A method for solving first order fuzzy differential equation

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1 Available online at ttp://ijim.srbiau.ac.ir/ Int. J. Industrial Matematics (ISSN ) Vol. 5, No. 3, 2013 Article ID IJIM-00250, 7 pages Researc Article A metod for solving first order fuzzy differential equation O. Sedagatfar, P. Darabi, S. Moloudzade Abstract In tis paper, a new approac for solving first-order fuzzy differential equation (F DE) wit fuzzy initial value under strongly generalized H-differentiability is considered. Te idea of te presented approac is constructed based on te extending 0-cut 1-cut solution of original F DE. First, under H-differentiability te solutions of fuzzy differential equations in 0-cut 1-cut cases are found convex combination of tem, is considered. Ten we coose te initial interval tat 0-cut of original problem is positive. Te obtained convex combination on tis interval is te solution of F DE. Keywords : Fuzzy differential equation (FDE); Interval differential equation; Strongly generalized H-differentiability. 1 Introduction e topic of fuzzy differential equation (F DE) T as been rapidly growing in recent years. Kel Byatt [19] applied te concept of (F DE) to te analysis of fuzzy dynamical problems. Te (F DE) te initial value problem (Caucy problem) were rigorously treated by Kaleva [17, 18], Seikkala [23], He Yi [13], Kloeden [20] by oter researcers (see [3, 9, 10, 12, 16]). Te numerical metods for solving F DE are introduced in [1, 2, 4, 5]. Bede [7] applied te concept of Strongly generalized H- differentiability to solving first order linear fuzzy differential equations Allaviranloo et. al. [6] proposed a metod to obtain analytical solutions for F DE under strongly generalized H- differentiability. Corresponding autor. sedagtfar@yaoo.com.com Department of Matematics, Sar-e-rey Branc, Islamic Azad University, Sar-e-rey, Iran. Department of Matematics, Science Researc Branc, Islamic Azad University, Teran, Iran. Department of Matematics, Science Researc Branc, Islamic Azad University, Teran, Iran. Te idea of te presented approac is constructed based on te extending 0-cut 1-cut solution of original F DE. Obviously, 0-cut of F DE is interval differential equation or ordinary differential equation. First, under H-differentiability F DE as been divided in two differential equations solutions of eac fuzzy differential equations in 0- cut 1-cut cases are found. Ten in eac cases of differentiability, te initial interval tat 0-cut of original problem is positive, is found. Te obtained convex combination on tis interval,is an solution of F DE. Te structure of tis paper is organized as follows. In Section 2, some basic definitions notations wic will be used are brougt. In Section 3, first order fuzzy differential equation is introduced te proposed approac is given in detail. In Section 4, te proposed metod is illustrated by solving several examples. Conclusion is drawn in Section

2 252 O. Sedagatfar, et al /IJIM Vol. 5, No. 3 (2013) Basic Definitions Notations Definition 2.1 An arbitrary fuzzy number is represented by an ordered pair of functions (u(r), u(r)) for all r [0, 1], wic satisfy te following requirements [15]. u(r) is a bounded left continuous nondecreasing function over [0, 1]; u(r) is a bounded left continuous nonincreasing function over [0, 1]; u(r) u(r), 0 r 1. Let E be te set of all upper semi-continuous normal convex fuzzy numbers wit bounded r-level intervals. Tis means tat if ṽ E ten te r- level set [v] r = {s v(s) r}, is a closed bounded interval wic is denoted by [v] r = [v(r), v(r)] for r (0, 1] [v] 0 = r (0,1] [v] r. Two fuzzy numbers ũ ṽ are called equal ũ = ṽ, if u(s) = v(s) for all s R or [u] r = [v] r for all r [0, 1]. Lemma 2.1 [22] If ũ, ṽ E, ten for r (0, 1], were [u + v] r = [u(r) + v(r), u(r) + v(r)], [u.v] r = [min k r, max k r ], k r = {u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)}. Te Hausdorff distance between fuzzy numbers given by D : E E R + {0}, D(u, v) = sup max{ u(r) v(r), u(r) v(r) }, r [0,1] were u = (u(r), u(r)), v = (v(r), v(r)) E is utilized (see [8]). Ten it is easy to see tat D is a metric in E were for all u, v, w, e E as te following properties (see [21]). D(u w, v w) = D(u, v), D(k u, k v) = k D(u, v), k R, D(u v, w e) D(u, w) + D(v, e), (D, E) is a complete metric space. Definition 2.2 [14]. Let f : R E be a fuzzy valued function. If for arbitrary fixed t 0 R ε > 0, δ > 0 suc tat t t 0 < δ = D(f(t), f(t 0 )) < ε, f is said to be continuous. Definition 2.3 Let x, y E. If tere exists z E suc tat x = y + z, ten z is called te H-difference of x y it is denoted by x y. In tis paper we consider te following definition of differentiability for fuzzy-valued functions wic was introduced by Bede et.al. [8] investigate by Calco-Cano et.al. [11]. Definition 2.4 Let f : (a, b) E x 0 (a, b). We say tat f is strongly generalized H- differentiable at x 0. If tere exists an element f (x 0 ) E, suc tat: (1) for all > 0 sufficiently near to 0, f(x 0 + ) f(x 0 ), f(x 0 ) f(x 0 ) suc tat te following limits old. lim 0 + f(x 0+) f(x 0 ) = lim 0 + f(x 0) f(x 0 ) = f (x 0 ) (2) for all < 0 sufficiently near to 0, f(x 0 ) f(x 0 + ), f(x 0 ) f(x 0 ) suc tat te following limits old. lim 0 + f(x 0) f(x 0 +) = lim 0 + f(x 0 ) f(x 0 ) = f (x 0 ) In te special case wen f is a fuzzy-valued function, we ave te following results. Teorem 2.1 [11]. Let f : R E be a function denote f(t) = (f(t; r), f(t; r)), for eac r [0, 1]. Ten (1) if f is differentiable in te first form (1) in definition 2.4 ten f(t; r) f(t; r)) are differentiable functions f (t) = (f (t; r), f (t; r)). (2) if f is differentiable in te second form (2) in definition 2.4 ten f(t; r) f(t; r)) are differentiable functions f (t) = (f (t; r), f (t; r)). Te principal properties of te H-derivatives in te first form (1), some of wic still old for te second form (2), are well known can be found in [17] some properties for te second form (2) can be found in [11]. Notice tat we say fuzzy-valued function f is (I)- differentiable if satisfies in te first form (1) in Definition 2.4 we say f is (II)-differentiable if satisfies in te second form (2) in Definition 2.4.

3 O. Sedagatfar, et al /IJIM Vol. 5, No. 3 (2013) First Order Fuzzy Differential Equations In tis section, we are going to investigate solution of F DE. Consider te following first order fuzzy differential equation: { y (t) = f(t, y(t)) (3.1) ỹ(t 0 ) = ỹ 0, were f : [a, b] E E is fuzzy-valued function, ỹ 0 E strongly generalized H- differentiability is also considered wic is defined in Definition 2.4 Now, we describe our propose approac for solving F DE (3.1). First, we sall solve F DE (3.1) in sense of 1-cut 0-cut as a follows: { (y ) [1] (t) = f [1] (t, y(t)), y [1] (t 0 ) = ỹ [1] 0, t (3.2) 0 [0, T ]. { (y ) [0] (t) = f [0] (t, y(t)), y [0] (t 0 ) = ỹ [0] 0, t 0 [0, T ]. (3.3) If Eq. (3.2) Eq. (3.3) be a crisp differential equation we can solve it as usual, oterwise, if Eq. (3.2) Eq. (3.3) be an interval differential equation we will solve it by stefanini et. al. s metod is proposed discussed in [24]. Notice tat te solutions of differential equation (3.2) Eq. (3.3) are presented wit notation y [1] (t) y [0] (t) respectively. Ten unknown ỹ(t) must be determined, tis approac leads to obtain y(t) = [y(t; r), y(t; r)] = (3.4) [(1 r)y [0] (t) + ry [1] (t), (1 r)y [0] (t) + ry [1] (t)] Pleased notice tat, we assumed te 0-cut 1-cut solutions are differentiable. Consequently, based on type of differentiability we ave two following cases: Case I. Suppose tat ỹ(t) in Eq. (3.3) is (I)- differentiable, ten we get: y (t) = [y (t; r), y (t; r)] (3.5) Consider Eq. (3.5) original F DE (3.1), ten we ave te following for all r [0, 1]: y (t; r) = f(t; r), t 0 t T, y (t; r) = f(t; r), t 0 t T. (3.6) Terefore (1 r)(y [0] ) (t) + r(y [1] ) (t) = (1 r)(f [0] )(t) + r(f [1] )(t), (1 r)(y [0] ) (t) + r(y [1] ) (t) = (1 r)(f [0] )(t) + r(f [1] )(t), y(t 0 ; r) = (1 r)y [0] (t 0 ) + ry [1] (t 0 ), y(t 0 ; r) = (1 r)y [0] (t 0 ) + ry [1] (t 0 ). Ten we ave y [1] (t 0 ) = y 0 [1], y [1] (t 0 ) = y 0 [1]. (3.7) (3.8) (3.9) Indeed, we will find y [0] (t), y [0] (t), y [1] (t), y [1] (t) by solving ODEs (3.8), (3.9). Hence, solution of original F DE (3.1) is derived based on 1-cut solution 0-cut solution as follows: ỹ(t) = [y(t; r), y(t; r)] = [(1 r)y [0] (t) + ry [1] (t), (1 r)y [0] (t) + ry [1] (t)], were for all 0 r 1 t [0, T ] suc tat: y(t; r) = (1 r)y [0] (t) + ry [1] (t), y(t; r) = (1 r)y [0] (t) + ry [1] (t). Case II. Suppose tat ỹ(t) in Eq. (3.3) is (II)- differentiable, ten we get: y (t) = [y (t; r), y (t; r)] (3.10)

4 254 O. Sedagatfar, et al /IJIM Vol. 5, No. 3 (2013) Similarly, ODEs (3.8) (3.9) can be rewritten in sense of (II)-differentiability as following: y [1] (t 0 ) = y 0 [1], y [1] (t 0 ) = y 0 [1]. (3.11) (3.12) Finally, by solving above ODEs (3.11) (3.12) y [0] (t), y [0] (t), y [1] (t), y [1] (t) are determined follows we can drive solution of original F DE (3.1) in sense of (II)-differentiability by using y(t; r) = (1 r)y [0] (t) + ry [1] (t), y(t; r) = (1 r)y [0] (t) + ry [1] (t), for all 0 r 1 t [0, T ]. 4 Examples In tis section, some examples are given to illustrate our metod we sow tat our approac is coincide wit te exact solutions. Example 4.1 Let consider te following F DE: { y (t) = y(t) + ã, y(0; r) = [r, 2 r], ã = [r 1, 1 r] 0 r 1. (4.13) Case I. Suppose tat ỹ(t) is (I)-differentiable. Te exact solution of above system is: ỹ(t) = [(2r 1)e t r + 1, (2r 3)e t r 1]. Based on ODEs (3.8) (3.9), y [0] (t) = y [0] (t) 1, y [1] (0) = 1, y [1] (0) = 1. By solving ODEs (4.16) (4.17), we get: y [0] (t) = e t + 1, y [0] (t) = 3e t 1, y [1] (t) = e t, y [1] (t) = e t. (4.15) y(t) = (1 r)( e t + 1) + re t = e t (2r 1) r + 1, y(t) = (1 r)(3e t 1) + re t = r e t (2r 3) 1, ỹ(t) = [e t (2r 1) r + 1, r e t (2r 3) 1], were y(t) as valid level sets for t > 0 y(t) is (I)-differentiable on t > 0, ten y(t) is a solution for original problem on t > 0. Case II. Suppose tat ỹ(t) is (II)- differentiable. Te exact solution of above system is: ỹ(t) = [e t r+(2r 2)/e t +1, r+e t (2r 2)/e t 1]. Based on ODEs (3.8) (3.9), y [0] (t) = y [0] (t) + 1, y [0] (t) = y [0] (t) 1, y [0] (0) = 0, y [0] (0) = 2, (4.16) y [0] (t) = y [0] (t) + 1, y [0] (0) = 0, (4.14) y [1] (0) = 1, (4.17) y [0] (0) = 2, y [1] (0) = 1.

5 O. Sedagatfar, et al /IJIM Vol. 5, No. 3 (2013) By solving ODEs (4.16) (4.17), we get: y [0] (t) = e t r + (2r 2)/e t + 1, y [0] (t) = r + e t (2r 2)/e t 1, y [1] (t) = e t, y [1] (t) = e t. y(t) = (1 r)(e t r + (2r 2)/e t + 1) + re t (1/2)e t ) = e t r + (2r 2)/e t + 1, y(t) = (1 r)(r + e t (2r 2)/e t 1) + re t = r + e t (2r 2)/e t 1 ỹ(t) = [e t r + (2r 2)/e t + 1, r + e t (2r 2)/e t 1]. were y(t) as valid level sets for [0, log(2)] y(t) is (II)-differentiable on t > 0, ten y(t) is a solution for te original problem on [0, log(2)]. Example 4.2 Let consider te following F DE: { y (t) = y(t), (4.18) y(0; r) = [1 + r, 5 r], 0 r 1. Case I. Suppose tat ỹ(t) is (I)-differentiable. Te exact solution of above system is: ỹ(t) = [3e t + (r 2)e t, 3e t (r 2)e t ]. Based on ODEs (3.11) (3.12), we get: y [0] (0) = 1, y [0] (0) = 5, y [1] (0) = 2, y [1] (0) = 4. (4.19) (4.20) By solving ODEs (4.21) (4.22), we get: y [0] (t) = 3e t 2e t, y [0] (t) = 3e t + 2e t, y [1] (t) = 3e t e t, y [1] (t) = 3e t + e t. y(t) = (1 r)(3e t 2e t ) + r(3e t e t ) = 3e t + (r 2)e t, y(t) = (1 r)(3e t + 2e t ) + r(3e t + e t ) = 3e t (r 2)e t ỹ(t) = [3e t + (r 2)e t, 3e t (r 2)e t ], were y(t) as valid level sets for t > 0 y(t) is (I)-differentiable on t > 0, ten y(t) is a solution for te original problem on t > 0. Case II. Suppose tat ỹ(t) is (II)- differentiable. Te exact solution of above system is: ỹ(t) = [(1 + r)e t, (5 r)e t ]. Based on ODEs (3.11) (3.12), we get: y [0] (0) = 1, y [0] (0) = 5, y [1] (0) = 2, y [1] (0) = 4. By solving ODEs (4.21) (4.22), we get: y [0] (t) = e t, y [0] (t) = 5e t, y [1] (t) = 2e t, y [1] (t) = 4e t. (4.21) (4.22)

6 256 O. Sedagatfar, et al /IJIM Vol. 5, No. 3 (2013) y(t) = (1 r)e t + 2re t = (1 + r)e t, y(t) = 5(1 r)e t + 4re t = (5 r)e t ỹ(t) = [(1 + r)e t, (5 r)e t ], were y(t) as valid level sets for t > 0 y(t) is (II)-differentiable on t > 0, ten y(t) is a solution for original problem on t > 0. 5 Conclusion A new metod for solving first order fuzzy differential equations F DE wit fuzzy initial value under strongly generalized H-differentiability was considered. Te idea of te presented approac was constructed based on te extending 0-cut 1-cut solution of original F DE. First, under H- differentiability te solutions of fuzzy differential equations in 0-cut 1-cut cases were found ten convex combination of tem was considered. Ten we found out te initial interval tat 0-cut of original problem was positive. Te obtained convex combination on tis interval was a solution of F DE. Using 0-cut 1-cut solutions we sow tat te discussed metod can be applied to solve te fuzzy differential equation. References [1] S. Abbasby, T. Allaviranloo, Numerical solutions of fuzzy differential equations by taylor metod, Comput. Metods Appl. Mat. 2 (2002) [2] S. Abbasby, T. Allaviranloo, Óscar López-Pouso, J. J. Nieto, Numerical Metods for Fuzzy Differential Inclusions, Comput. Mat. Appl. 48 (2004) [3] T. Allaviranloo, S. Salasour, A new approac for solving first order fuzzy differential equations, IPMU. (2010) [4] T. Allaviranloo, N. Amady, E. Amady, Numerical solution of fuzzy differential equations by predictorcorrector metod, Infor. Sci. 177 (2007) [5] T. Allaviranloo, N. Amady, E. Amady, A metod for solving n-t order fuzzy linear differential equations, Comput. Mat. Appl. 86 (2009) [6] T. Allaviranloo, N. A, Kiani, M. Barkordari, Toward te exiistence uniqueness of solution of second-order fuzzy differential equations, Information Sciences 179 (2009) [7] B. Bede, I. J. Rudas, A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inform. Sci. 177 (2007) [8] B. Bede, S. G. Gal, Generalizations of te differetiability of fuzzy-number-valued functions wit applications to fuzzy differential equation, Fuzzy Sets Systems 151 (2005) [9] J. J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy sets Systems 110 (2000) [10] J. J. Buckley, T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets Systems 105 (1999) [11] Y. Calco-cano, H. Roman-Flores, On new solutions of fuzzy differential equations, Caos, Solitons Fractals 38 (2006) [12] W. Congxin, S. Siji, Existence teorem to te Caucy problem of fuzzy differential equations under compactness-type conditions, Inf. Sci. 108 (1998) [13] O. He, W. Yi, On fuzzy differential equations, Fuzzy Sets Systems 24 (1989) [14] M. Friedman, M. Ming, A. Kel, Numerical solution of fuzzy differential integral equations, Fuzzy Sets System 106 (1999) [15] M. Friedman, M. Ming, A. Kel, Fuzzy linear systems, Fuzzy Set Systems 96 (1998) [16] L. J. Jowers, J. J. Buckley, K. D. Reilly, Simulating continuous fuzzy systems, Inf. Sci. 177 (2007)

7 O. Sedagatfar, et al /IJIM Vol. 5, No. 3 (2013) [17] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Systems 24 (1987) [18] O. Kaleva, Te Caucy problem for fuzzy differential equations, Fuzzy Sets Systems 35 (1990) [19] A. Kel, W. J. Byatt, Fuzzy differential equations, In:Proceedings of te International Conference on Cybernetics Society, Tokyo (1978) [20] P. Kloeden, Remarks on peano-like teorems for fuzzy differential equations, Fuzzy Sets Systems 44 (1991) [21] M. L. Puri, D. Ralescu, Fuzzy rom variables, J. Mat. Anal. Appl. 114 (1986) [22] D. Ralescu, A survey of te representation of fuzzy concepts its applications, In: M. M. Gupta, R. K. Ragade, R. R. Yager, Eds., Advances in Fuzzy Set Teory Applications (Nort-Holl, Amsterdam (1979) [23] S. Seikkala, On te fuzzy initial value problem, Fuzzy Sets Systems 24 (1987) [24] L. Stefanini, B. Bede, Generalized Hukuara differentiability of interval-valued functions interval differential equations, Nonlinear Anal. 71 (2009) Pejman Darabi was born in Teran-Iran in He received B.Sc M.Sc degrees in applied matematics from Sabzevar Teacer Education University, science researc Branc, IAU to Teran, respectively, is P.D degree in applied matematics from science researc Branc, IAU, Teran, Iran in Main researc interest include numerical solution of Fuzzy differential equation systems, ordinary differential equation, partial differential equation, fuzzy linear system similar topics. Saeid Moloudzade was born in te Nagade-Iran in He received B.Sc degree in matematics M.Sc degree in applied matematics from Payam-e-Noor University of Nagade, science researc Branc, IAU to Teran, respectively, is P.D degree in applied matematics from Science Researc Branc, IAU, Teran, Iran in He is researc interests include fuzzy matematics, especially, on numerical solution of fuzzy linear systems, fuzzy differential equations similar topics. Omolbanin Sedagatfar was born in te Teran, Sar-e-rey in Se received B.Sc M.Sc degree in applied matematics from Zaedan Branc, Teran - Science Researc Branc, IAU, respectively, is P.D degree in applied matematics from science researc Branc, IAU, Teran, Iran in Se is a Assistant Prof. in te department of matematics at Islamic Azad University, Sar-e-rey, Teran in Iran. Se is researc interests include numerical solution of fuzzy integro-differential equations fuzzy differential equation similar topics.