[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY SOLVING FUZZY DIFFERENTIAL EQUATIONS USING RUNGE-KUTTA FOURTH ORDER GILL METHOD D.Paul Dhayabaa * J.Chisty kigsto * Associate Pofesso & Picipal PG ad Reseach Depatmet of Mathematics Bishop Hebe College (Autoomous) Tiuchiappalli -620 07Idia Assistat Pofesso PG ad Reseach Depatmet of Mathematics Bishop Hebe College (Autoomous) Tiuchiappalli -620 07Idia ABSTRACT I this pape a umeical solutio fo the fist ode fuzzy diffeetial equatios by usig fouth ode Ruge-kutta Gill method is cosideed.the applicability ad accuacy of the poposed method has bee demostated by a example ad the covegece of the method has bee studied with a tiagula fuzzy umbe. KEYWORDS: Fuzzy Diffeetial Equatios Ruge-kutta fouth ode Gill MethodTiagula Fuzzy umbe. INTRODUCTION Fuzzy diffeetial equatios ae a atual way to model dyamical systems ude ucetaity.fist ode liea fuzzy diffeetial equatio is oe of the simplest fuzzy diffeetial equatioswhich 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] discussed diffeetiatio with fuzzy featues.m.l.pui D.A.Ralescu[24] ad R.Goetschel W.Voxma[0]these authos cotibuted towads the diffeetial of fuzzy fuctios.the fuzzy diffeetial equatio ad iitial value poblems wee extesively studied by O.Kaleva[56] ad by S.Seikkala[25].Recetly may eseach papes ae focused o umeical solutio of fuzzy iitial value poblems.numeical Solutio of fuzzy diffeetial equatios has bee itoduced by M.Ma M.Fiedma A.Kadel [9] though Eule method ad S.AbbasbadyT.Allahvialoo [] by Taylo s method.ruge Kutta methods have also bee studied by authos [222].V.NimalaN.SaveethaS.Chethupadiya discussed o umeical solutio of fuzzy diffeetial equatios by Ruge-Kutta method with highe ode deivative appoximatios[2].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[3].fouth ode Ruge-kutta Gill method was poposed by Oliveia.S.C [7]ad also it was studied by R.Poalagusamy S.Sethilkuma [23].Followig by the itoductio this pape is ogaised as follows:i sectio 2 some basic esults of fuzzy umbes ad defiitios of fuzzy deivative ae give.i sectio 3 the fuzzy iitial value poblem is beig discussed.sectio 4 descibes the geeal stuctue of the fouth ode Ruge-kutta Gill method.i sectio 5the fouth ode Ruge-kutta Gill 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 followed by the coclusio give i the last sectio. PRELIMINARIES Defiitio:(FUZZY NUMBER) A abitay fuzzy umbe is epeseted by a odeed pai of fuctios ( ( ) ( )) the followig coditios. i) () u is a bouded left cotiuous o-deceasig fuctio ove u u fo all 0 0 with espect to ay. which satisfy http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [844]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 ii) u () is a bouded ight cotiuous o-deceasig fuctio ove u fo all the the -level set is \ ( ) };0 iii) ( u( ) ( )) Clealyu 0 { x \ u( x) 0} 0 0 with espect to ay. u { x u x is compact which is a closed bouded iteval ad we deote by u ( u( ) u( )) Defiitio: (TRIANGULAR FUZZY NUMBER) A tiagula fuzzy umbe u is a fuzzy set i E that is chaacteized by a odeed tiple such that u u ; u ad ul uc u 0 l The membeship fuctio of the tiagula fuzzy umbe u is give by x ul ; ul x u uc ul u( x) ; x uc u x ; uc x u u uc. we have : () u 0 if u l 0; (2) u 0 if u l 0; (3) u 0 if u c 0 ;ad (4) u0 if u c 0 Defiitio: ( - Level Set) c u l u c. ( u u u ) l c 3 with 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 0 Defiitio: (Seikkala Deivative) The Seikkala deivative '( ) y t of a fuzzy pocess is defied by [ '( povided that this equatio defies a fuzzy umbe as i [25] y '( t) y t; ) y '( t; ) ] t I 0 R http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [845]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 Lemma: W If the sequece of o-egative umbe 0 m positive costats A ad B the 0 Lemma: If the sequece of o-egative umbes 0 0 N fo the give satisfy W A W B A W A W B A 0 N W m N V 0 satisfy W W A max{ W V } B V V A max{ W V } Bfo the give positive costats A ad B the U W V 0 N A U A U B A we have 0 0 N whee A 2A ad B 2B. Lemma Let F( t u v ) ad G( t u v ) belog to C'( R F ) ad the patial deivatives of F ad G be bouded ove F abitaily fixed 0 D y t y ( t ) h L C) 0 2 ( ( ) ) ( 2 ad C Max G t y( t ; ) y( t ; ) 0 Theoem N N N Let F( t u v ) ad G( t u v) belog to C'( R ) F R the fo whee L is a boud of patial deivatives of F ad G abitaily fixed 0 the umeical solutios of y( t ; ) Y( t ; ) ad Y( t ; ) uifomly i t. Theoem ad the patial deivatives of F ad G be bouded ove y( t ; ) ad R F covege to the exact solutios the fo Let F( t u v) ad G( t u v) belog to C'( RF ) ad the patial deivatives of F ad G be bouded ove RF ad 2Lh. The fo abitaily fixed 0 the iteative umeical solutios of umeical solutios y( t; ) ad y( t; ) i t 0 t t N whe j. ( j y ) ( t ; ) ad ( j) y ( t ; ) covege to the http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [846]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 FUZZY INITIAL VALUE PROBLEM A fist-ode fuzzy iitial value diffeetial equatio is give by y( t) f ( t y( t)) t t0 T y( t0) y0 whee y is a fuzzy fuctio of fuzzy deivative of y ad t y( t ) y 0 0 f ( t y) is a fuzzy fuctio of the cisp vaiable is a tiagula o a tiagula shaped fuzzy umbe. t ad the fuzzy vaiable y y ' is the (3.) We deote the fuzzy fuctio y by y [ y( t)] [ y( t; ) y( t; )] [ y y].it meas that the -level set of () yt fo t t T is 0 yt [ y( t0; ) y( t0; )] ( 0) ( 0 ] we wite f ( t y) [ f ( ) f( t y) ] ty ad f ( t y) F[ t y y ] f ( t y) G [ t y y] because of y f ( t y ) we have f ( t y( t); ) F [ t y( t; ) y( t; )] (3.2) 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) ; )] ( 0 ] (3.5) whee f t y( t); mi{ f ( t u) u y( t ) } (3.6) f t y( t); max{ f ( t u) u y( t) }. (3.7) http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [847]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 Defiitio 3. A fuctio f : R RF is said to be fuzzy cotiuous fuctio if fo a abitay fixed 0 such that t to D f ( t) f ( t ) exists. 0 t 0 R ad 0 The fuzzy fuctio cosideed ae cotiuous i metic D ad the cotiuity of f ( t y( t); ) guaatees the existece of the defiitio of ( ( ); ) f t y t fo ad t t T 0 0 [0]. Theefoe the fuctios G ad F ca be defiite too. FOURTH ORDER RUNGE-KUTTA GILL METHOD Fouth ode Ruge-kutta Gill method was poposed fo appoximatig the solutio of the iitial value poblem y( t) f ( t y( t)) y( t0) y0. The basis of all Ruge-Kutta methods is to expess the diffeece betwee the value of y at t ad t m as y y w k (4.) i i i0 whee w i s ae costat fo all i ad i k hf ( t a h y c k ) (4.2) i i ij j j Icease 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 i tems( i > 0) to obtai a highe ode of accuacy without a coespodig icease i evaluatios of f but with the additio of evaluatios of f. The fouth ode Ruge-kutta Gill method fo step + which was poposed by Oliveia.S.C [7] is give by y( t ) y( t) k (2 2) k2 (2 2) k3 k 4 6 Whee k hf ( t y( t )) (4.3) (4.4) k hf ( t a h y( t ) a k ) (4.5) 2 k hf ( t ( a a ) h y( t ) a k a k ) 3 2 3 2 3 2 k hf ( t ( a a ) h y( t ) a k a k ) 4 4 5 4 2 5 3 (4.6) (4.7) The paametes a a2 a3 a4 a 5 ae chose to make y close to y( t ).The value of paametes ae a 2 3 4 5 2 a 2 2 a 2 a 2 a 2 FOURTH ORDER RUNGE-KUTTA GILL METHOD FOR SOLVING FUZZY DIFFERENTIAL EQUATIONS http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [848]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 Let the exact solutio[ Y( t)] [ Y( t; ) Y( t; )] is appoximated by some [ y( t)] [ y( t; ) y( t; )] at which the solutios is calculated ae Fom 4.3to 4.7 we defie T t h 0 N t t ih i 0 ; 0 i N the gid poits k( t y( t )) (2 2) k2( t y( t )) y( t ) y( t ) 6 (2 2) k3( t y( t )) k4( t y( t )) (5.) whee k h F[ t y( t ) y( t )] (5.2) h k2 hf[ t y( t ) k( t y( t )) y( t ) k( t y( t ))] (5.3) 2 2 2 h k3 hf[ t y( t ) ( ( )) 2( ( )) 2 2 2 k t y t k t y t 2 y( t ) k( t y( t )) k2( t y( t ))] 2 2 2 (5.4) k4 hf[ t h y( t ) k2( t y( t )) k3( t y( t )) 2 2 y( t ) k2( t y( t )) k3( t y( t ))] 2 2 (5.5) ad k( t y( t )) (2 2) k2( t y( t )) y( t ) y( t ) 6 (2 2) k3( t y( t )) k4( t y( t )) (5.6) whee k hg[ t y( t ) y( t )] (5.7) h k2 hg[ t y( t ) k( t y( t )) y( t ) k( t y( t ))] (5.8) 2 2 2 http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [849]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 h k3 hg[ t y( t ) ( ( )) 2( ( )) 2 2 2 k t y t k t y t 2 y( t ) k( t y( t )) k2( t y( t ))] 2 2 2 (5.9) k4 hg[ t h y( t ) k2( t y( t )) k3( t y( t )) 2 2 y( t ) k2( t y( t )) k3( t y( t ))] 2 2 (5.0) we defie k( t y( t )) (2 2) k2( t y( t )) F[ t y( t )] 6 (2 2) k3( t y( t )) k4( t y( t )) (5.) (5.2) k( t y( t )) (2 2) k2( t y( t )) G[ t y( t )] 6 (2 2) k3( t y( t )) k4( t y( t )) Theefoe we have Y ( t ) ( ) [ ( )] Y t F t Y t (5.3) Y ( t ) ( ) [ ( )] Y t G t Y t (5.4) Ad y ( t ) ( ) [ ( )] y t F t Y t (5.6) y ( t ) ( ) [ ( )] y t G t Y t Clealy y( t ; ) ad y( t ; ) covege to Y( t ; ) ad Y( t ; ) wheeve h 0. (5.7) http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [850]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 NUMERICAL EXAMPLE Coside the fuzzy iitial value poblem y( t) y( t) t 0 y(0) (0.75 0.25 ;.5 0.5 ) (6.) The exact solutio is give by ( ) [( 0.75 0.25 ) t t Y t e (.5 0.5 ) e ] At t=we get Y( ) [ ( 0.75 0.25 ) e(.5 0. 5 ) e] 0 (6.2) (6.3) The values of exact ad appoximate solutio with h= 0. is give i Table :.The exact ad appoximate solutios obtaied by the poposed method is plotted i Figue:.The eos of exact ad appoximate solutios is plotted i Figue:2. Table: Exact ad Appoximate Solutios Exact Solutio (t=) Appoximate Solutio (h=0.) Y( t ) Y( t ) y( t ) y( t ) Eo Eo 2 0 2.0387 4.077423 2.03870 4.077420.563243e-006 3.26486e-006 0. 2.06668 3.94509 2.06667 3.94506.6535e-006 3.022270e-006 0.2 2.74625 3.805595 2.74624 3.805592.667459e-006 2.98053e-006 0.3 2.242583 3.669680 2.24258 3.669678.79567e-006 2.83837e-006 0.4 2.30540 3.533766 2.30538 3.533764.77675e-006 2.70962e-006 0.5 2.378497 3.397852 2.378495 3.397850.823783e-006 2.605405e-006 0.6 2.446454 3.26938 2.446452 3.26936.87589e-006 2.5089e-006 0.7 2.544 3.26024 2.54409 3.26022.928000e-006 2.396972e-006.0.8 2.582368 2.9900 2.582366 2.99008.98008e-006 2.292756e-006 0.9 2.650325 2.85496 2.650323 2.85494 2.03226e-006 2.88540e-006.0 2.78282 2.78282 2.78280 2.78280 2.084324e-006 2.084324e-006 http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [85]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 Figue: Figue:2 Exact ad Appoximate Solutio Eos of Exact ad Appoximate Solutios http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [852]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 CONCLUSION I this pape the umeical solutio of diffeetial equatios with fuzzy iitial values wee studied.the fouth ode Ruge-kutta gill method has bee applied fo fidig the umeical solutio of fist ode fuzzy diffeetial equatios usig tiagula 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. ACKNOWLEDGEMENT 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. REFERENCES [] S.AbbasbadyT.AllahVialoo(2002) Numeical Solutio of fuzzy diffeetial equatios by Taylo method Joual of Computatioal Methods i Applied Mathematics2(2)pp.3-24. [2] S.AbbasbadyT.Allah Vialoo(2004) Numeical solutio of fuzzy diffeetial equatios by Ruge- Kutta method Noliea studies.()pp. 7-29. [3] J.J.Buckley ad E.Eslami Itoductio to Fuzzy Logic ad Fuzzy Sets Physica-VelagHeidelbeg Gemay. 200. [4] J.J.Buckley ad E.Eslami ad T.Feuig Fuzzy Mathematics i Ecoomics ad Egieeig Physica- Velag Heidelbeg Gemay.2002. [5] J.C Butche(987) The Numeical Aalysis of odiay diffeetial equatios Ruge-Kutta ad Geeal Liea Methods New Yok: Wiley. [6] S.L.Chag ad L.A.Zadeh O Fuzzy Mappig ad Cotol IEEE Tas. Systems Ma Cybeet. 2 (972) 30-34. [7] D.DuboisH.Pade.(982) Towads fuzzy diffeetial calculus:pat3 Diffeetiatio Fuzzy sets ad systems 8 pp.2 25-233. [8] C.Duaisamy ad B.Usha Aothe appoach to solutio of Fuzzy Diffeetial Equatios Applied Mathematical scieces Vol.4 200o.6777-790 [9] D.GoekeJohso.(2000) Ruge Kutta with highe ode deivative Appoximatios Applied. Numeical Mathematics 34 pp.207-28. [0] R Goetschel ad W.Voxma Elemetay CalculusFuzzy sets ad systems8 (986) 3-43. [] R.Gethsi shamila & E.C.Hey Amithaaj Numeical Solutios of N th ode fuzzy iitial value poblems by No-liea tapezoidal method based o logathimic mea with step size cotol Iteatioal Joual of applied Mathematics & Statistical Scieces Vol 3Issue 3 july 204-24 [2] R.Gethsi shamila & E.C.Hey Amithaaj Numeical Solutios of N th ode fuzzy iitial value poblems by fouth ode Ruge-kutta Method based o Cetoidal mea IOSR joual of Mathematics Vol 6Issue 3(May-ju 203)pp 47-63 [3] R.Gethsi shamila & E.C.Hey Amithaaj Numeical Solutios of fist ode fuzzy iitial value poblems by o-liea tapezoidal fomulae based o vaiety of meas Idia joual of ReseachVol 3Issue-5 May 204. [4] R.Gethsi shamila & E.C.Hey Amithaaj Numeical Solutios of N 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 commuicatiosvol 2 Issue:8ISSN:232-869. [5] O.Kaleva (987) Fuzzy diffeetial equatios Fuzzy sets ad systems 24 pp.30-37. http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [853]
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 [6] O.Kaleva (990) The Cauchy poblem fo Fuzzy diffeetial equatios Fuzzy sets ad systems 35pp 389-396. [7] Oliveia.S.CEvaluatio of effectiveess facto of immobilized ezymes usig uge-kutta Gill method : how to solve mathematical udtemiatio at paticle cete poit?bio pocess Egieeig20(999) pp.85-87 [8] K.Kaagaaj ad M.Sambath Numeical solutio of Fuzzy Diffeetial equatios by thid ode Ruge- Kutta Method Iteatioal joual of Applied Mathematics ad Computatio Volume.2(4)pp -8200. [9] M.MaM. Fiedma M. Kadel (999) Numeical solutios of fuzzy diffeetial equatios Fuzzy sets ad System 05 pp. 33-38. [20] V.Nimala ad S.Chethupadia New Multi-Step Ruge Kutta Method fo solvig Fuzzy Diffeetial equatios Mathematical Theoy ad Modelig ISSN 2224-5804(Pape)ISSN 2225-0522 (olie)vol. No.320. [2] V.NimalaN.SaveethaS.Chethupadiya(200) Numeical 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 ApplicatiosIdia:MEPCO schlek Egieeig CollegeSivakasi Tamiladupp.3-34(ISBN:978-8-8424-649-0) [22] PalligkiisS.Ch.G.PapageogiouFamelisI.TH.(2009) Ruge-Kutta methods fo fuzzy diffeetial equatios Applied Mathematics Computatio 209pp.97-05. [23] R.Poalagusamy S.Sethilkuma A compaiso of Ruge kutta-fouth odes of vaiety of meas ad embedded meas o Multilaye Raste CNN simulatio Joual of Theoetical ad Applied Ifomatio Techology 2005-2007. [24] M. L. Pui ad D. A. Ralescu Diffeetials of Fuzzy Fuctios J. Math. Aal. Appl. 9 (983) 32-325. [25] S.Seikkala(987) O the Fuzzy iitial value poblem Fuzzy sets ad systems 24 pp.39-330. http: // www.ijest.com Iteatioal Joual of Egieeig Scieces & Reseach Techology [854]