete or vague. For example, the parameter values of functional relationships, or the initial conditions, may not be known precisely. Fuzzy differential equations (FDE) were first formulated by Kaleva [22] and Seikkala [27] in time dependent form. Kaleva had formulated fuzzy differential equations, in term of Hukuhara derivative [22]. Buckley and Feuring [10] have given a very general formulation of a fuzzy first-order initial value problem. They first find the crisp solution, fuzzify it and then check to see if it satisfies the FDE. For a solution x(t) to FDE, the use of the Hukuhara derivative suffers a grave disadvantage in so far as x(t) has the property that diam(x(t)) is non-decreasing in t, [9]. This renders the FDEs unsuitable for modeling and simulation. To overcome this difficulty, first, Aubin [5] and then Hu¨llermeier [19] introduced the notion of the fuzzy differential inclusion(FDI) relation. Hu¨llermeier [19] introduced a numerical algorithm to solve general FDIs. Diamond has extended some theoretic notions such as periodicity, Lyapunov stability and attraction [14,15]. Numerical methods for FDE are considered in [23,1,20,6], and for fuzzy differential inclusions recently in [2]. Recently Majumdar [24] has devised a numerical algorithm, which is easy to implement and named Crystalline algorithm (see, for example, [17,29,28,26]), for the one dimensional FDIs. In this paper we are going to propose a new numerical method for computing approximations of the set of all solutions to a FDI, which is named ‘‘Tuning of reachable set (TRS)’’. *
Corresponding author. Tel.: +98 9121305326; fax: +98 2813664099. E-mail addresses: [email protected] (S. Abbasbandy), [email protected] (J.J. Nieto), [email protected] (M. Alavi).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.018
1338
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
In Section 2, we briefly review FDIs. In Section 3, we present our method (TRS algorithm) to solve FDIs. A numerical example is proposed in Section 4 and conclusions in Section 5.
2. Fuzzy differential inclusion One way to model uncertainty and vagueness in a dynamic system is to replace functions and initial values in the problem x_ ðtÞ ¼ F ðt; xðtÞÞ; ð1Þ xð0Þ ¼ x0 by set-valued functions and initial sets, which leads to a differential inclusion x_ ðtÞ 2 F ðt; xðtÞÞ; xð0Þ 2 X 0 ;
ð2Þ
where F : ½0; T R ! 2R n f;g is a set-valued function and X 0 R is compact and convex. A function x : ½0; T ! R is a solution to (2), if it is absolutely continuous and satisfies (2) almost everywhere. Let v denote the set of all solutions to (2). Particularly, we are interested in the so-called reachable sets X ðtÞ :¼ fxðtÞjx 2 vg. The reachable set X(t) is the set of possible solution of (1) at time t 2 J = [0, T]. A reasonable generalization of this approach which takes vagueness into account is to replace ordinary or crisp sets by fuzzy sets, i.e., (2) becomes a fuzzy differential inclusion [5,7,8] ( x_ 2 Fe ðt; xðtÞÞ; ð3Þ e0 xð0Þ 2 X e 0 2 FðRÞ, where FðRÞ is the set of all fuzzy subsets of R. with a fuzzy function Fe : J R ! FðRÞ and a fuzzy set X Also x_ ðtÞ is the usual crisp derivative of the (crisp) differentiable function x(t) with respect to t. In this paper we propose a numerical method for finding the reachable set X(t). Let E be the space of all upper semi-continuous normal fuzzy convex fuzzy sets on R, with compact support. The Hausdorff distance between two non-empty sets A; B R is given as d H ðA; BÞ ¼ maxfqðA; BÞ; qðB; AÞg; where q(A,B) = supa2Adist(a, B) and dist(a, B) = infb2Bja bj. e B e2E For two fuzzy sets A; e BÞ e e ¼ sup df e d~H ð A; H ð½ Aa ; ½ Ba Þ a2ð0;1
e 2 E is a nondefines a distance measure, where for a 2 [0, 1], [Æ]a denotes the level set of a fuzzy set. Each level set of X empty convex compact subset of R. Theorem 1 [19]. Suppose that the fuzzy function Fe : J R ! E is continuous in t and satisfies the Lipschitz condition d~H ð Fe ðt; xÞ; Fe ðt; yÞÞ 6 Ljx yj e ðtÞ associated with e on J R with a Lipschitz constant L > 0. Let e v be the set of all solutions of (3). The reachable set X v is a normal, upper semi-continue and compactly supported fuzzy set for all t 2 J. If Fe is also concave, i.e., a Fe ðt; xÞ þ b Fe ðt; yÞ Fe ðt; ax þ byÞ e ðtÞ 2 E. for any numbers a, b > 0, a + b = 1, then X We call a function xa : J ! R an a-solution to (3), if it is absolutely continuous and satisfies x_ ðtÞ 2 F a ðt; xðtÞÞ; e 0 ; xð0Þ 2 ½ X a
ð4Þ
almost everywhere on J, where Fa(t, x(t)) is the a-cut of the fuzzy set Fe ðt; xðtÞÞ. The set of all a-solutions to (4) is denoted va and the a-reachable set Xa(t) is defined as Xa(t) :¼ {x(t) : x 2 va}.
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
1339
3. Approximation of fuzzy reachable sets Given a set X(t) of possible states at time t, we are interested in obtaining information on the set X(t + Dt). Since the solution (3), xa(t), is absolutely continuous on [0, T] then xa(t) is of bounded variation on [0, T], i.e., $M > 0 such that for any partition 0 = t0 < t1 < < tn = T of the interval [0, T] with step size D = ti+1 ti, i¼n X
jxa ðti Þ xa ðti1 Þj 6 M.
i¼1
Let m ¼ max jxa ðti Þ xa ðti1 Þj; i¼1;2;...;n
M , n
6 ! 0 as Dt ! 0. On the other hand let Xa(ti) = [xi,1, xi,2] then the difference between xi,1 and then m therefore m xi1,1 or xi,2 and xi1,2 is small for enough large n. Refer to (4) by Euler scheme x_ a ðti Þ ’
xa ðti þ DtÞ xa ðti Þ ; Dt
hence we have xa ðti þ DtÞ xa ðti Þ 2 F a ðti þ Dt; xa ðti þ DtÞÞ. Dt
4. Tuning of reachable set (TRS algorithm) Here we introduce an easy algorithm, for solving FDIs, that estimates a-reachable set Xa(ti). TRS Algorithm 1. 2. 3. 4. 5.
Fix a in [0, 1] and choose a small positive number . Xa(0) = [x0,1, x0,2]. Repeat steps 4–6 for i = 1, . . ., n. xi2:¼ xi1,2; Fa(ti, xi,2) = [F1, F2]. x x x x If i;2 Dti1;2 ; i;2 Dti1;1 \ ½F 1 ; F 2 ¼ /, then xi,2 is greater than an upper bound of Xa(ti) and while this inequality is true take xi,2 = xi,2 , since there does not exist a trajectory xa ðtÞ 2 e v a such that xa(ti) = xi,2, and xa(ti1) 2 Xa(ti1);elsexi,2 is less than an upper bound of Xa(ti) and while this inequality is true take xi,2 = xi,2 + , since there exists a trajectory xa ðtÞ 2 vea such that xa(ti) = xi,2, and xa(ti1) 2 Xa(ti1). 6. xi,1 :¼ xi1,1; Fa(ti,xi,1) = [F1, F2]. x x x x 7. If i;1 Dti1;2 ; i;1 Dti1;1 \ ½F 1 ; F 2 ¼ /, then xi,1 is less than a lower bound of Xa(ti) and while this inequality is true take xi,1 = xi,1 + , since there does not exist a trajectory xa ðtÞ 2 e v a such that xa(ti) = xi,1, and xa(ti1) 2 Xa(ti1);elsexi,1 is greater than a lower bound of Xa(ti) and while this inequality is true take xi,2 = xi,2 , since there exists a trajectory xa ðtÞ 2 e v a such that xa(ti) = xi,1, and xa(ti1) 2 Xa(ti1).
Example [15]. Consider the fuzzy differential inclusion on J = [0, T] ( e cos t; x_ ðtÞ 2 X ðtÞ þ W e xð0Þ 2 X 0 ; e and X e 0 are symmetric triangular fuzzy numbers whose supports are the compact interval [1, 1], hence where W e ¼ ½a 1; 1 þ a and ½ X f0 ¼ ½a 1; 1 þ a. The a-solution set is given, for t P 0, by ½W a a 1 e þ xð0Þ 1 ½ W e et ¼ va . xðtÞ 2 ðsin t þ cos tÞ½ W a a 2 2 Now we use the TRS algorithm, let Y(t) be the approximation of X(t). The least square error at t = 1, 2, 5, 10 and a 2 {0, 0.1, . . ., 1}, i.e., (see Figs. 1 and 2)
1340
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
0.4 0.2
0
2
4
6
8
10
12
–0.2 –0.4 Fig. 1. Estimation of upper bound and lower bound of e v 0.5 .
1
0.8
0.6
0.4
0.2
–0.4
–0.2
0
0.2
0.4
e ð0.5Þ. Fig. 2. Solid and dot lines represent, respectively, the exact and approximation of X
i¼10 i¼10 X X ðX ai ðtj Þ Y ai ðtj ÞÞ2 þ ðX ai ðtj Þ Y ai ðtj ÞÞ2 i¼0
i¼0
for Dt = 0.01 is equal to 0.001024, 0.219118, 0.080826, 0.114582, respectively.
5. Conclusions In this paper, we have outlined a new numerical method for solving one-dimensional fuzzy differential equation based on inclusion, TRS algorithm. In this algorithm we tune fuzzy a-reachable set, since xa(t) is of bounded variation. Numerical example shows the efficiency of implemented algorithm. According to our experiment, TRS algorithm can be modified for solving FDIs with higher dimension. The implementation of such method will be a relevant aspect of our future research.
Acknowledgments The research of J.J. Nieto was partially supported by Ministerio de Educacio´n y Ciencia and FEDER, project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, project PGIDIT02PXIC20703PN.
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
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References [1] Abbasbandy S, Allah Viranloo T. Numerical solution of fuzzy differential equation by Runge–Kutta method. Nonlinear Stud 2004;11:117–29. ´ scar, Nieto JJ. Numerical methods for fuzzy differential inclusions. Comput [2] Abbasbandy S, Allah Viranloo T, Lo´pez-Pouso O Math Appl 2004;48:1633–41. [3] Agarwal RP, ORegan D, Lakshmikantham V. Viability theory and fuzzy differential equations. Fuzzy Set Syst 2005;151:563–80. [4] Ammar EE, Hussein ML. Radiotherapy problem under fuzzy theoretic approach. Chaos, Solitons & Fractals 2003;18:739–44. [5] Aubin JP. Fuzzy differential inclusions. Problems Control Inform Theory 1990;19:55–67. [6] Babolian E, Sadeghi H, Javadi Sh. Numerically solution of fuzzy differential equations by Adomian method. Appl Math Comput 2004;149:547–57. [7] Baı˘dosov VA. Differential inclusions with a fuzzy right-hand side. Dokl Akad Nauk SSSR (Russian) 1989;309:781–3. [8] Baı˘dosov VA. Fuzzy differential inclusions. J Appl Math Mech 1990;54:8–13. [9] Bobylev VN. A possibilistic argument for irreversibility. Fuzzy Set Syst 1990;34:73–80. [10] Buckley JJ, Feuring T. Fuzzy differential equations. Fuzzy Set Syst 2000;110:43–54. [11] Caldas M, Jafari S. h-Compact fuzzy topological spaces. Chaos, Solitons & Fractals 2005;25:229–32. [12] Datta DP. The golden mean, scale free extension of real number system, fuzzy sets and 1/f spectrum in physics and biology. Chaos, Solitons & Fractals 2003;17:781–8. [14] Diamond P. Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations. IEEE Trans Fuzzy Syst 1999;7:734–40. [15] Diamond P. Stability and periodicity in fuzzy differential equations. IEEE Trans Fuzzy Syst 2000;8:583–90. [16] Feng G, Chen G. Adaptive control of discrete-time chaotic systems: a fuzzy control approach. Chaos, Solitons & Fractals 2005;23:459–67. [17] Gira˜o PM, Kohn RV. Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature. Numer Math 1994;67:41–70. [18] Guo M, Xue X, Li R. Impulsive functional differential inclusions and fuzzy population models. Fuzzy Set Syst 2003;138:601–15. [19] Hu¨llermeier E. An approach to modeling and simulation of uncertain dynamical systems. Int J Uncertain Fuzz Knowledge-based Syst 1997;5:117–37. [20] Hu¨llermeier E. Numerical methods for fuzzy initial value problems. Int J Uncertain Fuzz Knowledge-based Syst 1999;7:439–61. [21] Jiang W, Guo-Doug Q, Bin D. H1 Variable universe adaptive fuzzy control for chaotic system. Chaos, Solitons & Fractals 2005;24:1075–86. [22] Kaleva O. Fuzzy differential equations. Fuzzy Set Syst 1987;24:301–17. [23] Ma M, Friedman M, Kandel A. Numerical solution of fuzzy differential equations. Fuzzy Set Syst 1999;105:133–8. [24] Majumdar KK. One dimensional fuzzy differential inclusions. J Intell Fuzzy Syst 2003;13:1–5. [25] Park JH. Intuitionistic fuzzy metric spaces. Chaos Solitons & Fractals 2004;22:1039–46. [26] Rotstein HG, Brandon S, Novick-Cohen A, Nepomnyashchy A. Phase field equations with memory: the hyperbolic case. SIAM J Appl Math 2001;62:264–82. [27] Seikkala S. On the fuzzy initial value problem. Fuzzy Set Syst 1987;24:319–30. [28] Ushijima TK, Yazaki S. Convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature V = K a. SIAM J Numer Anal 2000;37:500–22. [29] Ushijima TK, Yazaki S. Convergence of a crystalline approximation for an area-preserving motion. J Comput Appl Math 2004;166:427–52. [30] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36. [31] Elnaschie MS. On a fuzzy Ka˜hler-like manifold which is consistent with the two-slit experiment. Int J Nonlinear Sci Numer Simulat 2005;6(2):95–8. [32] Elnaschie MS. The concepts of E-infinity theory. Chaos, Solitons & Fractals 2004;22:495–511. [33] Tanaka Y, Mizuno Y, Kado T. Chaotic dynamics in the Friedman equation. Chaos, Solitons & Fractals 2005;24:407–22.
Tuning of reachable set in one dimensional fuzzy differential inclusions S. Abbasbandy
a,*
, Juan J. Nieto b, M. Alavi
c
a
Department of Mathematics, Imam Khomeini International University, P.O. Box 34194-288, Qazvin, Iran Departamento de Ana´lisis Matema´tico, Facultad de Matema´ticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Department of Mathematics, Science and Research Branch, Islamic Azad University, P.O. Box 14515-397, Tehran, Iran b
c
Accepted 29 March 2005
Abstract This paper presents a new numerical method for solving one dimensional fuzzy differential inclusions. An efficient algorithm to solve them in MAPLE has been devised, which is easy to implement. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy models are now used to study a variety of problems ranging from fuzzy topological spaces [11], fuzzy metric spaces [25], population models [18], the golden mean [12] to control chaotic systems [16,21], radiotherapy treatment [4], to fuzzy differential equations [3] and particle physics [30–33]. Knowledge about differential equations is often incomplete or vague. For example, the parameter values of functional relationships, or the initial conditions, may not be known precisely. Fuzzy differential equations (FDE) were first formulated by Kaleva [22] and Seikkala [27] in time dependent form. Kaleva had formulated fuzzy differential equations, in term of Hukuhara derivative [22]. Buckley and Feuring [10] have given a very general formulation of a fuzzy first-order initial value problem. They first find the crisp solution, fuzzify it and then check to see if it satisfies the FDE. For a solution x(t) to FDE, the use of the Hukuhara derivative suffers a grave disadvantage in so far as x(t) has the property that diam(x(t)) is non-decreasing in t, [9]. This renders the FDEs unsuitable for modeling and simulation. To overcome this difficulty, first, Aubin [5] and then Hu¨llermeier [19] introduced the notion of the fuzzy differential inclusion(FDI) relation. Hu¨llermeier [19] introduced a numerical algorithm to solve general FDIs. Diamond has extended some theoretic notions such as periodicity, Lyapunov stability and attraction [14,15]. Numerical methods for FDE are considered in [23,1,20,6], and for fuzzy differential inclusions recently in [2]. Recently Majumdar [24] has devised a numerical algorithm, which is easy to implement and named Crystalline algorithm (see, for example, [17,29,28,26]), for the one dimensional FDIs. In this paper we are going to propose a new numerical method for computing approximations of the set of all solutions to a FDI, which is named ‘‘Tuning of reachable set (TRS)’’. *
Corresponding author. Tel.: +98 9121305326; fax: +98 2813664099. E-mail addresses: [email protected] (S. Abbasbandy), [email protected] (J.J. Nieto), [email protected] (M. Alavi).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.018
1338
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
In Section 2, we briefly review FDIs. In Section 3, we present our method (TRS algorithm) to solve FDIs. A numerical example is proposed in Section 4 and conclusions in Section 5.
2. Fuzzy differential inclusion One way to model uncertainty and vagueness in a dynamic system is to replace functions and initial values in the problem x_ ðtÞ ¼ F ðt; xðtÞÞ; ð1Þ xð0Þ ¼ x0 by set-valued functions and initial sets, which leads to a differential inclusion x_ ðtÞ 2 F ðt; xðtÞÞ; xð0Þ 2 X 0 ;
ð2Þ
where F : ½0; T R ! 2R n f;g is a set-valued function and X 0 R is compact and convex. A function x : ½0; T ! R is a solution to (2), if it is absolutely continuous and satisfies (2) almost everywhere. Let v denote the set of all solutions to (2). Particularly, we are interested in the so-called reachable sets X ðtÞ :¼ fxðtÞjx 2 vg. The reachable set X(t) is the set of possible solution of (1) at time t 2 J = [0, T]. A reasonable generalization of this approach which takes vagueness into account is to replace ordinary or crisp sets by fuzzy sets, i.e., (2) becomes a fuzzy differential inclusion [5,7,8] ( x_ 2 Fe ðt; xðtÞÞ; ð3Þ e0 xð0Þ 2 X e 0 2 FðRÞ, where FðRÞ is the set of all fuzzy subsets of R. with a fuzzy function Fe : J R ! FðRÞ and a fuzzy set X Also x_ ðtÞ is the usual crisp derivative of the (crisp) differentiable function x(t) with respect to t. In this paper we propose a numerical method for finding the reachable set X(t). Let E be the space of all upper semi-continuous normal fuzzy convex fuzzy sets on R, with compact support. The Hausdorff distance between two non-empty sets A; B R is given as d H ðA; BÞ ¼ maxfqðA; BÞ; qðB; AÞg; where q(A,B) = supa2Adist(a, B) and dist(a, B) = infb2Bja bj. e B e2E For two fuzzy sets A; e BÞ e e ¼ sup df e d~H ð A; H ð½ Aa ; ½ Ba Þ a2ð0;1
e 2 E is a nondefines a distance measure, where for a 2 [0, 1], [Æ]a denotes the level set of a fuzzy set. Each level set of X empty convex compact subset of R. Theorem 1 [19]. Suppose that the fuzzy function Fe : J R ! E is continuous in t and satisfies the Lipschitz condition d~H ð Fe ðt; xÞ; Fe ðt; yÞÞ 6 Ljx yj e ðtÞ associated with e on J R with a Lipschitz constant L > 0. Let e v be the set of all solutions of (3). The reachable set X v is a normal, upper semi-continue and compactly supported fuzzy set for all t 2 J. If Fe is also concave, i.e., a Fe ðt; xÞ þ b Fe ðt; yÞ Fe ðt; ax þ byÞ e ðtÞ 2 E. for any numbers a, b > 0, a + b = 1, then X We call a function xa : J ! R an a-solution to (3), if it is absolutely continuous and satisfies x_ ðtÞ 2 F a ðt; xðtÞÞ; e 0 ; xð0Þ 2 ½ X a
ð4Þ
almost everywhere on J, where Fa(t, x(t)) is the a-cut of the fuzzy set Fe ðt; xðtÞÞ. The set of all a-solutions to (4) is denoted va and the a-reachable set Xa(t) is defined as Xa(t) :¼ {x(t) : x 2 va}.
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
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3. Approximation of fuzzy reachable sets Given a set X(t) of possible states at time t, we are interested in obtaining information on the set X(t + Dt). Since the solution (3), xa(t), is absolutely continuous on [0, T] then xa(t) is of bounded variation on [0, T], i.e., $M > 0 such that for any partition 0 = t0 < t1 < < tn = T of the interval [0, T] with step size D = ti+1 ti, i¼n X
jxa ðti Þ xa ðti1 Þj 6 M.
i¼1
Let m ¼ max jxa ðti Þ xa ðti1 Þj; i¼1;2;...;n
M , n
6 ! 0 as Dt ! 0. On the other hand let Xa(ti) = [xi,1, xi,2] then the difference between xi,1 and then m therefore m xi1,1 or xi,2 and xi1,2 is small for enough large n. Refer to (4) by Euler scheme x_ a ðti Þ ’
xa ðti þ DtÞ xa ðti Þ ; Dt
hence we have xa ðti þ DtÞ xa ðti Þ 2 F a ðti þ Dt; xa ðti þ DtÞÞ. Dt
4. Tuning of reachable set (TRS algorithm) Here we introduce an easy algorithm, for solving FDIs, that estimates a-reachable set Xa(ti). TRS Algorithm 1. 2. 3. 4. 5.
Fix a in [0, 1] and choose a small positive number . Xa(0) = [x0,1, x0,2]. Repeat steps 4–6 for i = 1, . . ., n. xi2:¼ xi1,2; Fa(ti, xi,2) = [F1, F2]. x x x x If i;2 Dti1;2 ; i;2 Dti1;1 \ ½F 1 ; F 2 ¼ /, then xi,2 is greater than an upper bound of Xa(ti) and while this inequality is true take xi,2 = xi,2 , since there does not exist a trajectory xa ðtÞ 2 e v a such that xa(ti) = xi,2, and xa(ti1) 2 Xa(ti1);elsexi,2 is less than an upper bound of Xa(ti) and while this inequality is true take xi,2 = xi,2 + , since there exists a trajectory xa ðtÞ 2 vea such that xa(ti) = xi,2, and xa(ti1) 2 Xa(ti1). 6. xi,1 :¼ xi1,1; Fa(ti,xi,1) = [F1, F2]. x x x x 7. If i;1 Dti1;2 ; i;1 Dti1;1 \ ½F 1 ; F 2 ¼ /, then xi,1 is less than a lower bound of Xa(ti) and while this inequality is true take xi,1 = xi,1 + , since there does not exist a trajectory xa ðtÞ 2 e v a such that xa(ti) = xi,1, and xa(ti1) 2 Xa(ti1);elsexi,1 is greater than a lower bound of Xa(ti) and while this inequality is true take xi,2 = xi,2 , since there exists a trajectory xa ðtÞ 2 e v a such that xa(ti) = xi,1, and xa(ti1) 2 Xa(ti1).
Example [15]. Consider the fuzzy differential inclusion on J = [0, T] ( e cos t; x_ ðtÞ 2 X ðtÞ þ W e xð0Þ 2 X 0 ; e and X e 0 are symmetric triangular fuzzy numbers whose supports are the compact interval [1, 1], hence where W e ¼ ½a 1; 1 þ a and ½ X f0 ¼ ½a 1; 1 þ a. The a-solution set is given, for t P 0, by ½W a a 1 e þ xð0Þ 1 ½ W e et ¼ va . xðtÞ 2 ðsin t þ cos tÞ½ W a a 2 2 Now we use the TRS algorithm, let Y(t) be the approximation of X(t). The least square error at t = 1, 2, 5, 10 and a 2 {0, 0.1, . . ., 1}, i.e., (see Figs. 1 and 2)
1340
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
0.4 0.2
0
2
4
6
8
10
12
–0.2 –0.4 Fig. 1. Estimation of upper bound and lower bound of e v 0.5 .
1
0.8
0.6
0.4
0.2
–0.4
–0.2
0
0.2
0.4
e ð0.5Þ. Fig. 2. Solid and dot lines represent, respectively, the exact and approximation of X
i¼10 i¼10 X X ðX ai ðtj Þ Y ai ðtj ÞÞ2 þ ðX ai ðtj Þ Y ai ðtj ÞÞ2 i¼0
i¼0
for Dt = 0.01 is equal to 0.001024, 0.219118, 0.080826, 0.114582, respectively.
5. Conclusions In this paper, we have outlined a new numerical method for solving one-dimensional fuzzy differential equation based on inclusion, TRS algorithm. In this algorithm we tune fuzzy a-reachable set, since xa(t) is of bounded variation. Numerical example shows the efficiency of implemented algorithm. According to our experiment, TRS algorithm can be modified for solving FDIs with higher dimension. The implementation of such method will be a relevant aspect of our future research.
Acknowledgments The research of J.J. Nieto was partially supported by Ministerio de Educacio´n y Ciencia and FEDER, project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, project PGIDIT02PXIC20703PN.
S. Abbasbandy et al. / Chaos, Solitons and Fractals 26 (2005) 1337–1341
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