Two Points Hybrid Block Method for Solving First Order Fuzzy Differential Equations

Transcription

1 Journal of Soft Computing and Appliations 2016 No.1 (2016) Available online at Volume 2016, Issue 1, Year 2016 Artile ID jsa-00083, 11 Pages doi: /2016/jsa Researh Artile Two Points Hybrid Blok Method for Solving First Order Fuzzy Differential Equations Tiaw Kah Fook 1, Zarina Bibi Ibrahim 1* (1) Institute for Mathematial Researh, Department of Mathematis, Faulty of Siene, Universiti Putra Malaysia, Serdang, Selangor, Malaysia. Copyright 2016 Tiaw Kah Fook and Zarina Bibi Ibrahim. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. Abstrat In this paper, hybrid blok method is obtained by ombining the Blok Bakward Differentiation Formulas () with blok Simpson method for the numerial solution of first order Fuzzy Differential Equations (FDEs). The fuzzy version of the methods is disussed in detail under Seikkala differentiability onept. Numerial results obtained by the hybrid blok method are presented and ompare with the solution obtained by, Bakward Differential Formula (BDF) and Euler method. Several numerial problems are presented to illustrate the effiieny of the proposed hybrid method. Keywords: Fuzzy differential equations; Seikkala differentiability onept; Blok bakward differentiation formula; Blok Simpson, Bakward differentiation formula, Euler method. 1 Introdution In this paper, hybrid blok method is developed for initial value problems (IVPs) for First order Fuzzy Differential Equations (FDEs) of the form { y = f(t, y, y ), t [t 0, t n ] y(t 0 ) = y 0, y (1.1) (0) = y 0 Prodution and hosting by ISPACS GmbH. * Corresponding Author. address: zarinabb@upm.edu.my, Tel:

2 Journal of Soft Computing and Appliations 2016 No.1 (2016) In many ases of modeling the real world phenomena, information about the behavior of a dynamial system is unertain. In order to obtain a more realisti model, these unertainties have to be taken into aount. Fuzzy Differential Equation (FDEs) is a powerful tool for modeling unertainty and proessing vague or subjetive information in mathematial models. Fuzzy model is also adequate for some real-world phenomena. In reent years, FDEs system an be found in wide varieties of sientifi and engineering appliations. For example, modeling the deay of the biohemial oxygen demand in water by Diniz et al. (2001), biology population models by Mengshu et al. (2003), a fuzzy delay differential equation model for HIV dynamis in mediine by Jafelie et al. (2009), geneti programming by Kumaresan et al. (2011), modeling of hydrauli differential servo ylinders in engineering by Bensik et al. (2012), fuzzy environment by Mondal et al. (2016), and Kuo et al. (2016) used FDEs to solve the dynami vehile routing problem. Due to the large potential of fuzzy differential equation involving in these fields, it has beome the subjet for many researh projets. Therefore, many numerial methods have been developed for solving FDEs (1.1). Some of the numerial methods inlude the use of Taylor series method by Abbasbandy et al. (2002), the Runge-Kutta method by Abbasbandy et al. (2004), the preditor-orretor method by Allahviranloo et al. (2009), the Euler method by Ahmad et al. (2011), diagonally impliit blok bakward differentiation formulas by Zawawi et al. (2012), Khastan et al. (2015) studied some properties of the solutions to first order linear fuzzy differential equations under differential inlusions approah, exponentially fitted Runge Kutta method by Karimi et al. (2016), and numerial solution of first order fuzzy differential equations by Simpson s Rule in Devi et al. (2016). In this paper, we are interested in applying the hybrid blok method. The existing hybrid method in Timothy et al. (2012) is based on olloation of the differential system and interpolation of the approximate at the grid and off-grid points, but in this paper the hybrid method is defined as the ombination of blok bakward differentiation formulas (Ibrahim 2006; Ibrahim et al. 2007; Ibrahim et al. 2008) and blok Simpson (Adegboye et al. 2014). 2 Preliminaries In this setion, some general definitions of fuzzy numbers in Allahvinranloo et al. (2009) whih will be used throughout the paper are presented. Definition 2.1. A fuzzy number u is a fuzzy subset of real line with a normal, onvex and upper semi ontinuous membership funtion of bounded support. We onsider R, the lass of fuzzy subset of real axis, u: R [0, 1] satisfied the following properties: 1. u is a fuzzy onvex set, whih is u(tx + (1 t)y) min {u(x), u(y)} for all value x, y [0,1]; 2. supp u = x R; u(x) > 0 is ompat; 3. u is normal, that is, x 0 R for whih u(x 0 ) = 1; 4. u is a upper semi ontinuous on R. Definition 2.2. Let I be a real interval. A mapping y I E is alled a fuzzy proess. We denote its r-level set by: [y(t)] r = [y r (t), y r (t)] t I, r (0,1] (2.2) The Seikkala derivative y (t) of a fuzzy proess y is defined by [y (t)] r = (y r ) (t), (y r ) (t)], r (0,1] (2.3)

3 Journal of Soft Computing and Appliations 2016 No.1 (2016) However, fuzzy numbers will be almost always triangular or trapezoidal shaped fuzzy numbers. Let T be the set of all triangular or trapezoidal shaped fuzzy numbers and u T. Definition 2.3. Let [u] r be the losed bounded intervals and denote by [u] r = [u(r), u(r)] (2.4) [u] r = {x: u(x) r}, 0 r 1 (2.5) whih satisfy the three onditions: a) u(r) is a bounded left ontinuous inreasing funtion (0; 1). b) u(r) is a bounded right ontinuous dereasing funtion (0; 1). ) u(r) u(r). Definition 2.4. Let F: T E and t 0 T R. F is differentiable at t 0, if 1) there exist an element F (t 0 ) E, suh that for all h > 0 suffiiently small, there are F(t 0 + h) F(t 0 ), F(t0) F(t 0 h) and the limits (in D-matri) F(t lim 0 +h) F(t 0 ) F(t = lim 0 ) F(t 0 h) = F (t h 0 + h h 0 + h 0 ) (2.6) or 2) there exist an element(t 0 ) E, suh that for all h < 0 suffiiently small, there are F(t 0 + h) F(t 0 ), F(t 0 ) F(t 0 h) and the limits F(t lim 0 +h) F(t 0 ) F(t = lim 0 ) F(t 0 h) = F (t h 0 h h 0 h 0 ) (2.7) where the relation (2.6) is the lassial definition of the fuzzy H-derivation (or differentiability in the sense of Hukuhara). Definition 2.5. Let x, y R. If there exist z R suh that x = y + z, then z is alled the Hukuhara differene of x and y and is denoted by x y. Note that x y x + ( 1)y. 3 Literature Review 3.1. Review of Formulaton of Blok Bakward Differential Formula In this setion, we review the formulation of proposed by Ibrahim (2006); Ibrahim et al. (2007); Ibrahim et al. (2008). The general approah in omputing the values of y n+1 and y n+2 simultaneously is disussed in this setion. The blok bakward differential formula is presented by Ibrahim et al. (2007), the two values of y n+1 and y n+2 were omputed simultaneously in a blok with eah blok ontaining two points. From equation (1.1), [t 0, t n ] is divided into five points whih are x n 2, x n 1, x n, x n+1, x n+2, where h is a onstant step size. The formula is derived using interpolating polynomial p k (x) whih interpolates the values y n, y n 1,, y n k+1 of the funtion f at the interpolating points x n, x n 1,,x n k+1, in terms of Lagrange polynomial the formula was defined as follow with eah blok ontains two points, k P k (x) = j=0 L k,j (x)f(x n+1 j ) (3.8) where L k,j (x) = k (x x n+1 i ) i=0 (x n+1 j x n+1 i ) i j for eah j = 0,1,, k. (3.9) The equation generated from the Lagrange polynomial is as below,

4 Journal of Soft Computing and Appliations 2016 No.1 (2016) (x x n )(x x n+1 )(x x n+2 ) p(x n + sh) = (x n 1 x n )(x n 1 x n+1 )(x n 1 x n+2 ) y n 1 + (x x n 1)(x x n+1 )(x x n+2 ) (x n x n 1 )(x n x n+1 )(x n x n+2 ) y n (x x n 1 )(x x n )(x x n+2 ) + (x n+1 x n 1 )(x n+1 x n )(x n+1 x n+2 ) y n+1 (x x n 1 )(x x n )(x x n+1 ) + (x n+2 x n 1 )(x n+2 x n )(x n+2 x n+1 ) y (3.10) n+2 Next, let s = x x n and substitute x = x h n+1 into (3.10), we obtain the formula for the first point. y n+1 = 1 y 3h n 1 + 2y n 2 y 3 n+2 + 2hf n+1 (3.11) Then we let s = x x n and substitute x = x h n+2 in (3.10), we will get y n+2 = 2 y 11 n 1 9 y 11 n + 18 y 11 n hf 11 n+2 (3.12) 3.2. Review of Formulation of Blok Simpson The blok Simpson formula given by Adegboye et al. (2014) is used to further orret the values of y n+1 and y n+2 where it is derived in the form of power series, t+m 1 y(x) = a j x j j=0 a R, j = o(1)t + m 1, Y C m (a, b) P(x) (3.13) where a j s are the parameter to be determined, t and m are points of interpolation and olloation respetively. Speifially, t = 2 and m = k + 1 yield the following system of equations: t+m 1 j=0 a j x j = y n+j j = 2 (3.14) The proposed ontinuous formulation takes the form: y(x) = α 2 (x)y n+2 + h{β 0 (x)f n + β 1 (x)f n+1 + β 2 (x)f n+2 } (3.15) The blok Simpson formula was proposed by Adegboye et al. (2014) for solving ODEs. The formula is modified into fuzzy version of blok Simpson to solve first order FDEs. y n+1 = y n + h ( 5 f 12 n + 8 f 12 n+1 1 f 12 n+2) y n+2 = y n + h (f } (3.16) 3 n + 4f n+1 + f n+2 ) In addition, the hybrid method is developed by ombining the formula and the fuzzy version of blok Simpson to obtain numerial result for the fuzzy initial value problem. 4 Implementation of the Hybrid Method In this setion, we will modify the in Ibrahim (2006) and blok Simpson in Adegboye et al. (2014) into fuzzy version of and blok Simpson. Then, the hybrid method is developed by ombining the fuzzy version of formula and the fuzzy version of blok Simpson to obtain numerial result for the fuzzy initial value problem. Hybrid method = + Blok Simpson Let Y = [Y, Y] be the exat solution and y = [y, y] be the approximate solution of the fuzzy initial value problem.

5 Journal of Soft Computing and Appliations 2016 No.1 (2016) We onsider the definition 2.2 in (2.2) and (2.3), where [Y (t)]r = [Y (t, r), Y (t, r)] (4.17) The grid points at whih the solution is alulated are T t 0, t N i = t 0 + nh, 0 n N. Modified the formula and Blok Simpson formula in (11, 12, 16) into fuzzy version, we onfigure the general hybrid blok method by ombining the and blok Simpson as follow, y n+1 (r) = 1 y 3 n 1(r) + 2y n (r) 2 y 3 n+2(r) + 2hf n+1 [t, y(t, r), y(t, r)] y n+2 (r) = 2 y 11 n 1(r) 1 y 9 n(r) + 18 y 11 n+1(r) + 6 hf 11 n+2 [t, y(t, r), y(t, r)] } (4.18) y n+1 (r) = y n (r) + h ( 5 12 f n [t, y(t, r), y(t, r)] f n+1 [t, y(t, r), y(t, r)] 1 12 f n+2 [t, y(t, r), y(t, r)]) y n+2 (r) = y n (r) + h } 3 ( f n [t, y(t, r), y(t, r)] + 4f n+1 [t, y(t, r), y(t, r)] + f n+2 [t, y(t, r), y(t, r)] ) y n+1 (r) = 1 y 3 n+1 (r) + 2y n (r) 2 y 3 n+2 (r) + 2hf n+1 [t, y(t, r), y(t, r)] (4.19) y n+2 (r) = 2 y 11 n 1 (r) 1 y 18 (r) 9 n + y 11 n+1 (r) + 6 hf 11 n+2 y n+1 (r) = y n (r) + h ( 5 f 12 n [t, y(t, r), y(t, r)] + 8 f 12 n+1 [t, y(t, r), y(t, r)] 1 f 12 n+2 [t, y(t, r), y(t, r)]) y n+2 (r) = y n (r) + h (f 3 n [t, y(t, r), y(t, r)] + 4f n+1 [t, y(t, r), y(t, r)] + f n+2 [t, y(t, r), y(t, r)]) } (4.21) The following omputations are arried out to obtain the approximations: i. Compute [y n+1,n+2 (r), y n+1,n+2 (r)] using the preditor formula. [t, y(t, r), y(t, r)]} (4.20) ii. Compute the orreted values of [y n+1,n+2 (r), y n+1,n+2 (r)] using Fuzzy formula in (4.18, iii. 4.20). Compute the further orreted values of [y n+1,n+2 formula in (4.19, 4.21). (r), y n+1,n+2 (r)] using Fuzzy blok Simpson Note that the approximation to [y n+1 (r), y n+1 (r)] in formula (4.18, 4.20) is omputed using pass values [y n 2 (r), y n 2 (r)], [y n 1 (r), y n 1 (r)] and [y n (r), y n (r)] as well as predited values for [f n+1 (t, y(t, r), y(t, r)), f n+1 (t, y(t, r), y(t, r))] and future values [y n+2 (r), y n+2 (r)]. Then, the approximation to [y n+1 (r), y n+1 (r)] is further orreted by formula (4.19, 4.21) using the pass and future values obtained from (4.18, 4.20) in eah iteration. For this reason, we an say that the method is fully impliit. 5 Numerial Examples and Disussion In this setion, the fuzzy initial value problem is presented in Problem 1-2. For these numerial examples, errors between exat solution and approximation are shown in the tables for hybrid blok method and method. The notation used in the tables and figures take the following meaning: h: Step size r: Fuzzy numbers with fuzzy bounded r-level interval Y: Lower bounded exat solution Y: Upper bounded exat solution

6 Journal of Soft Computing and Appliations 2016 No.1 (2016) y: Lower bounded approximate solution y: Upper bounded approximate solution Error: absolute error The absolute error formula, onsidered in table 1 and table 4, is as follows: Error = y Y + y Y. Problem 1 Ahmad et al. (2011): onsider the initial value problem { y (t) = y(t) t [0, 1] y(0) = ( r, r). Exat solution at t = 1, where 0 r 1 is given by, Y (1; r) = [( r)e t, ( r)e t ], Solution: Aording to (2.3), the problem of FDEs an be redued to a system of ODEs as follow; y (t) = f(t, y(t)), y (t) = f(t, y(t)), y(t 0 ) = r y(t 0 ) = r The theoretial exat solution and the numerial solutions with different step sizes h are shown in Figures 1 and 2. r Exat Hybrid Euler y Figure 1: Exat solutions and approximate solutions for Problem 1 with h = 0.1

7 Journal of Soft Computing and Appliations 2016 No.1 (2016) r Exat Hybrid Euler Figure 2: Exat solutions and approximate solutions for Problem 1 with h = 0.01 y log max error 4 2 HYBRID EULER log h Figure 3: Error of hybrid method and at r = 1 with different step sizes Problem 2 Allahviranloo et al. (2007): onsider the initial value problem. y (t) = y(t) + t + 1, t 0 { y(0) = [ r, r] Exat solution at t = 1, where 0 r 1 is given by, Y(t; r) = [t + ( r)e t, t + ( r)e t ]

8 Journal of Soft Computing and Appliations 2016 No.1 (2016) Solution: Aording to the definition in (2.3), the problem of FDEs an be redued to a system of ODEs as follow; y (t) = f(t, y(t)), y(t 0 ) = r y (t) = f(t; y(t)), y(t 0 ) = r The theoretial exat solution and the numerial solutions with different step sizes h are shown in Figures 4 and 5. r Exat Hybrid Euler Figure 4: Exat solutions and approximate solutions for Problem 2 with h = 0.1 r y Exat Hybrid Euler y Figure 5: Exat solutions and approximate solutions for Problem 2 with h = 0.01

9 Journal of Soft Computing and Appliations 2016 No.1 (2016) log max error 4 2 HYBRID EULER Figure 6: Error of hybrid method and at r = 1 with different step sizes Figure 1, 2, 4 and 5 are the omparisons between the exat and approximate solutions of hybrid method, and Euler method with different step sizes. Based on the numerial solution, the numerial results showed that the proposed method provides an aurate solution at smaller step size for both methods. From the Figure 3 and Figure 6, we an observe that the errors obtained by hybrid method are smaller ompare to and Euler method. Therefore, hybrid method is a suitable iterative method to solve Fuzzy Differential Equations. 6 Conlusion In this paper, the hybrid method is developed for solving first order fuzzy differential equation under Seikkala differentiability onept. Errors between the approximate solutions and the exat solutions were omputed and the numerial results are obtained. After omparing the approximate solution obtained by the proposed method with the exat solution, the numerial results demonstrate the effiieny of hybrid methods for solving fuzzy differential equations. Aknowledgements We thank Institute for Mathematial Researh (INSPEM) and the Department of Mathematis, Universiti Putra Malaysia for the finanial support under Fundamental Grant Sheme (FRGS) FR. Conflit of Interests The authors delare that there is no onflit of interests regarding the publiation of this paper. log h

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