Commun Nonlinear Sci Numer Simulat 17 (2012) 3788–3794

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Numerical solution of hybrid fuzzy differential equations using improvedpredictor–corrector method

Hyunsoo Kim a, Rathinasamy Sakthivel b,⇑a College of Applied Science, Kyung Hee University, Yongin 446-701, South Koreab Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea

a r t i c l e i n f o

Article history:Received 2 September 2010Received in revised form 14 December 2011Accepted 5 February 2012Available online 17 February 2012

Keywords:Hybrid fuzzy differential equationsNumerical solutionImproved predictor–corrector methodFuzzy initial value problem

1007-5704/$ - see front matter � 2012 Elsevier B.Vdoi:10.1016/j.cnsns.2012.02.003

⇑ Corresponding author. Tel.: +82 31 299 4527; faE-mail address: krsakthivel@yahoo.com (R. Sakth

a b s t r a c t

The hybrid fuzzy differential equations have a wide range of applications in science andengineering. This paper considers numerical solution for hybrid fuzzy differential equa-tions. The improved predictor–corrector method is adapted and modified for solving thehybrid fuzzy differential equations. The proposed algorithm is illustrated by numericalexamples and the results obtained using the scheme presented here agree well with theanalytical solutions. The computer symbolic systems such as Maple and Mathematicaallow us to perform complicated calculations of algorithm.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The topic of fuzzy differential equations has been rapidly growing in recent years. The fuzzy differential equations werefirst formulated by Kaleva [10] and Seikkala [16]. The applications of fuzzy differential equations in other fields and relatedmathematical tools and techniques could be found in [6,7]. The investigation of numerical solutions of fuzzy differentialequations plays an important role in science and engineering. A variety of numerical methods have been developed for find-ing the solution of fuzzy initial value problems. Some numerical methods for fuzzy initial value problems under the Huku-hara differentiability concept such as the fuzzy Euler method, Runge–Kutta method, Adams–Bashforth method, Taylormethod, linear programming method and Nostrum method are presented in [8,9,11–15]. A numerical method based onthe collocation method is proposed in [4] for solving nth-order linear differential equations with fuzzy initial conditions.Numerical methods for fuzzy differential inclusions are considered in [1]. More recently, the differential transformationmethod with the concept of generalized H-differentiability is implemented for solving the fuzzy initial value problems in[3]. Another powerful and more convenient numerical technique called the predictor–corrector method was proposed in[2] for solving the fuzzy differential equations. The predictor–corrector method is based on Adams–Bashforth three stepmethod as a predictor and an iteration of Adams–Moulton two step method as a corrector. More recently, an improved pre-dictor–corrector method is introduced in [5], which is based on the explicit three-step method as a predictor and an iterationof the implicit two-step method as a corrector. It is shown that the result of the improved predictor–corrector method ismore accurate than one in [2].

On the other hand, hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior. Thehybrid systems are devoted to modeling, design, and validation of interactive systems of computer programs and continuoussystems. The differential equations containing fuzzy valued functions and interaction with a discrete time controller are

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x: +82 31 290 7033.ivel).

H. Kim, R. Sakthivel / Commun Nonlinear Sci Numer Simulat 17 (2012) 3788–3794 3789

named as hybrid fuzzy differential equations [13]. This paper deals with numerical solution of hybrid fuzzy differential equa-tions. In order to obtain numerical solutions for fuzzy equations under Hukuhara differentiability, it is not necessary to re-write the numerical methods for the ordinary differential equations in fuzzy setting. Instead one can use the numericalmethods directly for the ordinary differential equations by the use of characterization theorems. In the present paper, wemodify and implement the improved predictor–corrector method based on the Hukuhara derivative to obtain the solutionsof hybrid fuzzy differential equations. Further, we show that the solutions obtained by improved predictor–corrector methodis more accurate and well agree with the exact solutions.

2. Preliminaries

In this section the most basic notations used in fuzzy calculus are presented. There are various definitions for the conceptof fuzzy numbers [10,16,17]. We start by defining the fuzzy number.

Definition 2.1. A fuzzy number u is a fuzzy subset of the real line with a normal, convex and upper semicontinuousmembership function of bounded support.

Definition 2.2. A fuzzy number u is completely determined by any pair u ¼ ðu; �uÞ of functions uðaÞ; �uðaÞ; 0 6 a 6 1 whichsatisfy the following requirements:

(i) uðaÞ is a bounded monotonic increasing left continuous function over ½0;1�, with respect to any a.(ii) �uðaÞ is a bounded monotonic decreasing left continuous function over ½0;1�, with respect to any a.

(iii) uðaÞ 6 �uðaÞ; 0 6 a 6 1.

Let E denote the family of fuzzy numbers. For 0 < a 6 1, define the a�level set ½u�a ¼ fx 2 R : uðxÞP ag is a closedbounded interval which is denoted by ½u�a ¼ ½uðaÞ; �uðaÞ�. We refer to uðaÞ and �uðaÞ as the lower and upper branches on urespectively.

Let I be a real interval. A mapping y : I ! E is called a fuzzy process, and its a�level set is denoted by½yðtÞ�a ¼ ½yðaÞðtÞ; �yðaÞðtÞ�; t 2 I;a 2 ð0;1�.

Triangular fuzzy number is a fuzzy set U in E that is characterized by an ordered triple ðxl; xc; xrÞ 2 R3 with xl6 xc

6 xr suchthat ½U�0 ¼ ½xl; xr� and ½U� ¼ fxcg. The a� level set of a triangular fuzzy number U is given by ½U�a ¼ ½xc � ð1� aÞðxc � xlÞ;xc þ ð1� aÞðxr � xcÞ�, for any 0 6 a 6 1.

Let dH be the Hausdorff metric for non-empty compact sets in R. Then supremum metric d1 on E is defined byd1ðu;vÞ ¼ supfdHð½u�a; ½v �aÞ : 0 6 a 6 1g. The space ðE; dÞ is a complete metric space.

A mapping F : T ! E is Hukuhara differentiable at t0 2 T � R if for some h0 > 0 the Hukuhara differences Fðt0 þ DtÞ�hFðt0Þ;Fðt0Þ�hFðt0 � DtÞ exist in E for all 0 < Dt < h0 and if there exists an F 0ðt0Þ 2 E such that

limDt!0þ

d1Fðt0 þ DtÞ�hFðt0Þ

Dt; F 0ðt0Þ

� �¼ 0 and lim

h!0þd1

Fðt0Þ�hFðt0 � DtÞDt

; F 0ðt0Þ� �

¼ 0:

The fuzzy set F 0ðtÞ is called the Hukuhara derivative of F at t0 (see [10,16]).By the definition of the metric d, all the a-level set mappings ½Fð�Þ�a are Hukuhara differentiable at t0; t0 2 T with Huku-

hara derivatives ½F 0ðt0Þ�a for each a 2 T when F : T ! E is Hukuhara differentiable at t0 with Hukuhara derivative F 0ðt0Þ.

Definition 2.3. The fuzzy integral

Z b

ayðtÞdt; 0 6 a 6 b 6 1;

is defined by

Z b

ayðtÞdt

" #a

¼Z b

ayaðtÞdt;

Z b

a

�yaðtÞdt

" #

provided that the Lebesgue integrals on the right-hand side exist.If F : T ! E is Hukuhara differentiable and its Hukuhara derivative F 0 is integrable over ½0;1�, then

FðtÞ ¼ Fðt0Þ þR t

t0F 0ðsÞds; for all 0 6 t0 6 t 6 1.

Definition 2.4. A mapping y : I ! E is called a fuzzy process. We denote ½yðtÞ�a ¼ ½yaðtÞ; �yaðtÞ�; t 2 I; 0 < a 6 1. The Seikkaladerivative y0ðtÞ of a fuzzy process y is defined by ½y0ðtÞ�a ¼ ½ðyaÞ0ðtÞ; ð�yaÞ0ðtÞ�; 0 < a 6 1 provided that this equation in factdefines a fuzzy number y0ðtÞ 2 E.

If y : I! E is Seikkala differentiable and its Seikkala derivative y0 is integrable over ½0;1�, then yðtÞ ¼ yðt0Þ þR t

t0

y0ðsÞds; for all t0; t 2 I.

3790 H. Kim, R. Sakthivel / Commun Nonlinear Sci Numer Simulat 17 (2012) 3788–3794

3. Hybrid fuzzy differential equations

In this paper, we consider the hybrid fuzzy differential equation

y0ðtÞ ¼ f ðt; yðtÞ; kkðykÞÞ; t 2 ½tk; tkþ1�; k ¼ 0;1;2 . . .

yðt0Þ ¼ y0;ð1Þ

where ftkg1k¼0 is strictly increasing and unbounded, yk denotes yðtkÞ; f : ½t0;1Þ� E� E! E is continuous and each kk : E! E iscontinuous. A solution to (1) will be a function y : ½t0;1Þ ! E satisfying (1). For k ¼ 0;1;2; . . ., let fk : ½tk; tkþ1� � E! E, wherefkðt; ykðtÞÞ ¼ f ðt; ykðtÞ; kkðykÞÞ.

The hybrid fuzzy differential equation (1) can be written as

y0ðtÞ ¼

y00ðtÞ ¼ f ðt; y0ðtÞ; k0ðy0ÞÞ ¼ f0ðt; y0ðtÞÞ; t0 6 t 6 t1;

y01ðtÞ ¼ f ðt; y1ðtÞ; k1ðy1ÞÞ ¼ f1ðt; y1ðtÞÞ; t1 6 t 6 t2;

..

.

y0kðtÞ ¼ f ðt; ykðtÞ; kkðykÞÞ ¼ fkðt; ykðtÞÞ; tk 6 t 6 tkþ1;

..

.

8>>>>>>>><>>>>>>>>:

ð2Þ

The solution of (1) can be expressed as

yðtÞ ¼

y0ðtÞ; t0 6 t 6 t1;

y1ðtÞ; t1 6 t 6 t2;

..

.

ykðtÞ; tk 6 t 6 tkþ1;

..

.

8>>>>>>>><>>>>>>>>:

ð3Þ

By applying Bede’s characterization theorem proposed in [14], the following characterization theorem for hybrid fuzzy dif-ferential equation has been obtained:

Theorem 3.1 [14]. Consider the hybrid fuzzy differential equation (1) expanded as (2), where for k ¼ 0;1;2; . . ., eachfk : ½tk; tkþ1� � E! E is such that

(i) ½fkðt; yÞ�a ¼ ½ðf kÞaðt; ya; �yaÞ; ð�f kÞaðt; ya; �yaÞ�,(ii) ðf kÞa and ð�f kÞaare equicontinuous and uniformly bounded on any bounded set,

(iii) There exists an qk > 0 such that jðf kÞaðt; x1; y1Þ � ðf kÞaðt; x2; y2Þj 6 qk maxfjx2 � x1; jy2 � y1jg; ðt; x1; y1Þ 2 ½tk; tkþ1��R2; a 2 ½0;1�.Then (1) and the hybrid system of ordinary differential equations

ððykÞaðtÞÞ0 ¼ ðf kÞaðt; ðykÞaðtÞ; ð�ykÞaðtÞÞ;ðð�ykÞaðtÞÞ0 ¼ ð�f kÞaðt; ðykÞaðtÞ; ð�ykÞaðtÞÞ;ðykÞaðtkÞ ¼ ðyk�1ÞaðtkÞ; if k > 0; ðy0Þaðt0Þ ¼ ðy0Þa;ð�ykÞaðtkÞ ¼ ð�yk�1ÞaðtkÞ; if k > 0; ð�y0Þaðt0Þ ¼ ð�y0Þa;

8>>>><>>>>:

are equivalent.

4. Improved predictor–corrector method

In this section, we will present the improved predictor–corrector method for solving the fuzzy hybrid differential equa-tions. The improved predictor–corrector method algorithm is based on using the explicit three-step method as a predictor,and an iteration of the implicit two-step method as a corrector [5]. To numerically solve the hybrid system of ordinary dif-ferential system in ½t0; t1�; ½t1; t2�; . . . ; ½tk; tkþ1�; . . ., for a 2 ½0;1�, we will replace each interval ½tk; tkþ1� by a set of Nk þ 1 reg-ularly spaced grid points. The grid points on ½tk; tkþ1�will be tk;n ¼ tk þ nhk where hk ¼ tkþ1�tk

Nkand 0 6 n 6 Nk. Now we will give

algorithm to numerically solve the system in ½t0; t1�; ½t1; t2�; . . . ; ½tk; tkþ1�; . . ..

Algorithm. To approximate the solution of the following hybrid fuzzy initial value problem

y0kðtÞ ¼ f ðt; ykðtÞ; kkðykÞÞ; t 2 ½tk; tkþ1�; k ¼ 0;1;2 . . . ;

yaðtk;0Þ ¼ a0; yaðtk;1Þ ¼ a1; yaðtk;2Þ ¼ a2;

yaðtk;0Þ ¼ a3; yaðtk;1Þ ¼ a4; yaðtk;2Þ ¼ a5;

8><>:

H. Kim, R. Sakthivel / Commun Nonlinear Sci Numer Simulat 17 (2012) 3788–3794 3791

arbitrary positive integer Nk is chosen.

Step 1. Let hk ¼ tkþ1�tkN ,

waðtk;0Þ ¼ a0; waðtk;1Þ ¼ a1; waðtk;2Þ ¼ a2;

waðtk;0Þ ¼ a3; waðtk;1Þ ¼ a4; waðtk;2Þ ¼ a5:

Step 2. Let i ¼ 1.Step 3. Let

wð0Þaðtk;iþ2Þ ¼ waðtk;i�1Þ þ hk2 f aðtk;i�1;wðtk;i�1ÞÞ þ f aðtk;i;wðtk;iÞÞ þ 4f aðtk;iþ1;wðtk;iþ1ÞÞh i

;

wð0Þaðtk;iþ2Þ ¼ waðtk;i�1Þ þ hk2 f aðtk;i�1;wðtk;i�1ÞÞ þ f aðtk;i;wðtk;iÞÞ þ 4f aðtk;iþ1;wðtk;iþ1ÞÞh i

:

8><>:

Step 4. Let tk;iþ2 ¼ tk;0 þ ðiþ 2Þhk.Step 5. Let

waðtk;iþ2Þ ¼ waðtk;iÞ þ hk2

� �f aðtk;i;wðtk;iÞÞ þ hkf aðtk;iþ1;wðtk;iþ1ÞÞ þ hk

2

� �f aðtk;iþ2;wð0Þaðtk;iþ2ÞÞ

h i;

waðtk;iþ2Þ ¼ waðtk;iÞ þ hk2

� �f aðtk;i;wðtk;iÞÞ þ hkf aðtk;iþ1;wðtk;iþ1ÞÞ þ hk

2

� �f aðtk;iþ2;wð0Þaðtk;iþ2ÞÞ

h i:

8><>:

Step 6. Let i ¼ iþ 1.Step 7. If i 6 N � 2, go to Step 3.Step 8. The algorithm ends, and ðwaðtkþ1Þ;waðtkþ1ÞÞ approximates the value of ðYaðtkþ1Þ;Yaðtkþ1ÞÞ.

Remark 4.1. It should be mentioned that the convergence and stability of the improved predictor–corrector method forhybrid fuzzy differential equations can be proven as similar to those given in [5].

5. Example

Pederson and Sambandham [13,14] numerically solved the example below by using the Euler, Runge–Kutta and improvedEuler method. A similar example was recently considered in [15] for fuzzy hybrid systems using Adams–Bashforth methodand Adams–Moulton method. The method implemented in this paper gives better approximation.

Consider the following hybrid fuzzy initial value problem [14]

y0ðtÞ ¼ �yðtÞ þmðtÞkkðyðtkÞÞ; t 2 ½tk; tkþ1�; tk ¼ k; k ¼ 0;1;2; . . .

yð0Þ ¼ ½0:75þ 0:25a;1;1:125� 0:125a�;

�ð4Þ

where mðtÞ ¼ jsinðptÞj; k ¼ 0;1;2; . . ., and

kkðlÞ ¼0; if k ¼ 0l; if k 2 1;2; . . . :

�

Next using the algorithm given in Section 4, we will solve the hybrid fuzzy equations corresponding to (4) by the improvedpredictor–corrector method to obtain numerical solutions to (4).

Case I: When k = 0, the solution of (4) in the interval ½0;1�:When k ¼ 0, the hybrid fuzzy initial value problem (4) becomes

y0ðtÞ ¼ �yðtÞ; t 2 ½0;1�;yð0Þ ¼ ½0:75þ 0:25a;1;1:125� 0:125a�:

�ð5Þ

The fuzzy initial value problem (5) is equivalent to the following system of fuzzy initial value problem

yl 0ðtÞ ¼ �yrðtÞ;yc 0ðtÞ ¼ �ycðtÞ;yr 0ðtÞ ¼ �ylðtÞ; t 2 ½0;1�ylð0Þ ¼ 0:75þ 0:25a; ycð0Þ ¼ 1; yrð0Þ ¼ 1:125� 0:125a:

8>>><>>>:

ð6Þ

In order to solve the fuzzy system (5) by using the improved predictor–corrector method, we consider the following systemof fuzzy problem

y0ðtÞ ¼ �yðtÞ; t 2 ½0;1�yð0Þ ¼ ½ylð0Þ ¼ 0:75þ 0:25a; yrð0Þ ¼ 1:125� 0:125a�:

�ð7Þ

3792 H. Kim, R. Sakthivel / Commun Nonlinear Sci Numer Simulat 17 (2012) 3788–3794

The exact solution of (7) is given by

Table 1Compar

a

00.10.20.30.40.50.60.70.80.91.0

Table 2Compar

a

00.10.20.30.40.50.60.70.80.91.0

yðtÞ ¼ ½ylðtÞ ¼ ð�0:1875þ 0:1875aÞet þ ð0:9375þ 0:0625aÞe�t;

yrðtÞ ¼ �ð�0:1875þ 0:1875aÞet þ ð0:9375þ 0:0625aÞe�t �:

(ð8Þ

The exact solution (7) at t ¼ 0:1 can be written as

Yð0:1;aÞ ¼ ½ylð0:1Þ ¼ ð�0:1875þ 0:1875aÞe0:1 þ ð0:9375þ 0:0625aÞe�0:1;

yrð0:1Þ ¼ �ð�0:1875þ 0:1875aÞe0:1 þ ð0:9375þ 0:0625aÞe�0:1�:

(ð9Þ

Case II: When k = 1, the solution of (4) in the interval ½1;2�:When k ¼ 1 the above fuzzy problem (4) can be written as

y0ðtÞ ¼ �yðtÞ þ sinðptÞk1ðyðtÞÞ; t 2 ½1;2�yð0Þ ¼ ½0:75þ 0:25a;1;1:125� 0:125a�:

�ð10Þ

This is equivalent to the system of fuzzy differential equations

yl 0ðtÞ ¼ �yrðtÞ þ sinðptÞylðtÞ;yc 0ðtÞ ¼ �ycðtÞ þ sinðptÞycðtÞ;yr 0ðtÞ ¼ �ylðtÞ þ sinðptÞyrðtÞ; t 2 ½1;2�ylð0Þ ¼ 0:75þ 0:25a; ycð0Þ ¼ 1; yrð0Þ ¼ 1:125� 0:125a:

8>>><>>>:

ð11Þ

Similar to the previous Case. I, in order to solve the fuzzy system (10) we consider the following system of fuzzy equations

y0ðtÞ ¼ �yðtÞ þ sinðptÞk1ðyðtÞÞ; t 2 ½1;2�yð0Þ ¼ ½ylð0Þ ¼ 0:75þ 0:25a; yrð0Þ ¼ 1:125� 0:125a�:

�ð12Þ

The exact solution of (12) is

yðtÞ ¼ ½ylðtÞ ¼ e�0:31830988 cosðptÞðð�1:54665250þ 0:17185027aÞ sinhðtÞ þ ð0:34370055aþ 1:03110167Þ coshðtÞÞ;yrðtÞ ¼ �e�0:31830988 cosðptÞðð�1:54665250þ 0:17185027aÞ coshðtÞ þ ð0:34370055aþ 1:03110167Þ sinhðtÞÞ�:

(

ð13Þ

ison of exact and approximate solution in the interval [0,1].

Exact Impro.Pre–Cor Error Exact Impro.Pre–Cor Error

0.64106553 0.64532301 0.42574e�2 1.05550462 1.04954471 0.59599e�20.66744272 0.67127444 0.38317e�2 1.04043790 1.03507396 0.53639e�20.69381990 0.69722586 0.34059e�2 1.02537118 1.02060322 0.47679e�20.72019709 0.72317729 0.29801e�2 1.01030446 1.00613247 0.41719e�20.74657428 0.74912871 0.25544e�2 0.99523774 0.99166173 0.35760e�20.77295147 0.77508014 0.21286e�2 0.98017102 0.97719098 0.29800e�20.79932866 0.80103156 0.17029e�2 0.96510430 0.96272024 0.23840e�20.82570585 0.82698299 0.12771e�2 0.95003758 0.94824949 0.17880e�20.85208304 0.85293441 0.85137e�3 0.93497085 0.93377875 0.11921e�20.87846022 0.87888583 0.42561e�3 0.91990413 0.91930800 0.59613e�30.90483741 0.90483726 0.15340e�6 0.90483741 0.90483726 0.15420e�6

ison of exact and approximate solution in the interval [1,2].

Exact Impro.Pre–Cor Error Exact Impro.Pre–Cor Error

�0.46747585 �0.44282751 0.24648e�1 1.62890263 1.61645125 0.12451e�1�0.35878550 �0.33590573 0.22879e�1 1.52795513 1.51744514 0.10509e�1�0.25009515 �0.22898396 0.21111e�1 1.42700763 1.41843904 0.85685e�2�0.14140480 �0.12206219 0.19342e�1 1.32606013 1.31943294 0.66271e�2�0.03271446 �0.01514042 0.17574e�1 1.22511263 1.22042683 0.46857e�2

0.07597588 0.09178135 0.15805e�1 1.12416513 1.12142073 0.27443e�20.18466623 0.19870312 0.14036e�1 1.02321762 1.02241463 0.80299e�30.29335657 0.30562489 0.12268e�1 0.92227012 0.92340852 0.11384e�20.40204692 0.41254666 0.10499e�1 0.82132262 0.82440242 0.30798e�20.51073727 0.51946843 0.87311e�2 0.72037512 0.72539632 0.50211e�20.61942761 0.62639021 0.69625e�2 0.61942762 0.62639021 0.69625e�2

Fig. 1. Example at t = 0.1- the exact solution (solid graph) and the approximate solution by the improved predictor–corrector approach (diamond curve) inthe interval [0,1].

Fig. 2. Example at t = 1.1- the exact solution (solid graph) and the approximate solution by the improved predictor–corrector approach (diamond curve) inthe interval [1,2].

H. Kim, R. Sakthivel / Commun Nonlinear Sci Numer Simulat 17 (2012) 3788–3794 3793

When t ¼ 0:1, the exact solution is given by

Yð1:1;aÞ¼ ½ylð1:1Þ¼ e�0:31830988cosðp�1:1Þðð�1:54665250þ0:17185027aÞsinhð1:1Þþð0:34370055aþ1:03110167Þcoshð1:1ÞÞ;yrð1:1Þ¼�e�0:31830988cosðp�1:1Þðð�1:54665250þ0:17185027aÞcoshð1:1Þþð0:34370055aþ1:03110167Þsinhð1:1ÞÞ�:

(

ð14Þ

3794 H. Kim, R. Sakthivel / Commun Nonlinear Sci Numer Simulat 17 (2012) 3788–3794

The absolute error between the exact solution and the results obtained by improved predictor–corrector method is com-pared in Tables 1 and 2. The approximate solutions and the exact solutions are plotted in Figs. 1 and 2. for the interval [0,1]and [1,2] respectively. It is concluded that the results of the improved predictor–corrector method is very close to the exactsolutions which confirm the validity of this method.

6. Conclusion

In this paper, an improved predictor–corrector method is adapted and modified for solving the hybrid fuzzy differentialequations. Numerical examples and comparisons with exact solutions reveal that the proposed algorithm is capable of gen-erating accurate results. The future work in this direction is concerned with solving the hybrid fuzzy differential equationsunder generalized Hukuhara differentiability.

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