Third-order Composite Runge Kutta Method for Solving Fuzzy Differential Equations

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1 Global Joural of Pure ad Applied Mathematics. ISSN Volume Number (06) pp Research Idia Publicatios Third-order Composite Ruge Kutta Method for Solvig Fuzz Differetial Equatios A. Ramli a) R. R. Ahmad b) U. K. S. Di b) b) & A. R. Salleh School of Mathematical Scieces Facult of Sciece ad Techolog Uiversiti Kebagsaa Malasia 600 Bagi Selagor Selagor Darul Ehsa Malasia. Correspodig author: Abstract. I this paper a third-order composite Ruge Kutta method is applied for solvig fuzz differetial equatios based o geeralized Hukuhara differetiabilit. This stud iteds to explore the explicit methods which ca be improved ad modified to solve fuzz differetial equatios. Some defiitios ad theorem are reviewed as a basis i solvig fuzz differetial equatios. Some umerical examples are give to illustrate the accurac of the method. The comparisos with the existig method are also discussed. Based o the results the proposed method gives the better result. Hece the method ca be used to solve fuzz differetial equatios. Kewords: Ruge-Kutta Fuzz differetial equatios Hukuhara differetiabilit.. INTRODUCTION Differetial equatios pla a importat role i various fields such as phsics ecoom sciece ad egieerig. It is widel used b experts i this field to model problems i their stud. However i some cases iformatio regardig the phsical pheomea ivolved are ofte represeted ot b crisp values but istead the values are represeted b ucertait. Klir et al. [] defies the cocept of fuzz umbers which exist from ma pheomea that caot be measured with precise values. Nowadas fuzz differetial equatios (FDEs) have started to grow rapidl. The cocept of FDEs was first itroduced b Chag ad Zadeh []. Later Dubois ad Prade [] expaded the priciple approach i solvig FDEs. Kaleva [89] ad Seikkala [] maaged to solve FDEs with fuzz iitial value problem.

2 7 A. Ramli et al Ma et al. [6] first itroduced the classic Euler method as a umerical method to solve the problem of fuzz differetial equatios. Abbasbad ad Allahviraloo [] studied Talor method to solve the problem of fuzz differetial equatios. Subsequetl the field is growig rapidl with a variet of umerical methods to solve the problem of fuzz differetial equatios. The umerical methods such as Adam Bashford [] Ruge Kutta of order five [] block methods [6] ad Ruge- Kutta Method with Harmoic Mea of Three Quatities [] were ivestigated to fid solutios for fuzz differetial equatios problem. I this stud third-order composite Ruge-Kutta method [0] is applied for solvig fuzz differetial equatios. I secod sectio some ecessar prelimiaries is give. Trasformatio of FDEs ito the fuzz parametric form is show i the third sectio. Next the derivatio of third-order composite Ruge-Kutta method is preseted ad the stabilit of the proposed method is give i sectio five. Sectio six discussed the formulatio of the third-order composite Ruge-Kutta method for solvig FDEs. The proposed method is compared with Ruge-Kutta Method with Harmoic Mea of Three Quatities ad a umerical result is illustrated i sectio seve. The discussio is provided i sectio eight ad the coclusio is give i the last sectio.. PRELIMINARIES I this sectio some basic defiitios of fuzz umbers are reviewed: Defiitio : A fuzz umber is a fuzz subset of the real lie with ormal covex ad upper semi cotiuous membership fuctio of bouded support. A fuzz umber is determied b a pair ( ( r) ( r)) 0 r which satisf the three coditios:. r () is a bouded left cotiuous icreasig fuctio r [0].. r () is a bouded left cotiuous decreasig fuctio r [0].. ( r) ( r) 0 r. Defiitio : Let I be a real iterval. A mappig F : I process. We deote its r-level set of for I [ t0 T] r r r [ F( t)] [ ( t) ( t)] r [0]. The Seikkala derivatives '( t ) r r r [ F '( t)] [( )'( t)( )'( t)] r [0]. of a fuzz process is defied b E is called a fuzz Bede ad Stefaii [] itroduced the defiitio ad theorem of the geeralized Hukuhara as follows: Defiitio : Let F : T E( ) ad t 0 ( a b ). F is differetiable at t 0 if Form I. For all h 0 sufficietl close to 0 the Hukuhara differeces F( t0 h) F( t0) ad F( t0) F( t0 h) ad the limits (D-metric)

3 Third-order Composite Ruge Kutta Method 7 F( t0 h) F(t 0) F(t 0) F( t0 h) lim lim F'( t0). h 0 h h 0 h Form II. For all h 0 sufficietl close to 0 the Hukuhara differeces F( t0 h) F( t0) ad F( t0) F( t0 h) ad the limits (D-metric) F( t0 h) F(t 0) F(t 0) F( t0 h) lim lim F'( t0). h 0 h h 0 h Theorem : Let F : T E ( ) where t0 ( a b ) ad E is a fuzz fuctio ad r r r deote [ F( t)] [ ( t) ( t )] for each r [0]. The r r Case. If F is differetiable i the first form I the () t ad () t are differetiable fuctios ad r r r [ F '( t)] [( )'( t)( )'( t )]. r r Case. If F is differetiable i the first form II the () t ad () t are differetiable fuctios ad r r r [ F '( t)] [( )'( t)( )'( t )].. FUZZY DIFFERENTIAL EQUATIONS Mathematicall FDEs ca be defied as follows [7]: '( t) f ( t ( t )) () where is a fuzz fuctio of t f ( t ( t)) is fuzz fuctio of the crisp variable ad ' is a fuzz derivative of. If a iitial value ( t0) 0 is give a fuzz Cauch problem of first order is obtaied: '( t) f ( t ( t)) ( t0) 0 () It is possible to replace Eq.() b the followig equivalet sstem '( t) f ( t ( t)) F( t ) ( t0) 0 () '( t) f ( t ( t)) G( t ) ( t ) 0 0 where F( t ) mi f ( t ) () G( t ) max f ( t ). Based o Theorem the parametric form of two differet cases are as follows: Case : t r F t t r t r t r r '( ) ( ) ( ) ( 0 ) 0( ) '( t r) G t ( t r) ( t r) ( t r) ( r) 0 0 ()

4 76 A. Ramli et al Case : '( t r) G t ( t r) ( t r) ( t r) ( r) 0 0 '( t r) F t ( t r) ( t r) ( t r) ( r) where t [ t0 T] ad r [0]. 0 0 (6). THIRD-ORDER COMPOSITE RUNGE KUTTA METHOD Ahmad ad Yaacob [0] itroduced third-order composite Ruge-Kutta method for solvig ordiar differetial equatios. This method is the combiatio of the harmoic ad arithmetic meas of the Ruge-Kutta formulatios. The formulatio of the third-order composite of the arithmetic ad harmoic meas is show as follows: B usig the cocept of the arithmetic mea we have k f ( t ) f k f ( t a h a hk ) k f t a h a h k ) k h kk kk. k k k k Sice Eq. (7) ivolved the divisio of two series kk i i i (8) ki k i the it caot be solved b a direct substitutios. This problem is alleviated b crossmultiplig the terms b the commo deomiator ( k k)( k k) which ca be writte as TOP (9) BOTTOM where Top h[ k k ( k k ) k k ( k k )] BOTTOM ( k k )( k k ). Talor series expasio of t ( ) ma be writte as Talor hf h ff h ff f f h f f f f f ff... 6 Sice the error of the method ca be determied usig the expressio Error ( t ) we obtai TOP Error Talor BOTTOM which ca be writte as (7)

5 Third-order Composite Ruge Kutta Method 77 Error BOTTOM Talor BOTTOM TOP. (0) The coefficiets of the same terms i (0) up to the terms are compared ad two sets of parameter are obtaied a 0 a ad a a. The composite arithmetic harmoic mea Ruge Kutta formula ca be represeted as follows. h SET : k f ( t ) k f ( t ) k k k k f t h h f ( t h hk) h kk k k h k k k k k k k k k Sice k k.. () SET : k f ( t ) k f t h hk k k k f t h h h kk kk k k k k. Accordig to [0] SET gives the best result. Hece i this stud SET is used to solve fuzz differetial equatios. (). STABILITY ANALYSIS The stabilit of the third-order composite Ruge-Kutta method ca be calculated as follows: Cosider the test problem '. () B substitutig eq. () ito the test problem we have z( z) 0 0z 6z R( z) z. 0 /. () (0 z) ( z) The stabilit regios for the eq.() is illustrated i Figure ad Figure.

6 78 A. Ramli et al Figure : The D stabilit regio plotted for the third-order composite Ruge-Kutta method. Figure : The D stabilit regio plotted for the third-order composite Ruge-Kutta method.

7 Third-order Composite Ruge Kutta Method FUZZY CONFIGURATION FOR THIRD-ORDER COMPOSITE RUNGE KUTTA METHOD Let the exact solutio [ Y( t)] r [ Y( t; r) Y ( t; r )] is approximated b some[ ( t)] r [ ( t; r) ( t; r )]. The solutio is calculated b grid poits at h T t0 ti t0 N ih 0 i N () From (7) we defie ( t ; r) ( t ; r) w k ( t ( t ; r)) i i i ( t ; r) ( t ; r) w k ( t ( t ; r)) i i i where w i s are weighted value which are costats ki ( t ( t; r)) k( t ( t; r)) k ( t ( t; r)) i (7) whereb k ( t ( t; r)) mi { f ( t u) u [ ( t; r) ( t; r)]} k ( t ( t; r)) max { f ( t u) u [ ( t; r) ( t; r)]} k( t ( t; r)) mi f t h u u [ z ( t; r) z( t; r)] k( t ( t; r)) max f t h u u [ z ( t; r) z( t; r)] k( t ( t; r)) mi f t h u u [ z( t; r) z( t; r)] k( t ( t; r)) max f t h u u [ z( t; r) z( t; r)] where i the third-order composite Ruge-Kutta method for solvig FDEs is give as follows z( t ( t; r)) ( t; r) k( t ( t; r)) z( t ( t; r)) ( t; r) k( t ( t; r)) k( t ( t; r)) k( t ( t; r)) (9) z ( t ( t; r)) ( t; r) h k( t ( t; r)) k( t ( t; r)) z( t ( t; r)) ( t; r) h with the followig equatios (6) (8)

8 760 A. Ramli et al ( t ; r) ( t ; r) ( t ; r) ( t ; r) h k( t ( t; r)) k( t ( t; r)) k( t ( t; r)) k( t ( t; r)) k ( t ( t; r)) k ( t ( t; r)) k ( t ( t; r)) k ( t ( t; r)) h k( t ( t; r)) k( t ( t; r)) k( t ( t; r)) k( t ( t; r)) k ( t ( t; r)) k ( t ( t; r)) k ( t ( t; r)) k ( t ( t; r)) (0) 7. NUMERICAL EXAMPLES I this sectio the composite arithmetic harmoic mea Ruge Kutta is tested o the followig fuzz iitial value problems. EXAMPLE : Cosider the fuzz iitial value problem [6]. '( t) ( t) t [0] (0) ( r. 0. r). The exact solutio at t is Y( r) ( r) e. 0. r e. I this paper b cosiderig Case ( for h 0 ) i Theorem the the sstem of ODEs is give as follows: '( t) ( t) ( t ) r '( t) ( t) ( t ). 0. r. Error Y Y. 0 0 Table. Numerical results for Example usig third-order composite RK ad RK harmoic mea quatities with h 0.. r EXACT THIRD-ORDER COMPOSITE RK ERROR RK HARMONIC MEAN QUANTITIES ERROR

9 Third-order Composite Ruge Kutta Method 76 Table. Numerical results for Example usig third-order composite RK ad RK harmoic mea quatities with h 0.0. r EXACT THIRD-ORDER COMPOSITE RK ERROR RK HARMONIC MEAN QUANTITIES ERROR Table ad shows the error i the solutio b the third-order composite RK of two differet step size ( h 0.ad h 0.0 ) alog with the umerical result obtaied b the RK harmoic mea quatities. EXAMPLE : Cosider the fuzz iitial value problem [6]. '( t) tf ( t) t [0] (0) (.0 0. r e. 0. r e). The exact solutio at t is Y ( r).0 0. r e e. 0.r e e I this paper b cosiderig Case ( for h 0 ) i Theorem the the sstem of ODEs is give as follows: '( t) tf ( t. ( t)) ( t ).0 0. r e '( t) tf ( t. ( t)) ( t ). 0. r e Error Y Y. 0 0

10 76 A. Ramli et al Table. Numerical results for Example usig third-order composite RK ad RK harmoic mea quatities with h 0.. r EXACT THIRD-ORDER COMPOSITE RK ERROR RK HARMONIC MEAN QUANTITIES ERROR Table. Numerical results for Example usig third-order composite RK ad RK harmoic mea quatities with h 0.0. r EXACT THIRD-ORDER COMPOSITE RK ERROR RK HARMONIC MEAN QUANTITIES ERROR Table ad shows the approximated umerical result ad the compariso error with two differet step size ( h 0.ad h 0.0 ). 8. DISCUSSION Based o Table the umerical result obtaied through third-order composite Ruge- Kutta method produces umerical values approximatel close to the exact solutio i

11 Third-order Composite Ruge Kutta Method 76 compariso to the values obtaied through Ruge-Kutta Method with Harmoic Mea of Three Quatities. The differece of error betwee these two methods is b order oe. I Table as the step size decrease third-order composite Ruge-Kutta method shows outstadig result i compariso to Ruge-Kutta Method with Harmoic Mea of Three Quatities. The error is obtaied b order two. I Table third-order composite Ruge-Kutta method ields the umerical results slightl better tha Ruge-Kutta Method with Harmoic Mea of Three Quatities which mea the error betwee this two methods is almost the same. I Table with h=0.0 third-order composite Ruge-Kutta method gives a better umerical solutios tha Ruge-Kutta Method with Harmoic Mea of Three Quatities. The error is obtaied b order oe. 9. CONCLUSIONS I this paper we have applied the third-order composite Ruge-Kutta method for fidig the umerical solutio of fuzz differetial equatios. Compariso of the solutios of Example ad it proved that third-order composite Ruge-Kutta method gives better solutio tha Ruge-Kutta Method with Harmoic Mea of Three Quatities whe compared with the exact solutios. ACKNOWLEDGMENTS The authors wish to ackowledge fudig form UKM i this work through research grats FRGS//0/SG0/UKM/0/ PTS-0-60 DLP-0-0 ad MPhD scholarship from KPT. REFERENCES [] D. Dubois H. Prade 980 Fuzz Sets ad Sstem : Theor ad Applicatio New York: Academic Press. [] D. Paul Dhaabara & J. Christ Kigsto 0 A Method for Solvig Fuzz Differetial Equatios Usig Ruge-Kutta Method with Harmoic Mea of Three Quatities. Iteratioal Joural of Egieerig Sciece ad Iovative Techolog () [] G.J. Klir U.H. St Clair B. Yua 997 Fuzz Set Theor: Foudatios ad Applicatios NJ: Pretice Hall. [] J.H. Mathew ad K.D. Fik 999 Numerical Methods usig Matlab. th Editio. [] L. Stefaii B. Bede 009 Geeralized Hukuhara differetiabilit of iterval-valued fuctios ad iterval differetial equatios Noliear Aalsis 7-8. [6] M. Ma M. Friedma A. Kadel 999 Numerical solutios of fuzz differetial equatio Fuzz Sets ad Sstem 0-8.

12 76 A. Ramli et al [7] M. Puri D. Ralescu 98 Differetial of Fuzz Fuctios Joural of Mathematical Aalsis ad Applicatios [8] O. Kaleva 987 Fuzz Differetial Equatios Fuzz Sets ad Sstem [9] O. Kaleva 990 The Cauch problem for fuzz differetial equatios Fuzz Sets ad Sstem [0] R.R. Ahmad ad N. Yaacob 00 Third-order composite Ruge-Kutta method for stiff problems. Iteratioal Joural of Computer Mathematics. 8(0) :-6. [] S. Abbasbad T Allahviraloo 00 Numerical Solutios of Fuzz Differetial Equatios b Talor Method Joural of Computatioal Methods i Applied Mathematics -. [] S. Seikkala 987 O the fuzz iitial value problem Fuzz Sets ad Sstem 9-0. [] S. L. Chag L.A. Zadeh 97 O Fuzz Mappig ad Cotrol IEEE Tras. Sstem Ma Cberet 0-. [] T. Allahviraloo 00 Numerical Solutio of Fuzz Differetial Equatio b Adam-Bashforth two Step Method Joural of Applied Mathematics 6-7. [] T. Jaakumar D. Maheskumar K. Kaaraja 0 Numerical solutio of fuzz differetial equatios b Ruge Kutta method of order five Applied Mathematical Scieces [6] Z.A. Majid S. Mehkaroo M. Suleima 009 Block Method for Numerical Solutio of Fuzz Differetial Equatio Iteratioal Mathematical Forum BIOGRAPHICAL Amirah Ramli received her BSc Degree i Computatioal Mathematics ad MSc Degree i Mathematics from Uiversiti Malasia Tereggau. Curretl she is pursuig Doctor of Philosoph (Mathematics) at the Uiversiti Kebagsaa Malasia. Her field of studies is o fuzz differetial equatios with a focus o the umerical methods for solvig fuzz differetial equatios.