Abstract
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 Engineering
– Taylor & Francis
Published: Dec 1, 2012
Keywords: Hybrid method; Fuzzy Cauchy problem; Growth equations; Euler method; Adams-Bashforth method; Adams-Moulton method; Predictor-corrector method