Solving Fuzzy Differential Equations using Runge-Kutta third order method with modified contra-harmonic mean weights

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1 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 Solvig Fuzzy Diffeetial Equatios usig Ruge-Kutta thid ode method with modified cota-hamoic mea weights D.Paul Dhayabaa, J.Chisty kigsto Associate Pofesso & Picipal, PG ad Reseach Depatmet of Mathematics, Bishop Hebe College (Autoomous) Tiuchiappalli Assistat Pofesso, PG ad Reseach Depatmet of Mathematics, Bishop Hebe College (Autoomous) Tiuchiappalli Abstact I this pape a attempt has bee made to detemie a umeical solutio fo the fist ode fuzzy diffeetial equatios by usig Ruge-kutta thid ode method with modified cota-hamoic mea weights.the accuacy of the poposed method is illustated by a umeical example with a fuzzy iitial value poblem usig tapezoidal fuzzy umbe. Keywods Fuzzy Diffeetial Equatios, Thid ode Ruge-kutta method, Modified cota-hamoic mea, Tapezoidal fuzzy umbe 1.ITRODUCTIO The fuzzy diffeetial equatio cocept has bee most popula ad apidly gowig i the last few yeas. Fist ode liea fuzzy diffeetial equatio is oe of the simplest fuzzy diffeetial equatio,which appea i may applicatios. The cocept of fuzzy deivative was fist itoduced by S.L.Chag ad L.A.Zadeh i[6]. D.Dubois ad Pade [7] defied ad used the extesio piciple.othe methods have bee discussed by M.L.pui ad D.A.Ralescu[3] ad R.Goetschel ad W.Voxma[1] cotibuted towads the diffeetial of fuzzy fuctios.the fuzzy diffeetial equatio ad iitial value poblems wee extesively studied by O.Kaleva [15,16] ad by S.Seikkala[4].Recetly may eseach papes ae focused o umeical solutio of fuzzy iitial value poblems (FIVPS).umeical Solutio of fuzzy diffeetial equatio has bee itoduced by M.Ma, M.Fiedma, A.Kadel [18] though eule method ad by S.Abbasbady ad T.Allahvialoo [1] by taylo method.ruge Kutta methods have also bee studied by authos [,1].V.imala,.Saveetha,S.Chethupadiya discussed o umeical solutio of fuzzy diffeetial equatio by Ruge-Kutta method with highe ode deivative appoximatios[].r.gethsi shamila ad E.C.Hey Amithaaj discussed o umeical solutios of fist ode fuzzy iitial value poblems by o-liea tapezoidal fomulae based o vaiety of meas[13].ruge-kutta thid ode method with cota-hamoic mea fo stiff poblems was discussed by Osama Yusuf Ababeh,Rokiah Rozita[17].Followig by the itoductio this pape is ogaised as follows :I sectio, some basic esults of fuzzy umbes ad defiitios of fuzzy deivative ae give. I sectio 3, the fuzzy iitial value poblem is discussed. Sectio 4 descibes the Ruge-kutta thid ode method with modified cota-hamoic mea.i sectio 5, the Ruge-kutta thid ode with modified cota-hamoic mea method was poposed fo solvig fuzzy iitial value poblem ad the umeical examples ae povided to illustate the validity ad applicability of the ew method. Fially the coclusio is give fo the poposed method..prelimiaries Fuzzy umbe A abitay fuzzy umbe is epeseted by a odeed pai of fuctios ( ( ), ( )) followig coditios. i) () u is a bouded left cotiuous o-deceasig fuctio ove u is a bouded ight cotiuous o-deceasig fuctio ove ii) () 9 with espect to ay.,1 with espect to ay. u u fo all,1 which satisfy the

2 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 iii) ( u( ) u( )),1 Clealy,u { x \ u( x) } fo all the the -level set is \ ( ) }; u { x u x 1 is compact, which is a closed bouded iteval ad we deote by u ( u( ), u( )) Tapezoidal Fuzzy umbe A tapezoidal fuzzy umbe u is defied by fou eal umbes k l m, whee the base of the tapezoidal is the iteval [k, ] ad its vetices at x =, x = m. Tapezoidal fuzzy umbe will be witte as u = ( k,, m, ). The membeship fuctio fo the tapezoidal fuzzy umbe u = ( k,, m, ) is defied as the followig : we have : x k k x l l k u( x) 1 l x m x m x m (1) u if k () u if l (3) u if m ad (4) u if Defiitio: ( - Level Set) Let I be the eal iteval. A mappig y : I E is called a fuzzy pocess ad its - level Set is deoted by y( t) [ y( t; ), y( t; ) ], t I, 1 Defiitio: (Seikkala Deivative) The Seikkala deivative '( ) that this equatio defies a fuzzy umbe, as i [4]. Lemma: If the sequece of o-egative umbe costats A ad B, the Lemma: y t of a fuzzy pocess is defied by [ '( A 1 W A W B A 1, If the sequece of o-egative umbes m y '( t) y t; ), y '( t; ) ] t I, 1 povided W satisfy W 1 A W B, 1 fo the give positive m W, V satisfy W 1 W A max{ W, V } B, V 1 V A max{ W, V } B fo the give positive costats A ad B, the U W V, 93

3 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 A 1 U A U B A 1 we have, whee A 1 A ad B B. Lemma Let F( t, u, v ) ad G( t, u, v ) belog to abitaily fixed, 1, C ' ( R F ) ad the patial deivatives of F ad G be bouded ove R F.The fo D y t, y ( t ) h L C) ( ( 1) 1 ) (1 C Max G t, y( t; ), y( t 1; ),,1 Theoem Let F( t, u, v ) ad G( t, u, v) belog to fixed, 1, the umeical solutios of y( t 1; ) uifomly i t. Theoem Let F( t, u, v) ad G( t, u, v) belog to fo abitaily fixed 1 whee L is a boud of patial deivatives of F ad G, ad C ' ( RF ) ad the patial deivatives of F ad G be bouded ove R F.The fo abitaily y( t ; ) Y( t ; ) Y( t ; ) ad 1 covege to the exact solutios 1 ad 1 C ' ( RF ) ad the patial deivatives of F ad G be bouded ove RF ad Lh 1. The,the iteative umeical solutios of y( t; ) ad y( t; ) i t t t, whe j. 3.FUZZY IITIAL VALUE PROBLEM Coside a fist-ode fuzzy iitial value poblem ( j y ) ( t ; ) ad ( j) y ( t ; ) covege to the umeical solutios y( t) f ( t, y( t)), t t, T y( t) y (3.1) whee y is a fuzzy fuctio of t, f ( t, y ) is a fuzzy fuctio of the cisp vaiable ' t ' ad the fuzzy vaiable y, y ' is the fuzzy deivative of y ad y( t ) y is a tapezoidal o a tapezoidal shaped fuzzy umbe. We deote the fuzzy fuctio y by y y, yt [ y ( t; ), y ( t; )], (, 1], ( ) we wite f ( t, y) [ f ( t, y ), f( t, y) ] ad [ y].it meas that the -level set of f ( t, y) F [ t, y, y ], f ( t, y) G [ t, y, y], because of y f ( t, y ) we have 94 yt fo t t T is [ y ( t )] [ y ( t ; ), y ( t ; )], f t, y( t); F [ t, y( t ; ), yt ( ; ) ], (3.),

4 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 f t, y( t); G[ t, y( t ; ), y( t; ) ] (3.3) by usig the extesio piciple, we have the membeship fuctio f ( t, y( t) )( s ) sup{ y( t)( ) \ s f ( t, )}, s R (3.4) so the fuzzy umbe f t, y( t) follows that f ( t, y( t) ) [ f ( t, y( t); ), f ( t, y( t); )], (, 1] f t y t whee (, ( ); ) (3.5) mi f ( t, u) u y( t ) } (3.6) { f t, y( t); max f ( t, u) u y( t ) } (3.7) { Defiitio 3.1: A fuctio f : R RF such that t to D f ( t), f ( t ) exists. is said to be fuzzy cotiuous fuctio, if fo a abitay fixed t The fuzzy fuctio cosideed ae cotiuous i metic D ad the cotiuity of f ( t, y( t); ) of the defiitio of (, ( ); ) f t y t fo t t T ad,1, R ad, guaatees the existece [1]. Theefoe, the fuctios G ad F ca be defiite too. 4.THIRD ORDER RUGE-KUTTA METHOD WITH MODIFIED COTRA-HARMOIC MEA The thid ode Ruge-kutta method with modified cota-hamoic mea was poposed fo appoximatig the solutio of fist ode fuzzy iitial value poblem y( t) f ( t, y( t)) y( t) y. The basis of all Ruge-Kutta methods is to expess the diffeece betwee the value of y at t 1 ad t as 1 m y y w k i i i (4.1) whee w ' sae costat fo all i ad i i1 k hf ( t a h, y c k ) (4.) i i ij j j1 Iceasig of the ode of accuacy of the Ruge-Kutta methods have bee accomplished by iceasig the umbe of taylo s seies tems used ad thus the umbe of fuctioal evaluatios equied[5].the method poposed by Goeke.D ad Johso.O[9] itoduces ew tems ivolvig highe ode deivatives of f i the Ruge-Kutta k tems (i > ) to obtai a highe ode of accuacy without a coespodig icease i evaluatios of f, but with the additio of evaluatios of. Ruge-kutta thid ode method with modified cota-hamoic mea was discussed by Osama Yusuf Ababeh, ad Rokiah Rozita [17]. i Coside y( t ) yt 1 h k1 k 3( k k3 ) 4 k1 k k k3 whee k hf t y t 1, ( ) (4.3) (4.4) 95

5 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 k hf t a h, y( t ) a k (4.5) k hf t a h, y( t ) a k ad the paametes 1, 3 a a ae chose to make 1 y close to yt ( 1).The value of paametes ae a1 a (4.6) 4, (3 ) THIRD ORDER RUGE-KUTTA METHOD WITH MODIFIED COTRA-HARMOIC MEA FOR SOLVIG FUZZY DIFFERETIAL EQUATIOS [ y( t)] [ y( t; ), yt ( ; ) ] Fom 4.3to 4.6 we defie Let the exact solutio [ Y( t) ] [ Y( t; ), Y( t; ) ].The gid poits at which the solutios is calculated ae h T t, i,is appoximated by some t t ih ; i h k1 ( t, y( t, )) k ( t, y( t, )) 3( k ( t, y( t, )) k3 ( t, y( t, ))) y( t 1, ) y( t, ) 4 k1( t, y( t, )) k( t, y( t, )) k( t, y( t, )) k3( t, y( t, )) (5.1) whee k1 hf[ t, y( t, ), y( t, )] ad (5.) k hf[ t, y( t, ) k1( t, y( t, )), y( t, ) k1( t, y( t, ))] (5.3) k3 hf[ t (3 ), y( t, ) (3 ) k( t, y( t, )), (5.4) y( t, ) (3 ) k( t, y( t, ))] 1 h k1 ( t, y( t, )) k ( t, y( t, )) 3( k ( t, y( t, )) k3 ( t, y( t, ))) y( t 1, ) y( t, ) 4 k1( t, y( t, )) k( t, y( t, )) k( t, y( t, )) k3( t, y( t, )) whee k1 hg[ t, y( t, ), y( t, )] (5.6) k hg[ t, y( t, ) k1 ( t, y( t, )), y( t, ) k1 ( t, y( t, ))] (5.7) (5.5) 4 4 k3 hg[ t (3 ), y( t, ) (3 ) k( t, y( t, )), y( t, ) (3 ) k( t, y( t, ))] 1 (5.8) we defie h k1 ( t, y( t, )) k ( t, y( t, )) 3( k ( t, y( t, )) k3 ( t, y( t, ))) F( t, y( t, )) 4 k1( t, y( t, )) k( t, y( t, )) k( t, y( t, )) k3( t, y( t, )) (5.9) 96

6 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 Theefoe we have h k1 ( t, y( t, )) k ( t, y( t, )) 3( k ( t, y( t, )) k3 ( t, y( t, ))) G( t, y( t, )) 4 k1( t, y( t, )) k( t, y( t, )) k( t, y( t, )) k3( t, y( t, )) (5.1) Y ( t, ) (, ) [, (, )] 1 Y t F t Y t Y ( t, ) (, ) [, (, )] 1 Y t G t Y t (5.11) ad y ( t, ) (, ) [, (, )] 1 y t F t y t ] (5.1) y ( t, ) (, ) [, (, )] 1 y t G t y t Clealy y( t ; ) ad y( t ; ) covege to Y( t ; ) ad Y( t ; ) wheeve h 6.UMERICAL EXAMPLE Coside fuzzy iitial value poblem y( t) y( t), t y() (.8.15,1.1.1 ) (6.1) The exact solutio is give by At t=1we get t t Y( t, ) [(.8.15 ) e,(1.1.1 ) e ] Y(1, ) [(.8.15 ) e,(1.1.1 ) e], 1 The values of exact ad appoximate solutio with h=.1 is give i Table : 1. The exact ad appoximate solutios obtaied by the poposed method is plotted i Fig:1.The estimatio of Eo 1 ad Eo is plotted i Fig:. Exact Solutio t=1 Table:1 Appoximate Solutio (h=.1) Y( t ; ) Y( t ; ) y( t ; ) y( t ; ) 97 Eo 1 Eo , , e e , , e e , , e e , , e e , , e e , , e- 4.59e-

7 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, , , e e , , e e , , e e , , e e , , e e- Fig-1 (Appoximate & Exact) Fig- (Eo-1 & Eo-) ACKOWLEDGEMET I humbly ackowledge ad ecod my sicee gatitude to the Uivesity Gat Commissio (UGC) fo havig sactioed a mio eseach poject o the title Fuzzy Diffeetial Equatios.This study has eabled me to big out this pape. I also thak the maagemet of Bishop Hebe College fo thei suppot ad ecouagemet. 98

8 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 COCLUSIO I this pape the uge-kutta thid ode method with modified cota-hamoic mea has bee applied fo fidig the umeical solutio of fist ode fuzzy diffeetial equatios usig tapezoidal fuzzy umbe.the efficiecy ad the accuacy of the poposed method have bee illustated by a suitable example. Fom the umeical example it has bee obseved that the disceet solutios by the poposed method almost coicide with the exact solutios. REFERECES: [1] S.Abbasbady,T AllahVialoo, umeical Solutio of fuzzy diffeetial equatios by Taylo method, Joual of Computatioal Methods i Applied Mathematics(),pp (). [] S.Abbasbady,T.Allah Vialoo, umeical solutio of fuzzy diffeetial equatios by Ruge-Kutta method, oliea studies.11(1),pp (4). [3] J.J.Buckley ad E.Eslami, Itoductio to Fuzzy Logic ad Fuzzy Sets, Physica-Velag, Heidelbeg,Gemay. 1. [4] J.J.Buckley ad E.Eslami ad T.Feuig, Fuzzy Mathematics i Ecoomics ad Egieeig,Physica-Velag, Heidelbeg, Gemay.. [5] J.C Butche, The umeical Aalysis of Odiay Diffeetial equatios Ruge-Kutta ad Geeal Liea Methods, ew Yok: Wiley (1987). [6] S.L.Chag ad L.A.Zadeh, O Fuzzy Mappig ad Cotol, IEEE Tas. Systems Ma Cybeet., (197) [7] D.Dubois,H.Pade, Towads fuzzy diffeetial calculus:pat3, Diffeetiatio,Fuzzy sets ad systems 8, pp. 5-33(198). [8]C.Duaisamy ad B.Usha Aothe appoach to solutio of Fuzzy Diffeetial Equatios Applied Mathematical scieces Vol.4, o.16,777-79(1). [9] D.Goeke, Johso, Ruge Kutta with highe ode deivative Appoximatios Applied. umeical Mathematics 34, pp.7-18(). [1] R Goetschel ad W.Voxma, Elemetay Calculus,Fuzzy sets ad systems,18 (1986) [11]R,Gethsi shamila & E.C,Hey Amithaaj, umeical Solutios of th ode fuzzy iitial value poblems by o-liea Tapezoidal method based o logathimic mea with step size cotol Iteatioal Joual of applied Mathematics & Statistical Scieces Vol 3,Issue 3 july 14,11-4 [1] R,Gethsi shamila & E.C,Hey Amithaaj umeical Solutios of th ode fuzzy iitial value poblems by fouth ode Ruge-kutta Method based o Cetoidal mea IOSR joual of Mathematics Vol 6,Issue 3(May-ju 13),pp [13] R,Gethsi shamila & E.C,Hey Amithaaj, umeical Solutios of fist ode fuzzy iitial value poblems by o-liea Tapezoidal fomulae based o vaiety of Meas Idia joual of Reseach,Vol 3,Issue-5 May 14. [14] R,Gethsi shamila & E.C,Hey Amithaaj umeical Solutios of th ode fuzzy iitial value poblems by fouth ode Ruge-kutta Method based o Cota-hamoic Mea Iteatioal joual o ecet ad iovatio teds i computig ad commuicatios,vol Issue:8,ISS: [15] O.Kaleva, Fuzzy diffeetial equatios, Fuzzy sets ad systems 4 pp (1987). [16] O.Kaleva, The Cauchy poblem fo Fuzzy diffeetial equatios, Fuzzy sets ad systems 35,pp (199). [17] Osama Yusuf Ababeh ad Rokiah Rozita ew thid ode Ruge-kutta Based cotahamoic Mea fo stiff Poblems, Applied Mathematics Scieces,Vol 3,o.8, (9). 99

9 Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 [18] K.Kaagaaj ad M.Sambath umeical solutio of Fuzzy Diffeetial equatios by Thid ode Ruge-Kutta Method, Iteatioal joual of Applied Mathematics ad Computatio Volume.(4),pp 1-8(1). [19] M.Ma,M. Fiedma, Kadel. A, umeical solutios of fuzzy diffeetial equatios, Fuzzy sets ad System 15, pp (1999). [] V.imala ad S.Chethupadia, ew Multi-Step Ruge Kutta Method fo solvig Fuzzy Diffeetial equatios, Mathematical Theoy ad Modelig ISS 4-584(Pape),ISS 5-5 (olie)vol.1, o.3,11. [1]V.imala,.Saveetha,S.Chethupadiya, umeical Solutio of Fuzzy Diffeetial Equatios by Ruge-Kutta Method with Highe ode Deivative Appoximatios,Poceedigs of the Iteatioal cofeece o Emegig Teds i Mathematics ad Compute Applicatios,Idia:MEPCO schlek Egieeig College,Sivakasi Tamiladu,pp (ISB: ) (1) [] Palligkiis,S.Ch.,G.Papageogiou,Famelis,I.TH., Ruge-Kutta methods fo fuzzy diffeetial equatios,applied Mathematics Computatio, 9,pp.97-15(9). [3] M. L. Pui ad D. A. Ralescu, Diffeetials of Fuzzy Fuctios, J. Math. Aal. Appl., 91 pp (1983). [4] S.Seikkala, O the Fuzzy iitial value poblem, Fuzzy sets ad systems 4, pp (1987). 3