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Numerical Solutions of Fuzzy Differential Equations by Using Hybrid Methods

Numerical Solutions of Fuzzy Differential Equations by Using Hybrid Methods Fuzzy Inf. Eng. (2012) 4: 445-455 DOI 10.1007/s12543-012-0126-9 ORIGINAL ARTICLE Numerical Solutions of Fuzzy Differential Equations by Using Hybrid Methods P. Prakash · V. Kalaiselvi Received: 26 November 2010/ Revised: 10 August 2012/ Accepted: 18 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we study the numerical solution of fuzzy differential equa- tions by using hybrid Euler method and hybrid predictor-corrector method. These methods are used to increase the accuracy and the computing speed. Also examples are presented to illustrate the computational aspects of the above methods. Keywords Hybrid method · Fuzzy Cauchy problem · Growth equations · Euler method · Adams-Bashforth method · Adams-Moulton method · Predictor-corrector method 1. Introduction The theory of fuzzy differential equations has been investigated extensively in the original formulation as well as in an alternative framework, which leads to ordinary differential equations. The concept of differential equations in a fuzzy environment was first formulated by Kaleva [10]. The fuzzy differential equation is a particularly important topic from the theoretical [4, 5, 8, 11, 17] as well as the applied point of view [12, 14, 15], for example, in population models [9], in golden mean [6], syn- chronize hyperchaotic systems [18] and medicine [13]. In the last few years, the numerical solution of fuzzy differential and hybrid fuzzy differential equations has been studied by several authors [1-3]. Recently, the numerical solution of hybrid fuzzy differential equations by predictor-corrector method has been studied in [16]. The motivation of this paper is the numerical methods for solving the fuzzy differen- tial equations with the property of growth models. Numerical experiments show that P. Prakash () Department of Mathematics, Periyar University, Salem-36011, India email: pprakashmaths@gmail.com V. Kalaiselvi () Department of Mathematics, Hindustan University, Padur-603103, India email: kalaiselvi111@gmail.com 446 P. Prakash· V. Kalaiselvi (2012) the proposed methods for solving the fuzzy differential equations of the form y = f (t, y) = g(t, y)· y, t ∈ [t , T ], y(t ) = y , 0 0 where f and g are continuous functions, obviously increase the computing speed and accuracy. The structure of the paper is organized as follows. In Section 2, we introduce some basic definitions which will be used later in the paper. In Section 3 and 4, we apply hybrid Euler method and multi-step method for solving the exponential models of growth, then the proposed method is illustrated by solving examples in Section 5, and the conclusion is drawn in Section 6. 2. Preliminaries 2.1. Notations An arbitrary fuzzy number is represented by an ordered pair of functions (u(α), u(α)), 0 ≤ α ≤ 1, that satisfies the following requirements: 1) u(α) is a bounded left continuous nondecreasing function over [0, 1]. 2) u(α) is a bounded left continuous nonincreasing function over [0, 1]. 3) u(α) ≤ u(α), 0 ≤ α ≤ 1. Let E be the set of all upper semi-continuous normal convex fuzzy numbers with bounded α-level intervals. If v ∈ E, then the α-level set [v] = {s|v(s) ≥ α},0<α ≤ α α 1, is a closed bounded interval which is denoted by [v] = [v , v ]. Let I be a real interval. A mapping y : I → E is called a fuzzy process and its α α α α-level set is denoted by [y(t)] = [y (t), y (t)], t ∈ I,α ∈ (0, 1]. Triangular fuzzy numbers are those fuzzy sets U ∈ E which are characterized by l c r 3 l c r 0 l r an ordered triple (x, x , x ) ∈ R with x ≤ x ≤ x such that [U] = [x, x ] and 1 c [U] = {x }, then α c c l c r c [U] = [x − (1−α)(x − x ), x + (1−α)(x − x )] (1) for anyα ∈ [0, 1]. Definition 2.1 [7] The supremum metric d on E is defined by α α d (U, V) = sup{d ([U] , [V] ): α ∈ [0, 1]} ∞ H and (E, d ) is a complete metric space. Definition 2.2 [7] A mapping F : I → E is Hukuhara differentiable at t ∈ I ⊆ R if for some h > 0 the Hukuhara differences F(t +Δt) ∼ F(t ), F(t ) ∼ F(t −Δt), 0 0 h 0 0 h 0 exist in E for all 0 < Δt < h and if there exists an F (t ) ∈ E such that 0 0 lim d ((F(t +Δt) ∼ F(t ))/Δt, F (t )) = 0 ∞ 0 h 0 0 Δt→0 Fuzzy Inf. Eng. (2012) 4: 445-455 447 and lim d ((F(t ) ∼ F(t −Δt))/Δt, F (t )) = 0, ∞ 0 h 0 0 Δt→0 the fuzzy set F (t ) is called the Hukuhara derivative of F at t . 0 0 α α Recall that U ∼ V = W ∈ E are defined on level sets, where [U] ∼ [V] = h h [W] for all α ∈ [0, 1]. By consideration of definition of the metric d , all the level set mappings [F(.)] are Hukuhara differentiable at t with Hukuhara derivatives [F (t )] for each α ∈ [0, 1] when F : I → E is Hukuhara differentiable at t with 0 0 Hukuhara derivative F (t ). Definition 2.3 [7] The fuzzy integral y(t)dt, 0 ≤ a ≤ b ≤ 1, is defined by b b b y(t)dt = y (t)dt, y (t)dt , a a a provided the Lebesgue integrals on the right exist. Remark 2.1 If F : I → E is Hukuhara differentiable and its Hukuhara derivative F is integrable over [0, 1], then F(t) = F(t )+ F (s)ds for all values of t , t, where 0 ≤ t ≤ t ≤ 1. 0 0 Remark 2.2 The Seikkala derivative y (t) of a fuzzy process y is defined by α α  α [y (t)] = [(y ) (t), (y ) (t)],α ∈ (0, 1] provided the equation defines a fuzzy number y (t) ∈ E. Remark 2.3 If y : I → E is Seikkala differentiable and its Seikkala derivative y is integrable over [0, 1], then y(t) = y(t )+ y (s)ds for all values of t , t, where t , t ∈ I. 0 0 2.2. A Fuzzy Cauchy Problem Consider the first order fuzzy differential equation y = f (t, y), where y is a fuzzy function of t, f (t, y) is a fuzzy function of crisp variable t and fuzzy variable y and y is Hukuhara of Seikkala fuzzy derivative of y. If an initial value is given, a fuzzy Cauchy problem of first order is y = f (t, y), t ∈ [t , T ], (2) y(t ) = y . 0 0 Sufficient conditions for the existence of unique solution to Equation (2) are: • f is continuous, • Lipschitz condition d ( f (t, x), f (t, y)) ≤ Ld (x, y), L > 0. ∞ ∞ 448 P. Prakash· V. Kalaiselvi (2012) Single step and multi-step method to solve fuzzy Cauchy problem are as follows: • Euler method: α α y = y + h· f (t, y ), i i i+1 i (3) α α y = y + h· f (t, y ), i = 0, 1, 2,··· . i i i+1 i • Predictor-corrector method: 23 4 α 5 α α α α y = y + h f (t, y )− f (t , y )+ f (t , y ) , i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 (4) 23 α 4 5 α α α α y = y + h f (t, y )− f (t , y )+ f (t , y ) , i = 2, 3,··· i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 and 5 2 1 α α α α α y = y + h f (t , y )+ f (t, y )− f (t , y ) , i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 (5) 5 α 2 α 1 α α α y = y + h f (t , y )+ f (t, y )− (t , y ) , i = 1, 2,··· . i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 3. Hybrid Euler Method Consider the fuzzy differential equation of the form, y = f (t, y) = g(t, y)· y, t ∈ [t , T ], (6) y(t ) = y . 0 0 n n n n Here f ∈ C[[t , T ] × E , E ], g ∈ C[[t , T ] × E , E ]. (6) can be replaced by an 0 0 equivalent system α  α α α α α (y ) (t) = f (t, y(t)) = g (t, y(t))· y (t), y (t ) = y , (7) α α α α α (y ) (t) = f (t, y(t)) = g (t, y(t))· y (t), y (t ) = y . The above system can be solved by the following Euler’s method, α α α y = y + h· f (t, y ), i i i+1 i (8) α α y = y + h· f (t, y ), i = 0, 1, 2,··· , i i i+1 i where, h = t − t . By substituting z = In(y), (7) becomes i+1 i α  α (z ) (t) = g (t), (9) α α (z ) (t) = g (t). According to the Euler’s Method, (9) can be solved as α α α z = z + h· g (t ), i+1 i (10) α α α z = z + h· g (t ), i = 0, 1, 2,··· . i+1 i After replacing z = In(y), (10) becomes α α α y = y · exp(h· g (t, y )), i i i+1 i (11) α α α y = y · exp(h· g (t, y )), i = 0, 1, 2,··· . i i i+1 i Fuzzy Inf. Eng. (2012) 4: 445-455 449 On the other hand, we can achieve the numerical solution of Equations (7) and (9) α α α α by the Taylor series expansion for y (t), y (t) and z (t), z (t) around t with a step size of h as α 1 α α 2 α  3 y (t + h) = y (t )+ hf (t )+ h (y ) (t )+ o(h ), i i i i (12) α α α 1 2  3 y (t + h) = y (t )+ h f (t )+ h (y ) (t )+ o(h ), i = 0, 1, 2,··· i i i i and α α α 1 2 α  3 z (t + h) = z (t )+ hg (t )+ h (z ) (t )+ o(h ), i i i i (13) α α α α 1 2  3 z (t + h) = z (t )+ hg (t )+ h (z ) (t )+ o(h ), i = 0, 1, 2,··· . i i i i By comparing the Equations (12) with (8) and (13) with (10), we can get the local truncation errors , and, as α α 1 2 α = y (t + h)− y ≈ h (y ) (t ), i i 1 2 i+1 (14) α α α 1 2 = y (t + h)− y ≈ h (y ) (t ) 1 i i i+1 and 1 1 α α 2 α  2 α = z (t + h)− z ≈ h (z ) (t ) = h In(y ) (t ), i i i 2 2 i+1 (15) α α 1 2 α  1 2 α = z (t + h)− z ≈ h (z ) (t ) = h In(y ) (t ). i i i i+1 2 2 Then, the main part of local truncation error , can be obtained from (15), namely α α α 1 2 α = y (t + h)− y ≈ y [exp( h In(y ) (t ))− 1], i i 2 2 i+1 i+1 (16) α α α α 1 2 = y (t + h)− y ≈ y [exp( h In(y ) (t ))− 1]. 2 i i i+1 i+1 Here we use the differential expressions of α  α α α 2 In(y ) (t ) = [In(y )− 2In(y )+ In(y )]/h , i i−1 i−2 α α α α In(y ) (t ) = [In(y )− 2In(y )+ In(y )]/h i i−1 i−2 and α  α α α 2 (y ) (t ) = [y − 2y + y ]/h , i i−1 i−2 α  α α α 2 (y ) (t ) = [y − 2y + y ]/h . i i−1 i−2 From (12)-(16), we can find that, if the round-off error was ignored, the local trun- cation error based on the numerical method (11) is zero for the growth equations such as f (t, y) = c· y. But the Euler’s method (8) has the great error and even diverges for such cases. However, the numerical method (11) fails to solve (7) if its solutions y(t) include both positive and negative values, whereas the Euler’s method may succeed to obtain the numerical solution accurately. We can see from (14) and (16) that the numerical method (8) has less error for some part solutions of y(t ), but (11) has less error for other part solutions. Therefore, to overcome the shortcomings of (8) and 450 P. Prakash· V. Kalaiselvi (2012) (11) and decrease the local truncation error at each step, a hybrid method is proposed by merging them, namely, α α ⎪ y · exp(h· g (t, y )), if | | > | |, i i ⎨ 1 2 y = (17) ⎪ α α i+1 y + h· f (t, y ), if otherwise i i and α α y · exp(h· g (t, y )), if | | > | |, i i 1 2 α i y = (18) ⎪ α i+1 ⎪ α y + h· f (t, y ), if otherwise. i i 4. Hybrid Multi-step Method To decrease the local truncation error of the Euler’s method, the multi-step methods were proposed. The 3-step Adams-Bashforth and 2-step Adams-Moulton methods are given by 23 α 4 5 α α α y = y + h[ f (t, y )− f (t , y )+ f (t , y )], i i i−1 i−1 i−2 i−2 12 3 12 i+1 i (19) α α α α 23 4 α 5 y = y + h[ f (t, y )− f (t , y )+ f (t , y )] i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 and α α 5 α 2 α 1 y = y + h[ f (t , y )+ f (t, y )− f (t , y )], i+1 i+1 i i i−1 i−1 12 3 12 i+1 i (20) α α α α 5 2 1 α y = y + h[ f (t , y )+ f (t, y )− f (t , y )]. i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 The main term of the local truncation error , for (19) and (20) is 4 α 4 = Ch (y ) (t ), i−2 (21) 4 4 = Ch (y ) (t ). 1 i−2 Similarly, according to the operational process of (11), a set of new methods based on Adams-Bashforth and Adams-Moulton formula can be obtained as α α 23 α 4 5 α y = y · exp(h[ g (t, y )− g (t , y )+ g (t , y )]), i i i−1 i−1 i−2 i−2 12 3 12 i+1 i (22) α α α α 23 4 α 5 y = y · exp(h[ g (t, y )− g (t , y )+ g (t , y )]) i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 and α α 5 α 2 α 1 α y = y · exp(h[ g (t , y )+ g (t, y )− g (t , y )]), i+1 i+1 i i i−1 i−1 12 3 12 i+1 i (23) α α 5 α 2 α 1 y = y · exp(h[ g (t , y )+ g (t, y )− g (t , y )]). i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 The main term of local truncation error  , for (22) and (23) is α 4 α 4 = y · exp(Ch (y ) (t )− 1), i−2 i+1 (24) α α 4 4 = y · exp(Ch (y ) (t )− 1). 2 i−2 i+1 Here, the coefficient C has the value 3/8 for Adams-Bashforth method and -1/24 for Adams-Moulton method. Implicit methods are usually both more accurate and more Fuzzy Inf. Eng. (2012) 4: 445-455 451 stable than explicit methods, but they are more difficult to apply. In order to take ad- vantage of the beneficial properties of the implicit multi-step methods while avoiding the difficulties inherent in solving the implicit equations, predictor-corrector methods are proposed. The most popular predictor-corrector methods are Adams-Bashforth- Adams-Moulton method, where Adams-Bashforth and Adams-Moulton methods are implemented as the predictor and corrector, respectively. Numerical experiments show that the predictor-corrector methods based on (22) and (23) can improve the accuracy and increase the computing speed for (6) obvi- ously. However, (22) and (23) also fail to fuzzy differential equations including both positive and negative solutions. Compared (21) with (24), we can find that (22) and (23) offer less error than (19) and (20) for some fuzzy differential equations, while the situation is reverse for other cases. Predictor: ⎪ 23 4 ⎪ α α y · exp h g (t, y )− g (t , y ) i i i−1 i−1 ⎪ i ⎪ 12 3 ⎪ 5 ⎪ + g (t , y ) , if | |> |, i−2 i−2 1 2 y = (25) 23 4 α i+1 ⎪ ⎪ α α y + h f (t, y )− f (t , y ) ⎪ i i i−1 i−1 ⎪ i 12 3 ⎪ 5 ⎩ + f (t , y ) , if otherwise. i−2 i−2 23 4 ⎪ α α α ⎪ y · exp h g (t, y )− g (t , y ) i i i−1 i−1 ⎪ 12 3 ⎪ 5 ⎪ α + g (t , y ) , if | | >  |, i−2 i−2 1 2 α ∗ y = (26) i+1 23 α 4 α α y + h f (t, y )− f (t , y ) ⎪ i i i−1 i−1 12 3 ⎪ 5 α + f (t , y ) , if otherwise. i−2 i−2 Corrector: ⎪ 5 2 ⎪ α ∗ α y · exp h g (t , y )+ g (t, y ) i+1 i i ⎪ i+1 ⎪ i ⎪ 12 3 ⎪ − g (t , y ) , if | |> |, ⎪ i−1 i−1 1 2 y = (27) 5 2 i+1 ⎪ α α ∗ α y + h f (t , y )+ f (t, y ) ⎪ i+1 i i ⎪ i+1 ⎪ i 12 3 ⎪ 1 α ⎩ − f (t , y ) , if otherwise. i−1 i−1 ⎪ 5 2 ⎪ α α ∗ α y · exp h g (t , y )+ g (t, y ) i+1 i i ⎪ i i+1 ⎪ 12 3 ⎪ 1 − g (t , y )]), if | | >  |, i−1 i−1 1 2 y = (28) i+1 5 α 2 α ⎪ ∗ y + h f (t , y )+ f (t, y ) ⎪ i+1 i i i i+1 12 3 ⎪ 1 ⎩ − f (t , y ) , if otherwise. i−1 i−1 12 452 P. Prakash· V. Kalaiselvi (2012) Here we use the differential expressions of α 4 α α α α α 4 In(y ) (t ) = [In(y )− 4In(y )+ 6In(y )− 4In(y )+ In(y )]/h , i i−1 i−2 i−3 i−4 α 4 α α α α α 4 In(y ) (t ) = [In(y )− 4In(y )+ 6In(y )− 4In(y )+ In(y )]/h i i−1 i−2 i−3 i−4 and α 4 α α α α α 4 (y ) (t ) = [y − 4y + 6y − 4y + y ]/h , i i−1 i−2 i−3 i−4 α α α α α α 4 4 (y ) (t ) = [y − 4y + 6y − 4y + y ]/h . i i−1 i−2 i−3 i−4 5. Examples Example 5.1 [12] Consider the fuzzy initial value problem y (t) = ty, t ∈ [−1, 1], √ √ √ (29) ⎪ l c r y(−1) = (y , y , y ) = ( e− 0.5, e, e+ 0.5). 0 0 0 The exact solution of (29) is the following: For t < 0: A+ B A− B A− B A+ B l r c l r y(t) = y + y , Ay , y + y , (30) 0 0 0 0 0 2 2 2 2 where 2 2 1 (t −t )/2 A = e B = . (31) For t ≥ 0: 2 2 2 l t /2 c t /2 r t /2 y(t) = (y (0)e , y (0)e , y (0)e ). (32) The absolute value of relative error between the numerical solutions y and the exact values y(t ), i.e., |(y(t )− y )/y(t )| are calculated. A comparison of the absolute i i i i value of relative error obtained by the use of Euler method and by the use of hybrid Euler method at 0-level and 1-level are shown in Fig.1 and Fig.2. Solid lines are calculated from Euler method and dash-dotted lines are calculated from Hybrid Euler method. It shows that hybrid Euler method have a less error than Euler method. Fuzzy Inf. Eng. (2012) 4: 445-455 453 Relative error at 0−level 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 −1 −0.5 0 0.5 1 Fig. 1 “-” Relative error calculated from Euler method. “-·-” Relative error calculated from hybrid Euler method at 0-level Relative error at 1−level 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 −1 −0.5 0 0.5 1 Fig. 2 “-” Relative error calculated from Euler method. “-·-” Relative error calculated from hybrid Euler method at 1-level Example 5.2 [2] Consider the fuzzy initial value problem y (t) = −y+ t+ 1, t ∈ [0, 1], (33) ⎪ l c r y(0) = (y , y , y ) = (0.96, 1, 1.01). 0 0 0 The exact solution of (33) is t −t −t t −t y(t) = (t− 0.025e + 0.985e , t+ e , t+ 0.025e + 0.985e ). (34) relative error relative error 454 P. Prakash· V. Kalaiselvi (2012) A comparison of the absolute value of relative error obtained by the use of predictor- corrector method and by the use of hybrid predictor-corrector method at 0-level and 1- level are shown in Fig.3 and Fig.4. Solid lines are calculated from predictor-corrector method and dash-dotted lines are calculated from hybrid predictor-corrector method. It shows that hybrid predictor-corrector method has less error than a predictor-corrector one. Relative error at 0−level 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0.2 0.4 0.6 0.8 1 Fig. 3 “-” Relative error calculated from predictor-corrector method. “-·-” Relative error calculated from hybrid predictor-corrector method at 0-level −8 Relative error at 1−level x 10 1.4 1.2 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Fig. 4 “-” Relative error calculated from predictor-corrector method. “-·-” Relative error calculated from hybrid predictor-corrector method at 1-level relative error relative error Fuzzy Inf. Eng. (2012) 4: 445-455 455 6. Conclusion In this paper, we show that the hybrid numerical methods help to increase the ac- curacy and decrease the CPU time. These methods can be used to solve all fuzzy differential equations which are solved by the ordinary numerical methods. Acknowledgments This work is supported by Department of Science and Technology, New Delhi, India. References 1. Abbasbandy S, Allahviranloo T (2002) Numerical solutions of fuzzy differential equations by Taylor method. Journal of Computational Methods in Applied Mathematics 2: 113-124 2. Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equations by predictor-corrector method. Information Sciences 177: 1633-1647 3. Bede B (2008) Note on “Numerical solutions of fuzzy differential equations by predictor-corrector method”. Information Sciences 178: 1917-1922 4. Buckley J J, Feuring T (2000) Fuzzy differential equations. Fuzzy Sets and Systems 110: 43-54 5. Chalco-Cano Y, Roman-Flores H (2006) On new solutions of fuzzy differential equations. Chaos, Solitons and Fractals 38: 112-129 6. Datta D P (2003) The golden mean, scale free extension of real number system, fuzzy sets and 1/ f spectrum in physics and biology. Chaos, Solitons and Fractals 17: 781-788 7. Dubois D, Prade H (2000) Fundamentals of fuzzy sets. Kluwer Academic Publishers, USA 8. Georgiou D N, Nieto J J, Rosana Rodriguez-Lopez (2005) Initial value problems for higher-order fuzzy differential equations. Nonlinear Analysis 63: 587-600 9. Guo M, Xue X, Li R (2003) Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets and Systems 138: 601-615 10. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets and Systems 24: 301-317 11. Kaleva O (1990) The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems 35: 389-396 12. Ma M, Friedman M, Kandel A (1999) Numerical solution of fuzzy differential equations. Fuzzy Sets and Systems 105: 133-138 13. Nieto J J, Torres A (2003) Midpoints for fuzzy sets and their application in medicine. Artificial Intelligence in Medicine 27: 81-101 14. Pederson S, Sambandham M (2007 Numerical solution to hybrid fuzzy systems. Mathematical and Computer Modelling 45: 1133-1144 15. Pederson S, Sambandham M (2008) The Runge-Kutta method for hybrid fuzzy differential equation. Nonlinear Analysis: Hybrid Systems 2: 626-634 16. Prakash P, Kalaiselvi V (2009) Numerical solution of hybrid fuzzy differential equations by predictor- corrector method. International Journal of Computer Mathematics 86: 121-134 17. Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets and Systems 24: 319-330 18. Zhang H, Liao X, Yu J (2005) Fuzzy modeling and synchronization of hyperchaotic systems. Chaos, Solitons and Fractals 26: 835-843 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Numerical Solutions of Fuzzy Differential Equations by Using Hybrid Methods

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Fuzzy Inf. Eng. (2012) 4: 445-455 DOI 10.1007/s12543-012-0126-9 ORIGINAL ARTICLE Numerical Solutions of Fuzzy Differential Equations by Using Hybrid Methods P. Prakash · V. Kalaiselvi Received: 26 November 2010/ Revised: 10 August 2012/ Accepted: 18 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we study the numerical solution of fuzzy differential equa- tions by using hybrid Euler method and hybrid predictor-corrector method. These methods are used to increase the accuracy and the computing speed. Also examples are presented to illustrate the computational aspects of the above methods. Keywords Hybrid method · Fuzzy Cauchy problem · Growth equations · Euler method · Adams-Bashforth method · Adams-Moulton method · Predictor-corrector method 1. Introduction The theory of fuzzy differential equations has been investigated extensively in the original formulation as well as in an alternative framework, which leads to ordinary differential equations. The concept of differential equations in a fuzzy environment was first formulated by Kaleva [10]. The fuzzy differential equation is a particularly important topic from the theoretical [4, 5, 8, 11, 17] as well as the applied point of view [12, 14, 15], for example, in population models [9], in golden mean [6], syn- chronize hyperchaotic systems [18] and medicine [13]. In the last few years, the numerical solution of fuzzy differential and hybrid fuzzy differential equations has been studied by several authors [1-3]. Recently, the numerical solution of hybrid fuzzy differential equations by predictor-corrector method has been studied in [16]. The motivation of this paper is the numerical methods for solving the fuzzy differen- tial equations with the property of growth models. Numerical experiments show that P. Prakash () Department of Mathematics, Periyar University, Salem-36011, India email: pprakashmaths@gmail.com V. Kalaiselvi () Department of Mathematics, Hindustan University, Padur-603103, India email: kalaiselvi111@gmail.com 446 P. Prakash· V. Kalaiselvi (2012) the proposed methods for solving the fuzzy differential equations of the form y = f (t, y) = g(t, y)· y, t ∈ [t , T ], y(t ) = y , 0 0 where f and g are continuous functions, obviously increase the computing speed and accuracy. The structure of the paper is organized as follows. In Section 2, we introduce some basic definitions which will be used later in the paper. In Section 3 and 4, we apply hybrid Euler method and multi-step method for solving the exponential models of growth, then the proposed method is illustrated by solving examples in Section 5, and the conclusion is drawn in Section 6. 2. Preliminaries 2.1. Notations An arbitrary fuzzy number is represented by an ordered pair of functions (u(α), u(α)), 0 ≤ α ≤ 1, that satisfies the following requirements: 1) u(α) is a bounded left continuous nondecreasing function over [0, 1]. 2) u(α) is a bounded left continuous nonincreasing function over [0, 1]. 3) u(α) ≤ u(α), 0 ≤ α ≤ 1. Let E be the set of all upper semi-continuous normal convex fuzzy numbers with bounded α-level intervals. If v ∈ E, then the α-level set [v] = {s|v(s) ≥ α},0<α ≤ α α 1, is a closed bounded interval which is denoted by [v] = [v , v ]. Let I be a real interval. A mapping y : I → E is called a fuzzy process and its α α α α-level set is denoted by [y(t)] = [y (t), y (t)], t ∈ I,α ∈ (0, 1]. Triangular fuzzy numbers are those fuzzy sets U ∈ E which are characterized by l c r 3 l c r 0 l r an ordered triple (x, x , x ) ∈ R with x ≤ x ≤ x such that [U] = [x, x ] and 1 c [U] = {x }, then α c c l c r c [U] = [x − (1−α)(x − x ), x + (1−α)(x − x )] (1) for anyα ∈ [0, 1]. Definition 2.1 [7] The supremum metric d on E is defined by α α d (U, V) = sup{d ([U] , [V] ): α ∈ [0, 1]} ∞ H and (E, d ) is a complete metric space. Definition 2.2 [7] A mapping F : I → E is Hukuhara differentiable at t ∈ I ⊆ R if for some h > 0 the Hukuhara differences F(t +Δt) ∼ F(t ), F(t ) ∼ F(t −Δt), 0 0 h 0 0 h 0 exist in E for all 0 < Δt < h and if there exists an F (t ) ∈ E such that 0 0 lim d ((F(t +Δt) ∼ F(t ))/Δt, F (t )) = 0 ∞ 0 h 0 0 Δt→0 Fuzzy Inf. Eng. (2012) 4: 445-455 447 and lim d ((F(t ) ∼ F(t −Δt))/Δt, F (t )) = 0, ∞ 0 h 0 0 Δt→0 the fuzzy set F (t ) is called the Hukuhara derivative of F at t . 0 0 α α Recall that U ∼ V = W ∈ E are defined on level sets, where [U] ∼ [V] = h h [W] for all α ∈ [0, 1]. By consideration of definition of the metric d , all the level set mappings [F(.)] are Hukuhara differentiable at t with Hukuhara derivatives [F (t )] for each α ∈ [0, 1] when F : I → E is Hukuhara differentiable at t with 0 0 Hukuhara derivative F (t ). Definition 2.3 [7] The fuzzy integral y(t)dt, 0 ≤ a ≤ b ≤ 1, is defined by b b b y(t)dt = y (t)dt, y (t)dt , a a a provided the Lebesgue integrals on the right exist. Remark 2.1 If F : I → E is Hukuhara differentiable and its Hukuhara derivative F is integrable over [0, 1], then F(t) = F(t )+ F (s)ds for all values of t , t, where 0 ≤ t ≤ t ≤ 1. 0 0 Remark 2.2 The Seikkala derivative y (t) of a fuzzy process y is defined by α α  α [y (t)] = [(y ) (t), (y ) (t)],α ∈ (0, 1] provided the equation defines a fuzzy number y (t) ∈ E. Remark 2.3 If y : I → E is Seikkala differentiable and its Seikkala derivative y is integrable over [0, 1], then y(t) = y(t )+ y (s)ds for all values of t , t, where t , t ∈ I. 0 0 2.2. A Fuzzy Cauchy Problem Consider the first order fuzzy differential equation y = f (t, y), where y is a fuzzy function of t, f (t, y) is a fuzzy function of crisp variable t and fuzzy variable y and y is Hukuhara of Seikkala fuzzy derivative of y. If an initial value is given, a fuzzy Cauchy problem of first order is y = f (t, y), t ∈ [t , T ], (2) y(t ) = y . 0 0 Sufficient conditions for the existence of unique solution to Equation (2) are: • f is continuous, • Lipschitz condition d ( f (t, x), f (t, y)) ≤ Ld (x, y), L > 0. ∞ ∞ 448 P. Prakash· V. Kalaiselvi (2012) Single step and multi-step method to solve fuzzy Cauchy problem are as follows: • Euler method: α α y = y + h· f (t, y ), i i i+1 i (3) α α y = y + h· f (t, y ), i = 0, 1, 2,··· . i i i+1 i • Predictor-corrector method: 23 4 α 5 α α α α y = y + h f (t, y )− f (t , y )+ f (t , y ) , i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 (4) 23 α 4 5 α α α α y = y + h f (t, y )− f (t , y )+ f (t , y ) , i = 2, 3,··· i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 and 5 2 1 α α α α α y = y + h f (t , y )+ f (t, y )− f (t , y ) , i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 (5) 5 α 2 α 1 α α α y = y + h f (t , y )+ f (t, y )− (t , y ) , i = 1, 2,··· . i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 3. Hybrid Euler Method Consider the fuzzy differential equation of the form, y = f (t, y) = g(t, y)· y, t ∈ [t , T ], (6) y(t ) = y . 0 0 n n n n Here f ∈ C[[t , T ] × E , E ], g ∈ C[[t , T ] × E , E ]. (6) can be replaced by an 0 0 equivalent system α  α α α α α (y ) (t) = f (t, y(t)) = g (t, y(t))· y (t), y (t ) = y , (7) α α α α α (y ) (t) = f (t, y(t)) = g (t, y(t))· y (t), y (t ) = y . The above system can be solved by the following Euler’s method, α α α y = y + h· f (t, y ), i i i+1 i (8) α α y = y + h· f (t, y ), i = 0, 1, 2,··· , i i i+1 i where, h = t − t . By substituting z = In(y), (7) becomes i+1 i α  α (z ) (t) = g (t), (9) α α (z ) (t) = g (t). According to the Euler’s Method, (9) can be solved as α α α z = z + h· g (t ), i+1 i (10) α α α z = z + h· g (t ), i = 0, 1, 2,··· . i+1 i After replacing z = In(y), (10) becomes α α α y = y · exp(h· g (t, y )), i i i+1 i (11) α α α y = y · exp(h· g (t, y )), i = 0, 1, 2,··· . i i i+1 i Fuzzy Inf. Eng. (2012) 4: 445-455 449 On the other hand, we can achieve the numerical solution of Equations (7) and (9) α α α α by the Taylor series expansion for y (t), y (t) and z (t), z (t) around t with a step size of h as α 1 α α 2 α  3 y (t + h) = y (t )+ hf (t )+ h (y ) (t )+ o(h ), i i i i (12) α α α 1 2  3 y (t + h) = y (t )+ h f (t )+ h (y ) (t )+ o(h ), i = 0, 1, 2,··· i i i i and α α α 1 2 α  3 z (t + h) = z (t )+ hg (t )+ h (z ) (t )+ o(h ), i i i i (13) α α α α 1 2  3 z (t + h) = z (t )+ hg (t )+ h (z ) (t )+ o(h ), i = 0, 1, 2,··· . i i i i By comparing the Equations (12) with (8) and (13) with (10), we can get the local truncation errors , and, as α α 1 2 α = y (t + h)− y ≈ h (y ) (t ), i i 1 2 i+1 (14) α α α 1 2 = y (t + h)− y ≈ h (y ) (t ) 1 i i i+1 and 1 1 α α 2 α  2 α = z (t + h)− z ≈ h (z ) (t ) = h In(y ) (t ), i i i 2 2 i+1 (15) α α 1 2 α  1 2 α = z (t + h)− z ≈ h (z ) (t ) = h In(y ) (t ). i i i i+1 2 2 Then, the main part of local truncation error , can be obtained from (15), namely α α α 1 2 α = y (t + h)− y ≈ y [exp( h In(y ) (t ))− 1], i i 2 2 i+1 i+1 (16) α α α α 1 2 = y (t + h)− y ≈ y [exp( h In(y ) (t ))− 1]. 2 i i i+1 i+1 Here we use the differential expressions of α  α α α 2 In(y ) (t ) = [In(y )− 2In(y )+ In(y )]/h , i i−1 i−2 α α α α In(y ) (t ) = [In(y )− 2In(y )+ In(y )]/h i i−1 i−2 and α  α α α 2 (y ) (t ) = [y − 2y + y ]/h , i i−1 i−2 α  α α α 2 (y ) (t ) = [y − 2y + y ]/h . i i−1 i−2 From (12)-(16), we can find that, if the round-off error was ignored, the local trun- cation error based on the numerical method (11) is zero for the growth equations such as f (t, y) = c· y. But the Euler’s method (8) has the great error and even diverges for such cases. However, the numerical method (11) fails to solve (7) if its solutions y(t) include both positive and negative values, whereas the Euler’s method may succeed to obtain the numerical solution accurately. We can see from (14) and (16) that the numerical method (8) has less error for some part solutions of y(t ), but (11) has less error for other part solutions. Therefore, to overcome the shortcomings of (8) and 450 P. Prakash· V. Kalaiselvi (2012) (11) and decrease the local truncation error at each step, a hybrid method is proposed by merging them, namely, α α ⎪ y · exp(h· g (t, y )), if | | > | |, i i ⎨ 1 2 y = (17) ⎪ α α i+1 y + h· f (t, y ), if otherwise i i and α α y · exp(h· g (t, y )), if | | > | |, i i 1 2 α i y = (18) ⎪ α i+1 ⎪ α y + h· f (t, y ), if otherwise. i i 4. Hybrid Multi-step Method To decrease the local truncation error of the Euler’s method, the multi-step methods were proposed. The 3-step Adams-Bashforth and 2-step Adams-Moulton methods are given by 23 α 4 5 α α α y = y + h[ f (t, y )− f (t , y )+ f (t , y )], i i i−1 i−1 i−2 i−2 12 3 12 i+1 i (19) α α α α 23 4 α 5 y = y + h[ f (t, y )− f (t , y )+ f (t , y )] i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 and α α 5 α 2 α 1 y = y + h[ f (t , y )+ f (t, y )− f (t , y )], i+1 i+1 i i i−1 i−1 12 3 12 i+1 i (20) α α α α 5 2 1 α y = y + h[ f (t , y )+ f (t, y )− f (t , y )]. i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 The main term of the local truncation error , for (19) and (20) is 4 α 4 = Ch (y ) (t ), i−2 (21) 4 4 = Ch (y ) (t ). 1 i−2 Similarly, according to the operational process of (11), a set of new methods based on Adams-Bashforth and Adams-Moulton formula can be obtained as α α 23 α 4 5 α y = y · exp(h[ g (t, y )− g (t , y )+ g (t , y )]), i i i−1 i−1 i−2 i−2 12 3 12 i+1 i (22) α α α α 23 4 α 5 y = y · exp(h[ g (t, y )− g (t , y )+ g (t , y )]) i i i−1 i−1 i−2 i−2 i+1 i 12 3 12 and α α 5 α 2 α 1 α y = y · exp(h[ g (t , y )+ g (t, y )− g (t , y )]), i+1 i+1 i i i−1 i−1 12 3 12 i+1 i (23) α α 5 α 2 α 1 y = y · exp(h[ g (t , y )+ g (t, y )− g (t , y )]). i+1 i+1 i i i−1 i−1 i+1 i 12 3 12 The main term of local truncation error  , for (22) and (23) is α 4 α 4 = y · exp(Ch (y ) (t )− 1), i−2 i+1 (24) α α 4 4 = y · exp(Ch (y ) (t )− 1). 2 i−2 i+1 Here, the coefficient C has the value 3/8 for Adams-Bashforth method and -1/24 for Adams-Moulton method. Implicit methods are usually both more accurate and more Fuzzy Inf. Eng. (2012) 4: 445-455 451 stable than explicit methods, but they are more difficult to apply. In order to take ad- vantage of the beneficial properties of the implicit multi-step methods while avoiding the difficulties inherent in solving the implicit equations, predictor-corrector methods are proposed. The most popular predictor-corrector methods are Adams-Bashforth- Adams-Moulton method, where Adams-Bashforth and Adams-Moulton methods are implemented as the predictor and corrector, respectively. Numerical experiments show that the predictor-corrector methods based on (22) and (23) can improve the accuracy and increase the computing speed for (6) obvi- ously. However, (22) and (23) also fail to fuzzy differential equations including both positive and negative solutions. Compared (21) with (24), we can find that (22) and (23) offer less error than (19) and (20) for some fuzzy differential equations, while the situation is reverse for other cases. Predictor: ⎪ 23 4 ⎪ α α y · exp h g (t, y )− g (t , y ) i i i−1 i−1 ⎪ i ⎪ 12 3 ⎪ 5 ⎪ + g (t , y ) , if | |> |, i−2 i−2 1 2 y = (25) 23 4 α i+1 ⎪ ⎪ α α y + h f (t, y )− f (t , y ) ⎪ i i i−1 i−1 ⎪ i 12 3 ⎪ 5 ⎩ + f (t , y ) , if otherwise. i−2 i−2 23 4 ⎪ α α α ⎪ y · exp h g (t, y )− g (t , y ) i i i−1 i−1 ⎪ 12 3 ⎪ 5 ⎪ α + g (t , y ) , if | | >  |, i−2 i−2 1 2 α ∗ y = (26) i+1 23 α 4 α α y + h f (t, y )− f (t , y ) ⎪ i i i−1 i−1 12 3 ⎪ 5 α + f (t , y ) , if otherwise. i−2 i−2 Corrector: ⎪ 5 2 ⎪ α ∗ α y · exp h g (t , y )+ g (t, y ) i+1 i i ⎪ i+1 ⎪ i ⎪ 12 3 ⎪ − g (t , y ) , if | |> |, ⎪ i−1 i−1 1 2 y = (27) 5 2 i+1 ⎪ α α ∗ α y + h f (t , y )+ f (t, y ) ⎪ i+1 i i ⎪ i+1 ⎪ i 12 3 ⎪ 1 α ⎩ − f (t , y ) , if otherwise. i−1 i−1 ⎪ 5 2 ⎪ α α ∗ α y · exp h g (t , y )+ g (t, y ) i+1 i i ⎪ i i+1 ⎪ 12 3 ⎪ 1 − g (t , y )]), if | | >  |, i−1 i−1 1 2 y = (28) i+1 5 α 2 α ⎪ ∗ y + h f (t , y )+ f (t, y ) ⎪ i+1 i i i i+1 12 3 ⎪ 1 ⎩ − f (t , y ) , if otherwise. i−1 i−1 12 452 P. Prakash· V. Kalaiselvi (2012) Here we use the differential expressions of α 4 α α α α α 4 In(y ) (t ) = [In(y )− 4In(y )+ 6In(y )− 4In(y )+ In(y )]/h , i i−1 i−2 i−3 i−4 α 4 α α α α α 4 In(y ) (t ) = [In(y )− 4In(y )+ 6In(y )− 4In(y )+ In(y )]/h i i−1 i−2 i−3 i−4 and α 4 α α α α α 4 (y ) (t ) = [y − 4y + 6y − 4y + y ]/h , i i−1 i−2 i−3 i−4 α α α α α α 4 4 (y ) (t ) = [y − 4y + 6y − 4y + y ]/h . i i−1 i−2 i−3 i−4 5. Examples Example 5.1 [12] Consider the fuzzy initial value problem y (t) = ty, t ∈ [−1, 1], √ √ √ (29) ⎪ l c r y(−1) = (y , y , y ) = ( e− 0.5, e, e+ 0.5). 0 0 0 The exact solution of (29) is the following: For t < 0: A+ B A− B A− B A+ B l r c l r y(t) = y + y , Ay , y + y , (30) 0 0 0 0 0 2 2 2 2 where 2 2 1 (t −t )/2 A = e B = . (31) For t ≥ 0: 2 2 2 l t /2 c t /2 r t /2 y(t) = (y (0)e , y (0)e , y (0)e ). (32) The absolute value of relative error between the numerical solutions y and the exact values y(t ), i.e., |(y(t )− y )/y(t )| are calculated. A comparison of the absolute i i i i value of relative error obtained by the use of Euler method and by the use of hybrid Euler method at 0-level and 1-level are shown in Fig.1 and Fig.2. Solid lines are calculated from Euler method and dash-dotted lines are calculated from Hybrid Euler method. It shows that hybrid Euler method have a less error than Euler method. Fuzzy Inf. Eng. (2012) 4: 445-455 453 Relative error at 0−level 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 −1 −0.5 0 0.5 1 Fig. 1 “-” Relative error calculated from Euler method. “-·-” Relative error calculated from hybrid Euler method at 0-level Relative error at 1−level 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 −1 −0.5 0 0.5 1 Fig. 2 “-” Relative error calculated from Euler method. “-·-” Relative error calculated from hybrid Euler method at 1-level Example 5.2 [2] Consider the fuzzy initial value problem y (t) = −y+ t+ 1, t ∈ [0, 1], (33) ⎪ l c r y(0) = (y , y , y ) = (0.96, 1, 1.01). 0 0 0 The exact solution of (33) is t −t −t t −t y(t) = (t− 0.025e + 0.985e , t+ e , t+ 0.025e + 0.985e ). (34) relative error relative error 454 P. Prakash· V. Kalaiselvi (2012) A comparison of the absolute value of relative error obtained by the use of predictor- corrector method and by the use of hybrid predictor-corrector method at 0-level and 1- level are shown in Fig.3 and Fig.4. Solid lines are calculated from predictor-corrector method and dash-dotted lines are calculated from hybrid predictor-corrector method. It shows that hybrid predictor-corrector method has less error than a predictor-corrector one. Relative error at 0−level 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0.2 0.4 0.6 0.8 1 Fig. 3 “-” Relative error calculated from predictor-corrector method. “-·-” Relative error calculated from hybrid predictor-corrector method at 0-level −8 Relative error at 1−level x 10 1.4 1.2 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Fig. 4 “-” Relative error calculated from predictor-corrector method. “-·-” Relative error calculated from hybrid predictor-corrector method at 1-level relative error relative error Fuzzy Inf. Eng. (2012) 4: 445-455 455 6. Conclusion In this paper, we show that the hybrid numerical methods help to increase the ac- curacy and decrease the CPU time. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Dec 1, 2012

Keywords: Hybrid method; Fuzzy Cauchy problem; Growth equations; Euler method; Adams-Bashforth method; Adams-Moulton method; Predictor-corrector method

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