COMPARISON RESULTS OF LINEAR DIFFERENTIAL EQUATIONS WITH FUZZY BOUNDARY VALUES

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1 Journal of Science and Arts Year 8, No. (4), pp , 08 ORIGINAL PAPER COMPARISON RESULTS OF LINEAR DIFFERENTIAL EQUATIONS WITH FUZZY BOUNDARY VALUES HULYA GULTEKIN CITIL Manuscript received: ; Accepted paper:.09.07; Published online: Abstract. In this paper, comparison results of linear differential equations with fuzzy boundary conditions are examined. It is solved the problem of Liu [7] by using the solution method of Gasilov et al s []. Here, the solution is same as the solutions (,) and (,) of Liu for the case of positive constant coefficient and the solution is same as the solutions (,) and (,) of Liu for the case of negative constant coefficient. If the boundary values are symmetric triangular fuzzy numbers, the value of the solution at any time is a symmetric triangular fuzzy number. Several examples are solved about the worked problems. Keywords: Fuzzy boundary value problems; Boundary value problems; Second order fuzzy differential equations. INTRODUCTION There are several approaches to studying fuzzy differential equations [, 6, 9, 3, 6, 8]. The first approach was the use of Hukuhara derivative. This approach has a drawback: the solution becomes fuzzier as time goes by [3, 0]. Also, Bede [4] proved that a large class of fuzzy boundary value problems have not a solution under the Hukuhara derivative concept. The strongly generalized derivative was introduced [] and studied in [3, 5, 8, 4, 9]. Recently, Khastan and Nieto [5] have found solutions for a large enough class of fuzzy boundary value problems with the strongly generalized derivative. The second approach generate the fuzzy solution from the crips solution. This approach can be three ways. The first way is the extension principle, where firstly, the initial value is taken a real constant and crips problem is solved. Then, the real constant in the solution is replaced with the initial fuzzy value [6, 7]. The second way is the concept of differential inclusion, where by taking an α-cut of initial value, the differential equation is converted to a differential inclusion and the solution is accepted as the α-cut of fuzzy solution [3]. In the third way, fuzzy solution is considered to be a set of crips problems []. In this paper, a investigation is made on comparison results of linear differential equations with fuzzy boundary conditions. It is solved the problem of Liu [7] by using the solution method of Gasilov et al s []. Giresun University, Faculty of Sciences and Arts, Department of Mathematics, Giresun, Turkey. hulyagultekin55@hotmail.com ISSN:

2 34 Comparison results of. PRELIMINARIES.. DEFINITION A fuzzy number is a function u: R [0,] satisfying the following properties: i) u is normal, ii) u is convex fuzzy set, iii) u is upper semi-continuous on R, iv) cl{xεr u(x) > 0} is compact where cl denotes the closure of a subset [4]. Let R F denote the space of fuzzy numbers... DEFINITION Let u R F. The α -level set of u, denoted [u] α, 0<α, is If α =0, the support of u is defined [u] α = {xεr u(x) α}. [u] 0 = cl{xεr u(x) > 0}. α The notation, [u] = [u,u α ] denotes explicitly the α -level set of u. We refer to α u and u as the lower and upper branches of u, respectively [5]. The following remark shows when [u α,u α ] is a valid α -level set..3. REMARK The sufficient and necessary conditions for [u α,u α ] to define the parametric form of a fuzzy number as follows: i) u α is bounded monotonic increasing (nondecreasing) left-continuous function on (0,] and right-continuous for α= 0, ii) uα is bounded monotonic decreasing (nonincreasing) left-continuous function on (0,] and right-continuous for α= 0, iii) uα uα, 0 α [4]..4. DEFINITION If A is a symmetric triangular numbers with supports a, a, the level sets of [A] α is [A] α = a + a a a a α, a α [7].

3 Comparison results of SECOND ORDER FUZZY LINEAR DIFFERENTIAL EQUATIONS Consider the fuzzy boundary value problem x (t) + a (t)x + a (t)x = f(t) (3.) x(0) = A, x(t) = B, (3.) where A and B are fuzzy numbers, A = a cc + a, B = b cc + b, a cc and b cc denote the crips parts of A and B; a and b denote the uncertain parts. The problem is splited to following two problems: ) Associated crips problem: x (t) + a (t)x + a (t)x = f(t), (3.3) x(0) = a cc, x(t) = b cc. (3.4) ) Homogeneous problem with fuzzy boundary values: x (t) + a (t)x + a (t)x = 0, (3.5) x(0) = a, x(t) = b. (3.6) The solution of the problem (3.)-(3.) is of the form x(t) = x cc (t) + x uu (t), where x cc (t) is the solution of the non-homogeneous crips problem (3.3)-(3.4) and x uu (t) is the solution of the homogeneous problem (3.5)-(3.6). Let be x (t) and x (t) linear independent solutions of the differential equations (3.5), s(t) = [x (t) x (t) ], M = x (0) x (0) x (l) x (l), W(t) = s(t)m = [w (t), w (t)]. Then, the solution of the fuzzy boundary problem (3.)-(3.) is x(t) = x cc (t) + w (t)a + w (t)b [ ]. A. THE CASE OF POSITIVE CONSTANT COEFFICIENTS Consider the fuzzy boundary value problem x (t) = λx(t) (3.7) x(0) = A, x(l) = B (3.8) ISSN:

4 36 Comparison results of a+a where, λ > 0. Boundary conditions A = a,, a and B = b, b+b triangular fuzzy numbers, the α -level sets of [A] α and [B] α are respectively. ) Crips problem: [A] α = a + a a [B] α = b + The solution of the differential equation (3.9) is Using the boundary conditions (3.0) a a α, a α, α, b α,, b are symmetric x (t) = λx(t), (3.9) x(0) = a+a b+b, x(l) =. (3.0) x cc (t) = c e λt + c e λt. x cc (0) = c + c = a+a, x cc (l) = c e λl + c e λl = b+b and from this c = b+b a+a e λl, c e λl e λl = a+a e λl b+b e λl e λl are obtained. Then the solution of the boundary value problem (3.9)-(3.0) is obtained as x cc (t) = b+b a+a e λl e λt + a+a e λl b+b e λt, e λl e λl e λl e λl x cc (t) = e λl e λl e λ(t l) + e λ(t l) a+a + e λt e λt b+b. ) The problem with fuzzy boundary values: x(0) = a a a a, 0, x (t) = λx(t), (3.), x(l) =, 0, (3.)

5 Comparison results of 37 x (t) = e λt, x (t) = e λt are linear independent solutions for the differential equation (3.). Then, s(t) = e λt e λt, M = e λl e λl. Since M = W(t) = s(t)m e λl e λl e λl e λl, W(t) = e λl e λl e λt e λt e λl e λl W(t) = e λl e [ λl e λ(t l) + e λ(t l) e λt e λt] is obtained, where w (t) = e λl e λl e λ(t l) + e λ(t l), w (t) = e λl e λl e λt e λt. Then, the solution of the fuzzy boundary value problem (3.)-(3.) is x uu (t) = e λl e λl e λ(t l) + e λ(t l) a a a a, 0, + e λt e λt we have Since the solution of the fuzzy boundary value problem (3.7)-(3.8) x(t) = x(t) = x cc (t) + x uu (t),, 0,. e λl e λl e λ(t l) + e λ(t l) a, a+a, a + e λt e λt b, b+b, b. Then, the α-cut of the solution x(t) is obtained as [x(t)] α = x α (t), x α (t), [x(t)] α = e λl e λl e λ(t l) + e λ(t l) a + a a α, a a a α + e λt e λt b + α, b α. (3.3) ISSN:

6 38 Comparison results of 3. PROPOSITION If b b a a e λl > 0, the solution [x(t)] α = x α (t), x α (t) of (3.7)-(3.8) is a valid α-level set. Proof. Using w (t) > 0, w (t) > 0 and fuzzy arithmetic, we have If x α(t) α be [x(t)] α = e λl e λl e λ(t l) + e λ(t l) a + + e λt e λt b b b + α, e λ(t l) + e λ(t l) a + e λt e λt b α. Then, x α (t) = e λl e λl e λ(t l) + e λ(t l) a + + e λt e λt b + α, x α (t) = e λl e λl e λ(t l) + e λ(t l) a 0, x α(t) α + e λt e λt b α. a a α a a α a a α a a α 0 and x α (t) x α (t) 0, [x(t)] α is a valid α-level set. Then, it must e λt a a e λl + e λt a a e λl 0. From this, e λt e λt b b a a e λl + a a e λl b b 0, Let be f(t) = e λt b b a a e λl + a a e λl b b. f(0) = a a e λl e λl > 0 f (t) = λ b b a a e λl e λt > 0,

7 Comparison results of 39 if set. b b a a e λl > 0. Then, if b b a a e λl > 0, [x(t)] α = x α (t), x α (t) is a valid α-level 3. PROPOSITION For any t [0, l], the solution [x(t)] α = x α (t), x α (t) of (3.7)-(3.8) is a symmetric triangle fuzzy number. Proof. Since x (t) = e λl e λl e λ(t l) + e λ(t l) a+a + e λt e λt b+b = x (t) and x (t) x α (t) = e λl e λl e λ(t l) + e λ(t l) a a ( α) + e λt e λt ( α) = x α(t) x (t), the solution [x(t)] α = x α (t), x α (t) of (3.7)-(3.8) is a symmetric triangle fuzzy number. 3.3 EXAMPLE Consider the fuzzy boundary value problem x (t) = x(t), t 0, (3.4) x(0) =, 3,, x = 3, 7, 4 (3.5) where, 3, α = + α, α, 3, 7, 4 α = 3 + α, 4 α. )Crips problem: x (t) = x(t) (3.6) x(0) = 3, x = 7 (3.7) The solution of the differential equation (3.6) is x cc (t) = c e t + c e t. Using the boundary conditions (3.7), we have c = 7 3 e e e, c = 3 e 7 e e. ISSN:

8 40 Comparison results of Then, the solution of the boundary value problem (3.6)-(3.7) is x cc (t) = 7 3 e e e e t + 3 e 7 e e e t, x cc (t) = e e 3) The problem with fuzzy boundary values: 7 (et e t ) + 3 e t e t. x (t) = x(t), (3.8) x(0) =, 0,, x =, 0,. (3.9) x (t) = e t, x (t) = e t are linear independent solutions for the differential equation (3.8). Then s(t) = [e t e t ], M = e e and Since W(t) = s(t)m. we have M = e e e, e W(t) = e e [e t e t ] e, e W(t) = e e e t + e t e t e t, where w (t) = e e e t + e t, w (t) = e e (e t e t ). Then, the solution of the fuzzy boundary value problem (3.8)-(3.9) is Since x uu (t) = e e e t e t, 0, + (et e t ), 0,. x(t) = x cc (t) + x uu (t), the solution of the fuzzy boundary value problem (3.4)-(3.5) is

9 Comparison results of 4 x(t) = e e e t e t, 3, + (et e t ) 3, 7, 4. Then, the α-level set [x(t)] α = x α (t), x α (t) of the soluton x(t) is [x(t)] α = e e e t e t + α, α + (et e t ) 3 + α, 4 α. Using w (t) > 0, w (t) > 0 and fuzzy arithmetic, we have [x(t)] α = e e e t e t + α + (et e t ) 3 + α, Then, x α (t) = x α (t) = e t e t α + e λt e λt 4 α. e e e e e t e t + α + (et e t ) 3 + α e t e t α + e λt e λt 4 α According to the proposition 3., since > e, the solution [x(t)] α = x α (t), x α (t) of the fuzzy boundary value problem (3.4)-(3.5) is a valid α-level set and the solution is a symmetric triangle fuzzy number. B. THE CASE OF NEGATIVE CONSTANT COEFFICIENTS Consider the fuzzy boundary value problem x (t) = λx(t) (3.0) x(0) = A, x(l) = B (3.) a+a where, λ > 0. Boundary conditions A = a,, a and B = b, b+b, b are symmetric triangular fuzzy numbers, the α -level sets of [A] α and [B] α are [A] α = a + a a a a α, a α, respectively. [B] α = b + α, b α, ISSN:

10 4 Comparison results of ) Crips problem: x (t) = λx(t), (3.) x(0) = a+a b+b, x(l) =. (3.3) The solution of the differential equation is Using the boundary conditions (3.3) x cc (t) = c cos λt + c sin λt. x cc (0) = c = a+a, x cc (l) = a+a cos λl + c sin λl = b+b, c = b+b a+a cos λl are obtained. Then, the solution of the boundary value problem (3.)-(3.3) is b+b x cc (t) = a+a cos λt + a+a cos λl sin λt, l uπ λ, n N = a+a siu λt cos λl cos λt + b+b siu λt. ) The problem with fuzzy boundary values: x(0) = a a a a, 0, x (t) = λx(t) (3.4), x(l) =, 0, (3.5) x (t) = cos λt, x (t) = sin λt are linear independent solutions for the differential equation (3.4). Then, and s(t) = cos λt sin λt, M = cos λl sin λl W(t) = s(t)m.

11 Comparison results of 43 Since M = 0 cos λl, W(t) = cos λt 0 sin λt cos λl sin λl sin λl = cos λt sin λt cos λl sin λl sin λt sin λl sin λ(l t) = sin λl sin λt sin λl is obtained, where w (t) = siu λ(l t), w (t) = siu λt. Thus, x uu (t) = siu λ(l t) a a a a, 0, + siu λt, 0,. Since x(t) = x cc (t) + x uu (t), we have x(t) = siu λ(l t) The α-cut of the solution x(t) is a, a+a [x(t)] α = x α (t), x α (t) siu λt b+b, a + b,, b. [x(t)] α = siu λ(l t) 3.4 PROPOSITION a + a a a a siu λt α, a α + b + α, b α. (3.6) t [0, π], the solution [x(t)] α = x α (t), x α (t) of (3.0)-(3.) is no longer a valid α-level set as t > λ cot a a cos λl. a a ISSN:

12 44 Comparison results of Proof. For t [0, π], using w (t) > 0, w (t) > 0 and fuzzy arithmetic, we have sin λ(l t) [x(t)] α = a + sin λl siu λ(l t) a a a a a sin λt α + sin λl α + siu λt b α. b b b + α, Using sin λ(l t) = cos λt sin λl sin λt cos λl, is obtained, where it must be c = If x α(t) α x α (t) = a + a a α cos λt + csin λt, x α (t) = a a a α cos λt + csin λt a a b+ α a+ α cos λl 0, x α(t) α, c = a a b α a α cos λl 0 and x α (t) x α (t) 0, [x(t)] α is a valid α-level set. Then,. From this, and since t [0, π] a a cos λt + a a cos λl sin λt 0. cot λt a a cos λl, a a t cot a a cos λl a a is obtained. Then, the solution [x(t)] α = x α (t), x α (t) of (3.0)-(3.) is no longer a valid α-level set as t > λ cot a a cos λl. a a

13 Comparison results of PROPOSITION For any t [0, l], the solution [x(t)] α = x α (t), x α (t) of (3.0)-(3.) is a symmetric triangle fuzzy number. Proof. Since sin λ(l t) = cos λt sin λl sin λt cos λl, x (t) = a + a a α cos λt + csin λt, x α (t) = a a a α cos λt + csin λt is obtained, where c = a a b+ α a+ α cos λl, c = a a b α a α cos λl. and Then, x (t) = a+a cos λt + b+b a+a cos λl x (t) x α (t) = ( α) a a cos λt + a a sin λt = x (t) cos λl sin λt = x α (t) x (t). The solution [x(t)] α = x α (t), x α (t) of (3.0)-(3.) is a symmetric triangle fuzzy number. 3.6 EXAMPLE Consider the fuzzy boundary value problem x (t) = x(t), t 0, π (3.7) x(0) =, 3,, x π = 3, 7, 4 (3.8) ) Crips problem: x (t) = x(t) (3.9) x(0) = 3, x π = 7 (3.30) ISSN:

14 46 Comparison results of The solution of the differential equation (3.9) is x cc (t) = c cos(t) + c sin(t). Using the boundary values (3.30), we have x cc (0) = c = 3, x cc π = c = 7. Then, the solution of the boundary value problem (3.9)-(3.30) is x cc (t) = 3 cos(t) + 7 sin(t). ) The problem with fuzzy boundary values: x (t) = x(t) (3.3) x(0) =, 0,, x =, 0, (3.3) x (t) = cos(t), x (t) = sin(t) are linear independent solutions for the differential equation (3.3). Then, and s(t) = [cos(t) sin(t)], M = 0 0 W(t) = s(t)m W(t) = [cos(t) sin(t)] 0 0 W(t) = [cos(t) sin(t)] are obtained, where w (t) = cos(t), w (t) = sin(t). Then, the solution of the fuzzy boundary value problem (3.3)-(3.3) is x uu (t) = cos(t), 0, + sin(t), 0,. Since x(t) = x cc (t) + x uu (t), the solution of the fuzzy boundary value problem (3.7)-(3.8) is x(t) = cos(t), 3, + sin(t) 3, 7, 4.

15 Comparison results of 47 Then, the α-level set [x(t)] α = x α (t), x α (t) of the soluton x(t) is [x(t)] α = cos(t) + α, α + sin(t) 3 + α, 4 α. Using w (t) > 0, w (t) > 0 for t 0, π and fuzzy arithmetic, we have [x(t)] α = cos(t) + α + sin(t) 3 + α, cos(t) α + sin(t) 4 + α. Then, x α (t) = cos(t) + α + sin(t) 3 + α, x α (t) = cos(t) α + sin(t) 4 + α. According to the proposition 3.4, when t > cot ( ), the solution [x(t)] α = x α (t), x α (t) of the fuzzy boundary value problem (3.7)-(3.8) is not a valid α-level set. Also, the solution is a symmetric triangle fuzzy number. 4. CONCLUSIONS In this paper, it is solved the problem of Liu [7] by using the solution method of Gasilov et al s []. Here, the solution is same as the solutions (,) and (,) of Liu for the case of positive constant coefficient and the solution is same as the solutions (,) and (,) of Liu for the case of negative constant coefficient. In Gasilov et al s paper [], the α-cuts of the solution x(t) defined in the form [x(t)] α = x α (t), x α (t) = x cc (t) + x uu,α (t) is not a valid α-level set, where x uu,α (t) = ( α) x uu (t), x uu (t). This, it can be see on the examples in the paper []. REFERENCES [] Agarwal, R.P., Lakshmikantham, V., Nieto, J.J., Nonlinear Anal., 7, 859, 009. [] Bede, B., Gal, S.G., Fuzzy Sets and Systems, 47, 385, 004. [3] Bede, B., Gal, S.G., Fuzzy Sets and Systems, 5, 58, 005. [4] Bede, B., Fuzzy Sets and Systems, 57, 986, 006. [5] Bede, B., Rudas, I.J., Bencsik, A.L., Inform. Sci., 77, 648, 007. [6] Buckley, J.J., Feuring, T., Fuzzy Sets and Systems, 0, 43, 000. [7] Buckley, J.J., Feuring, T., Fuzzy Sets and Systems,, 47, 00. [8] Chalco-Cano, Y., Roman-Flores, H., Chaos, Solitons & Fractals, 38,, 008. [9] Chalco-Cano, Y., Roman-Flores, H., Fuzzy Sets and Systems, 60, 57, 009. ISSN:

16 48 Comparison results of [0] Diamond, P., Kloeden P., Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 994. [] Gasilov, N.A., Amrahov, S.E., Fatullayev, A.G., Appl. Math. Inf. Sci., 5, 484, 0. [] Gasilov, N.A., Amrahov, S.E., Fatullayev, A.G., Linear differential equations with fuzzy boundary values, arxiv: [cs.na], 0. [3] Hüllermeier, E., Int. J. Uncertain. Fuzz., Knowledge-Based System, 5, 7, 997. [4] Khastan, A., Bahrami, F., Ivaz, K., Boundary Value Problems, 009, 39574, 009. [5] Khastan, A., Nieto, J.J., Nonlinear Analysis, 7, 3583, 00. [6] Lakshmikantham, V., Nieto, J.J., Dyn. Contin., Discrete Impuls Syst., 0, 99, 003. [7] Liu, H.K., International Journal of Computational and Mathematical Sciences, 5,, 0. [8] Nieto, J.J., Rodriguez-Lopez, R., Inform. Sci., 77, 456, 007. [9] Nieto, J.J., Khastan, A., Ivaz, K., Nonlinear Anal. Hybrid Syst., 3, 700,