Solving Systems of Fuzzy Differential Equation

Transcription

1 International Mathematical Forum, Vol. 6, 2011, no. 42, Solving Systems of Fuzzy Differential Equation Amir Sadeghi 1, Ahmad Izani Md. Ismail and Ali F. Jameel School of Mathematical Sciences, Universiti Sains Malaysia USM, Penang, Malaysia Abstract In this paper, a solution procedure for the solution of the system of fuzzy differential equations ẋ(t) =α(a I n )[t(a I n )+I n ] 1 x(t), x(0) = x 0 where A is real n n matrix which does not have eigenvalues on R and x 0 is a vector consisting of n fuzzy numbers is developed. In addition, the necessary and sufficient condition for the existence of fuzzy solution is presented. An example is provided to show the effectiveness of the proposed theory. Keywords: Fuzzy number, Fuzzy linear system, Fuzzy differential equations, Function of matrices 1 Introduction In order to solve an n n fuzzy linear system of equation, Friedman et al. in [14] proposed a general model which replaces the original n n fuzzy linear system by 2n 2n crisp function linear system. Authors such as Abbasbandy et al. in [1], [2], Allahviranloo et al. in [3], [4], [5] and Dehghan and Hashemi [11] have extended the work of Friedman. There have been application of fuzzy differential equation (FDEs) to model problems where the degree of ambiguity is high [19]. FDEs can be studied by several approaches. The Hukuhara differentiability for fuzzy number valued functions was the first approach which has been utilized. Fuzzy differential equations were first formulated by Kaleva [18] and Seikkala [22] in time dependent form. A very general formulation of a fuzzy first-order initial value problem, has been given by Buckley and Feuring [9]. In their formulas, first the crisp solution is found, fuzzified and then it is compared to see whether it satisfies the FDEs. Also, Puri and Ralescu [21] have discussed several different methods. The 1 Corresponding author ( address: sadeghi.usm@gmail.com)

2 2088 A. Sadeghi et al existence and uniqueness of the solution of a fuzzy differential equation were investigated in [21]. Pearson [20] using complex number representation has studied linear fuzzy differential equations of the form ẋ(t) = Aẋ(t) where A is a real n n matrix and the initial condition x(0) is a vector of n fuzzy numbers. Otadi et. al. [19] have investigated the solution of a special system of linear homogenous FDEs of the form ẋ(t) = Aẋ(t) + Bx(t), with initial condition x(0) = x 0 where A and B are real n n matrix and the initial condition x 0 is a vector of fuzzy numbers. The embedding method was used to solve the system. Fard [13] has extended this approach and has solved non-homogenous FDEs of the form ẋ(t) =Aẋ(t)+Bx(t)+f(t), with initial condition x(0) = x 0 by using variational iteration method. We consider the initial problem ẋ(t) =α(a I n )[t(a I n )+I n ] 1 x(t), x(0) = x 0 where A is an n n matrix that has non-negative eigenvalues in real axis. This initial value problem was introduced by Davis and Higham in [10]. In our paper, x 0 is fuzzified and the embedding method is used to solve the fuzzified initial value problem. By using some special functions of matrix, the fuzzy solution of fuzzy differential equation is obtained. Furthermore, the necessary and sufficient condition for the existence of fuzzy solution x(t) is presented. The outline of the paper is as follows: In Section 2 we will present basic concepts. In Section 3 we will discuss the necessary and sufficient condition for the existence of solution of FDEs. A numerical example will be given in Section 4 and the conclusions will be presented in Section 5. 2 Preliminaries In this section, several basic concepts of fuzzy system, fuzzy differential equation and function of matrices are recalled. 2.1 Basic definitions A nonempty subset A of R is called convex if and only if (1 ξ)x + ξy A for every x, y A and ξ [0, 1] [18]. Definition 2.1: [18] A fuzzy number is a function u : R [0, 1] satisfying the following properties:

3 Systems of fuzzy differential equation u is normal, i.e. x 0 R with u(x 0 )=1; 2. u is a convex fuzzy set i.e u(ξx +(1 ξ)y) min{u(x),u(y)}, x, y R,ξ [0, 1]; 3. u is upper semi-continuous on R; 4. {x R : u(x) > 0} is compact, where A denotes the closure of A. Following [5], the family of fuzzy numbers will be denoted by E. For 0 <r 1, [u] r denotes {x R : u(x) >r} and[u] 0 denotes {x R : u(x) > 0}. It should be noted that for any 0 r 1, [u] r is a bounded closed interval. For u, v E and λ R, the sum u + v and the product λu can be defined by [u + v] r =[u] r +[v] r,[λu] r = λ[u] r,r [0, 1], where [u] r +[v] r means the addition of two intervals of R and [u] r means the product between a scalar and a subset of R. Arithmetic operation of arbitrary fuzzy numbers u =(u(r), u(r)), v =(v(r), v(r)) and λ R, can be defined as [14]: u = v iff u(r) = v(r) and u(r) = v(r), u + v =(u(r)+v(r), u(r)+v(r)), u v =(u(r) v(r), u(r) v(r)), (λu(r),λu(r)), λ 0, λu = (λu(r),λu(r)), λ < 0. (2.1) Note that crisp number σ is simply represented by u(r) =u(r) =σ, 0 r 1. In the following we recall some definitions concerning fuzzy differential equations. Definition 2.2: A mapping f : T E for some interval T R is called a fuzzy process. Therefore, its r-level set can be written as follows [13], [f(t)] r =[f r (t),f r +(t)] r, t T,r [0, 1]. (2.2) Definition 2.3: [18] A function f : T E is said to be Hukuhara differentiable at t 0 T, if there exists an element f (t 0 ) E such that for all h>0 sufficiently small. In other words there exist f(t 0 + h)hf(t 0 ),f(t 0 )Hf(t 0 h) such that f(t 0 + h)hf(t 0 ) f(t 0 )Hf(t 0 h) lim = lim h 0 + h h 0 + h = f (t 0 ) (2.3)

4 2090 A. Sadeghi et al Where w = uhv is Hukuhara difference which for u, v E is defined by u = v + w. If f : T E be Hukuhara differentiable then, the boundary function f+ r (t) and f + r (t) are (Seikkala) differentiable [22] and [f (t)] r =[(f r (t)), (f r + (t)) ] r, t T,r [0, 1]. (2.4) 2.2 Fuzzy linear systems Definition 2.4: [14] The n n system of linear equations n a ij x j = y i, i =1,...,n. (2.5) j=1 where the coefficient matrix A =(a ij ), 1 i, j n, is a crisp n n matrix and y i E,1 i n, is called a fuzzy system of linear equations (FLE). Two crisp n n linear systems for all i can be extended to an 2n 2n crisp linear system as follows: If a ij 0 for i, j =1,...,2n, then we can put s i,j = s i+n,j+n = a ij. If a ij < 0 for i, j =1,...,n, then we can put s i,j+n = s i+n,j = a ij. In other words, S 1,S 2, A = S 1 S 2, S 1 + S 2 = A =( a ij ), i,j =1,...,n, and x 1 x 1 x x = 2., x = x 2., y = y 2., y = y 2. y 1 y 1 x n x n y n y n By using matrix notation, an 2n 2n linear system of equation can be obtained as where S = ( S 1 S 2 S 2 S 1 ), X = ( x x ), and Y = ( y y ). SX = Y. (2.6) Theorem 2.1: [14] The unique solution X of Equation (2.6) is a fuzzy vector for arbitrary Y if and only if S 1 is nonnegative, i.e, (s ij ) 1 0. Theorem 2.2: [14] The matrix S is nonsingular if and if only the matrices S 1 S 2 and S 1 + S 2 are both singular.

5 Systems of fuzzy differential equation Function of matrix Suppose f(z) is an analytic function on a closed contour Γ which encircles spec(a), where spec(a) denote the set of eigenvalues of matrix A. A function of matrix can be defined by using Cauchy integral definition f(a) = 1 f(z)(zi A) 1 dz (2.7) 2πi Γ The entries of (zi A) 1 are analytic on Γ and f(a) is analytic in a neighborhood of spec(a) [15]. Matrix exponential, matrix logarithm and matrix pth root are examples of function of matrices and are defined as follows. Definition 2.5: (Exponential) [15] The exponential of A C n n, denoted by e A or exp(a), is defined by exp(a) = A i i=0. i! Definition 2.6: (Principal logarithm) [17] Suppose A is an n n matrix. A unique matrix X which satisfies the nonlinear matrix equation e X A = 0 such that the eigenvalues of X lie in the strip {z : π <Im(z) <π} is called the principal logarithm of A, and is denoted by X = log A. Definition 2.7: (principal pth root) [24] Let A be an n n matrix. A unique matrix X that satisfy the nonlinear matrix equation X p A = 0 such that all of eigenvalues of X lie in the segment {z : π/p < arg(z) <π/p}, is called the principal pth root of A and is denoted by X = A 1/p. 3 System of fuzzy ordinary differential equation In this section, the homogenous system of fuzzy differential equation ẋ(t) =α(a I n )[t(a I n )+I n ] 1 x(t), (3.1) x(0) = x 0 where ẋ(t) = dx, will be solved. In the following theorem the general solution of the dt nonhomogeneous equation is presented. Theorem 3.1: Let A C n n have non-negative real eigenvalues and α R. The

6 2092 A. Sadeghi et al non-homogeneous initial value ordinary differential equation problem ẋ(t) =α(a I n )[t(a I n )+I n ] 1 x(t)+f(t, x), (3.2) x(0) = x 0 has a unique solution x(t) =[t(a I n )+I n ] α x 0 + t 0 [ (t(a In )+I n )(s(a I n )+I n ) 1] α f(s, x)ds (3.3) Proof: For proving this theorem we utilize the following facts [16] 1. t 0 (A I n)[s(a I n )+I n ] 1 ds = log [t(a I n )+I n ], 2. log A commutes with [t(a I n )+I n ] 1. By considering equation (3.2), we can multiply the factor e α log[t(a In)+In] to both sides of equation (3.2) as follows (ẋ(t) α(a In )[t(a I n )+I n ] 1 x(t) ) e α log[t(a In)+In] α log[t(a In)+In] = f(t, x)e (3.4) Then we have d ( e α log[t(a I n)+i n] x(t) ) = f(t, x)e α log[t(a In)+In] (3.5) dt Integrating both sides, we have s 0 d dt Hence, it can be obtained that ( e α log[t(a I n)+i n] x(t) ) s dt = f(t, x)e α log[t(a In)+In] dt (3.6) e α log[s(a In)+In] x(s) x(0) = s 0 0 f(t, x)e α log[t(a In)+In] dt (3.7) x(s) =e α log[s(a In)+In] x(0) + Therefore, it can be obtained x(t) =[t(a I n )+I n ] α x 0 + t 0 s 0 f(t, x)e α log[s(a In)+In] e α log[t(a In)+In] dt (3.8) [ (t(a In )+I n )(s(a I n )+I n ) 1] α f(s, x)ds.

7 Systems of fuzzy differential equation 2093 For t = 1 we can obtain 1 x(1) = A α x 0 + A α [s(a I n )+I n ] α f(s, x)ds. 0 It should be emphasized that for homogenous case (f(x, t) = 0) Theorem 3.1 can be replaced by Theorem 2 in [10]. As we mentioned in Section 2 the derivative ẋ(t) ofa fuzzy process x(t) can be defined as ẋ(t, r) =(ẋ(t, r), ẋ(t, r)), 0 r 1 (3.9) provided that is equation defines a fuzzy number [22]. A is an n n real matrix and a vector made up of n fuzzy numbers described the initial condition x 0. We study a specific linear problem, where the parameters known and the initial condition of the system is fuzzy. It is well known that there is no inverse element for an arbitrary fuzzy number u E. In other words, there exists no element v E such that u + v = 0. In fact, for all non-crisp fuzzy number u E we have u +( u) 0 [13,19]. Thus, the system of linear fuzzy differential equation (3.1) cannot be equivalently replaced by the system of linear fuzzy equations equations ẋ(t, r) =α(a I n )[t(a I n )+I n ] 1 x(t, r), (3.10) x(0,r)=x 0 which had been considered. Consider the system of linear fuzzy differential equations (3.1). We are going to transform its n n system with coefficient matrix A into 2n 2n system with coefficient matrix S. Define matrix S =(s ij ) where can be determined as follows If a ij 0 for i, j =1,...,2n, then we can put s i,j = s i+n,j+n = a ij. If a ij < 0 for i, j =1,...,2n, then we can put s i,j+n = s i+n,j = a ij. while all the remaining s ij are taken zero. Using matrix notation the following system can be obtained ẋ(t, r) =α(s I 2n )[t(s I 2n )+I 2n ] 1 x(t, r) (3.11) x(0,r)=x 0

8 2094 A. Sadeghi et al where S and I 2n are 2n 2n matrices, and ẋ 1 (t, r) x 1 (t, r).. ẋ ẋ(t, r) = n (t, r) x, x(t, r) = n (t, r), x 0 = ẋ 1 (t, r) x 1 (t, r).. ẋ n (t, r) x n (t, r) x 01. x 0n x 01. x 0n. (3.12) Thus, the solution of fuzzy differential equation is x(t, r) =[t(s I 2n )+I 2n ] α x 0 (3.13) However, x(t, r) may still not be the appropriate fuzzy vector. The following theorem provides necessary and sufficient conditions for x(t, r) be a fuzzy vector. First we will show the matrix t(s I 2n )+I 2n must be positive definite. Theorem 3.2: [15] An arbitrary matrix is positive definite if and only if all its eigenvalues are positive. Theorem 3.3: If t((s 1 S 2 ) I n )+I n and t((s 1 + S 2 ) I n )+I n for t>0 are positive definite matrix then t(s I 2n )+I 2n is a symmetric positive definite matrix. Proof: Clearly t(s I 2n )+I 2n is a symmetric matrix. Suppose that t((s 1 S 2 ) I n )+I n and t((s 1 + S 2 ) I n )+I n are positive definite matrix. Let λ>0 be a eigenvalue of ( ) ts 1 +(1 t)i n ts 2 t(s I 2n )+I 2n =. (3.14) ts 2 ts 1 +(1 t)i n Then, there exist x 1, x 2 0 such that ( )( ts 1 +(1 t)i n ts 2 ts 2 ts 1 +(1 t)i n x 1 x 2 ) ( = λ x 1 x 2 ) (3.15) Thus, we have [ts 1 +(1 t)i n ] x 1 + ts 2 x 1 = λx 1 (3.16) ts 2 x 1 +[ts 1 +(1 t)i n ] x 2 = λx 2

9 Systems of fuzzy differential equation 2095 By addition and subtraction of (3.16), it can be obtained ts 1 (x 1 + x 2 ) ts 2 (x 1 + x 2 )+(1 t)(x 1 + x 2 )=λ(x 1 + x 2 ) ts 1 (x 1 x 2 )+ts 2 (x 1 x 2 )+(1 t)(x 1 x 2 )=λ(x 1 x 2 ) (3.17) or t(s 1 S 2 )(x 1 + x 2 )+(1 t)(x 1 + x 2 )=λ(x 1 + x 2 ) t(s 1 + S 2 )(x 1 x 2 )+(1 t)(x 1 x 2 )=λ(x 1 x 2 ) (3.18) Clearly, λ is the eigenvalue of (S 1 S 2 ) and (S 1 + S 2 ), which proves the theorem. Now, it is known that A α can be obtained by using A α = exp(α log A), where the log is the principal logarithm. For α =1/p, when p an integer, we have A 1/p. It should be mentioned that by using commutativity relation, it can be written (A α ) p = exp(α log A) p = exp(pα log A) = exp(log A) =A Therefore, A α is some pth root of A. To determine which root it is, we need to find its spectrum. The eigenvalues of A α are of the form e 1 p log λ, where λ is an eigenvalue of A. Now log λ = x + iy with y ( π,π), and so e 1 p log λ = e x/p e iy/p lies in the segment {z : π/p < arg(z) <π}. The spectrum of A α is therefore precisely that of A 1/p, so these two matrices are one and the same [16]. It is known that if positive definite matrix A has positive entries then A α has positive entries. Therefore, by using this point we have the fuzzy solution for fuzzy differential equation. 4 Numerical example Consider the equation ẋ(t) =α(a I n )[t(a I n )+I n ] 1 x(t) when ( ) 4 0 A = 1 2 (4.1) and the initial values are x 01 2 and x This can be done by setting, for example, ( ) ( ) x 01 (1 + r, 3 r) x(0; r) = = (4.2) (2r, 6 4r) x 02

10 2096 A. Sadeghi et al We have the following equation ( )[ ( ) ( )] ẋ(t) =α t + x(t) (4.3) Therefore, by using the embedding method (4.3) can be transformed to 1 ẋ 1 (t, r) x 1 (t, r) ẋ 2 (t, r) ẋ 1 (t, r) = α t x 2 (t, r) x 1 (t, r) (4.4) ẋ 2 (t, r) x 2 (t, r) Now, by integrating from this system of linear differential equation we can obtain α x 1 (t, r) x 1 (0,r) x 2 (t, r) x 1 (t, r) = exp log t x 2 (0,r) (4.5) x 1 (0,r) x 2 (t, r) x 2 (0,r) By utilizing the property of matrix exponential function A p = exp(p log A) we have α x 1 (t, r) r x 2 (t, r) x 1 (t, r) = t r (4.6) r x 2 (t, r) r For specific case t = 1 the (4.6) becomes to x 1 (1,r) x 2 (1,r) x 1 (1,r) = r 2r 3 r. x 2 (1,r) r It remain to compute αth root of a matrix. For t = 0.1, 0.2,...,1 the approximate solution for α =1/7 and α =3/5 are obtained in Tables 1 to 4. α 5 Conclusion A technique for approximating the solution of a time dependent system of fuzzy linear differential equations is presented. The necessary and sufficient condition for the existence of a fuzzy solution is proposed. The original fuzzy system of differential equations with matrix coefficient A is transformed by a 2n 2n crisp linear system of differential equations with matrix coefficient S. We have illustrated the proposed technique by an example.

11 Systems of fuzzy differential equation 2097 Table 1: The approximate solution x 1 (t, r) for α =1/7 t x 1 (t, r) x 1 (t, r) r r r r r r r r r r r r r r r r r r r r References [1] S. Abbasbandy, R. Ezzati, A. Jafarian, LU decomposition method for solving fuzzy system of equations, Appl. Math. Comput. 172 (2006) [2] S. Abbasbandy, A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric positive definite system of linear equations, Appl. Math. Comput. 171 (2005) [3] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Appl. Math. Comput. 155 (2004) [4] T. Allahviranloo, Successive overrelaxation iterative method for fuzzy system of linear equations, Appl. Math. Comput. 162 (2005) [5] T. Allahviranloo, M. Afshar Kermani, Solution of a fuzzy system of linear equation, Appl. Math. Comput. 175 (2006) [6] E. J. Allen, J. Baglama, and S. K. Boyd. Numerical approximation of the product of the square root of a matrix with a vector. Linear Algebra Appl, 310 (2000) [7] E.J. Allen, Stochastic differential equations and persistence time of two interacting populations, Dynamics Continuous Discrete Impul. Syst. 5 (1999) [8] B. Bede, Note on Numerical solutions of fuzzy differential equations by predictor-corrector method, Information Sciences, 178 (2008)

12 2098 A. Sadeghi et al Table 2: The approximate solution x 2 (t, r) for α =1/7 t x 2 (t, r) x 2 (t, r) r r r r r r r r r r r r r r r r r r r [9] J.J. Buckley and T. Feuring Fuzzy differential equations, Fuzzy Sets Syst, 110 (2000) [10] P. I. Davies and N. J. Higham, Computing f(a)b for matrix functions f. QCD and Numerical Analysis III, Springer-Verlag, Berlin, 47 (2005) [11] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems, Appl. Math. Comput. 175 (2006) [12] D. Dubois, H. Prade, Towards fuzzy differential calculus: Part3, differentiation, Fuzzy Sets Syst, 8 (1982) [13] O. S. Fard, An Iterative Scheme for the Solution of Generalized System of Linear Fuzzy Differential Equations, World Applied Sciences Journal, 7 (2009) [14] M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets Syst, 96 (1998) [15] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, [16] N. J. Higham. Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, [17] R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1994.

13 Systems of fuzzy differential equation 2099 Table 3: The approximate solution x 1 (t, r) for α =3/5 t x 1 (t, r) x 1 (t, r) r r r r r r r r r r r r r r r r r r r r [18] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst, 24 (1987) [19] M. Otadi, S. Abbasbandy and M. Mosleh, System of linear fuzzy differential equations, First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran, [20] D.W. Pearson, A property of linear fuzzy differential equations, Appl. Math. Lett 10 (1997) [21] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) [22] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets Syst, 24 (1987) [23] W.D. Sharp, E.J. Allen, Stochastic neutron transport equations for rod and plane geometries, Ann. Nuclear Energy 27 (2000) [24] M. I. Smith, A Schur algorithm for computing matrix pth root, SIAM J. Matrix Anal. Appl., 24 (2003), pp Received: March, 2011

14 2100 A. Sadeghi et al Table 4: The approximate solution x 2 (t, r) for α =3/5 t x 2 (t, r) x 2 (t, r) r r r r r r r r r r r r r r r r r r r r