A Method for Solving Fuzzy Differential Equations using fourth order Runge-kutta Embedded Heronian Means

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1 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 A Metod fo Solvig Fuzzy Diffeetial Equatios usig fout ode Ruge-kutta Embedded Heoia Meas D.Paul Dayabaa 1, J.Cisty kigsto 2 Associate Pofesso & Picipal, PG ad Reseac Depatmet of Matematics, Bisop Hebe College (Autoomous), Tiuciappalli, Idia 1 Assistat Pofesso, PG ad Reseac Depatmet of Matematics, Bisop Hebe College (Autoomous), Tiuciappalli, Idia 2 ABSTRACT: I tis pape a study as bee caied out to fid a umeical solutio fo te fist ode fuzzy diffeetial equatio by usig fout ode Ruge-kutta metod based o embedded eoia mea. Te accuacy ad applicability of te poposed metod is illustated by solvig a fuzzy iitial value poblem wit a tapezoidal fuzzy umbe. KEYWORDS: Fuzzy Diffeetial Equatios, Ruge-kutta fout Ode metod, Embedded Heoia mea, Tapezoidal Fuzzy Numbe I.INTRODUCTION I ecet yeas fuzzy diffeetial equatios leads a atual way to model dyamical systems ude ucetaity.fist ode liea fuzzy diffeetial equatios ae oe of te simplest fuzzy diffeetial equatio, wic appeas i may applicatios. Te cocept of fuzzy deivative was fist itoduced by S.L.Cag ad L.A.Zade i[6]. D.Dubois ad Pade [7] discussed diffeetiatio wit fuzzy featues.m.l.pui,d.a.ralescu [23] ad R.Goetscel, W.Voxma [10] maily cotibuted towads te diffeetial of fuzzy fuctios. Te fuzzy diffeetial equatio ad iitial value poblems wee extesively studied by O.Kaleva [15,16] ad by S.Seikkala[24].Recetly may eseac papes ae focused o umeical solutio of fuzzy iitial value poblems (FIVPS).Numeical Solutio of fuzzy diffeetial equatio as bee itoduced by M.Ma, M.Fiedma, A.Kadel [18] toug Eule metod ad S.Abbasbady ad T.Allavialoo [1] toug taylo s metod. Ruge Kutta metods ave also bee studied by autos [2,21].V.Nimala, N.Saveeta, S.Cetupadiya discussed umeical Solutio of fuzzy diffeetial equatio by Ruge-Kutta metod wit ige ode deivative appoximatios[20]. R.Getsi samila & E.C.Hey Amitaaj ivestigate umeical solutios of fist ode fuzzy iitial value poblems by o-liea Tapezoidal fomulae based o vaiety of Meas[13].Fout ode Ruge-kutta embedded eoia mea was studied by R.Poalagusamy ad S.Setilkuma [22]. Followig te itoductio i sectio 1,tis pape is ogaised as follows:i sectio 2, some basic esults of fuzzy umbes ad defiitios of fuzzy deivative ae give. I sectio 3,te fuzzy iitial value poblem is discussed. Sectio 4 descibes te stuctue of fout ode uge-kutta embedded eoia mea metod. I sectio 5, te poposed metod is applied fo solvig fuzzy diffeetial equatio poblems ad te umeical examples ae give to illustate te applicability of te poposed metod. Te coclusio was give i te last sectio. Copyigt to IJIRSET DOI: /IJIRSET

2 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology 2.1 FUZZY NUMBER (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 II.PRELIMINARIES A abitay fuzzy umbe is epeseted by a odeed pai of fuctios ( ( ), ( )) wic satisfy te followig coditios. i) u( ) is a bouded left cotiuous o-deceasig fuctio ove ii) u( ) is a bouded igt cotiuous o-deceasig fuctio ove iii) ( u( ) u( )) fo all 0,1 te te -level set is u { x \ u( x) } ;0 1 0,1 wit espect to ay. 0,1 wit espect to ay. u u fo all 0,1 Clealy,u 0 { x \ u( x) 0} is compact, wic is a closed bouded iteval ad we deote byu ( u( ), u( )). 2.2 TRAPEZOIDAL FUZZY NUMBER: A tapezoidal fuzzy umbe u is defied by fou eal umbes k l m, wee te base of te tapezoidal is te iteval [k, ] ad its vetices at x =l, x = m. Tapezoidal fuzzy umbe will be witte as u =( k, l, m, ). Te membesip fuctio fo te tapezoidal fuzzy umbe u = ( k, l, m, ) is defied as te followig : x k k x l l k u x 1 l x m x m x m we ave : (1) u 0 if k 0; (2) u 0 if l 0; (3) u 0 if m 0; ad (4) u 0 if 0; Let us deote R F by te class of all fuzzy subsets of R (i.e. u : R [0,1]) satisfyig te followig popeties: (i) ur F, u is omal, i.e. x 0 R wit u(x o ) = 1; (ii) ur F, u is covex fuzzy set (i.e. u(tx + (1 t) y) mi{u( x ),u( y )}, t[0,1], x, y R); (iii) ur F, u is uppe semi cotiuous o R; (iv) { xr; u( x ) > 0 } is compact, wee A deotes te closue of A. 2.3:Defiitio: ( - Level Set) Let I be te 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, :Defiitio: (Seikkala Deivative) Te Seikkala deivative '( ) y t of a fuzzy pocess is defied by [ '( povided tat tis equatio defies a fuzzy umbe, as i [24] y '( t) y t; ), y '( t; ) ] t I, 0 1 Copyigt to IJIRSET DOI: /IJIRSET

3 2.5:Lemma: ISSN(Olie) : ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology If te sequece of o-egative umbe 0 (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 A 1 positive costats A ad B, te W A W0 B A 1 2.6:Lemma: m W satisfy W 1 A W B, 0 N 1 fo te give m, 0 N If te sequece of o-egative umbesw 0,V 0 satisfy W 1 W A max{ W, V } B, V 1 V A max{ W, V } B fo te give positive costats A ad B, te U W V 0 N A 1 we ave, U A U0 B A 1 2.7:Lemma Let F( t, u, v ) ad G( t, u, v ) belog to te fo abitaily fixed, 0 1, 0 N wee A 1 2A ad B 2B. N, C ' ( R F ) ad te patial deivatives of F ad G be bouded ove R F. 0 2 D( y( t ), y ( t )) L(1 2C) wee L is a boud of patial deivatives 1 1 of F ad G, ad C Max G t, y( t ; ), y( t ; ), 0,1 N N N 1 2.8:Teoem Let F( t, u, v ) ad G( t, u, v) belog to C ' ( RF ) ad te patial deivatives of F ad G be bouded ove R F te fo abitaily fixed, 0 1, te umeical solutios of y( t 1; ) Y ( t 1; ) ad Y ( t 1; ) uifomly i t. 2.9:Teoem Let F( t, u, v) ad G( t, u, v) belog to y( t ; ) ad 1 covege to te exact solutios C ' ( RF ) ad te patial deivatives of F ad G be bouded ove F 2L 1.Te fo abitaily fixed 0 1,te iteative umeical solutios of covege to te umeical solutios y( t; ) ad y( t ; ) i t 0 t t N, we j. III. FUZZY INITIAL VALUE PROBLEM Coside a fist-ode fuzzy iitial value diffeetial equatio is give by y( t) f ( t, y( t)), t t0, T y( t0) y0 wee y is a fuzzy fuctio of ' t ', f ( t, y ) is a fuzzy fuctio of te cisp vaiable ' ' ( j y ) ( t ; ) ad te fuzzy deivative of y ad y( t ) y is a tapezoidal (o) a tapezoidal saped fuzzy umbe. 0 0 We deote te fuzzy fuctio y by y [ y, y].it meas tat te -level set of ( ) [ y( t) ] [ y( t; ), y( t; ) ], y t [ y( t0; ), y( t0; )], ( 0, 1], ( 0) we wite f ( t, y) [ f ( t, y ), f ( t, y) ] ad R ad ( j) y ( t ; ) t ad te fuzzy vaiable y, (3.1) y ' is y t fo t t T 0, is Copyigt to IJIRSET DOI: /IJIRSET

4 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 f ( t, y) F[ t, y, y ], f ( t, y) G [ t, y, y], because of y f ( t, y ) we ave 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 te extesio piciple, we ave te membesip fuctio f ( t, y( t) )( s ) sup{ y( t)( ) \ s f ( t, )}, s R (3.4) so te fuzzy umbe f t, y( t) follows tat ( ( ) wee f ( t, y( t); ) f t, y( t); = f t, y t ) [ f ( t, y( t); ), f ( t, y( t); )], ( 0, 1] (3.5) mi f ( t, u) u y( t ) } (3.6) { max f ( t, u) u y( t ) } (3.7) { Defiitio 3.1 A fuctio f : R RF is said to be fuzzy cotiuous fuctio, if fo a abitay fixed t R 0 ad 0, 0 suc tat t t o ( ), ( ) D f t f t exists. 0 Te fuzzy fuctio cosideed ae cotiuous i metic D ad te cotiuity of f ( t, y( t); ) guaatees te existece of te defiitio of (, ( ); ) be defiite too. f t y t fo t t T ad 0,1 0, [10]. Teefoe, te fuctios G ad F ca IV. FOURTH ORDER RUNGE-KUTTA METHOD WITH EMBEDDED HERONIAN MEAN Fout ode Ruge-kutta metod wit embedded eoia mea is cosideed fo appoximatig te solutio of fist ode fuzzy iitial value poblem y( t) f ( t, y( t)) ; y( t0) y0. Te basis of all Ruge-Kutta metods is to expess te diffeece betwee te value of y at t 1 ad t as m y y w k (4.1) 1 i i i0 wee w ' s ae costat fo all i ad i i1 k f ( t a, y c k ) (4.2) i i ij j j1 Iceasig of te ode of accuacy, te Ruge-Kutta metods ave bee accomplised by iceasig te umbe of Taylo s seies tems used ad tus te umbe of fuctioal evaluatios equied[5].te metod poposed by Goeke.D ad Joso.O[9] itoduces ew tems ivolvig ige ode deivatives of f i te Ruge-Kutta k tems( i > 0) to obtai a ige ode of accuacy witout a coespodig icease i evaluatios of f, but wit te additio of evaluatios of f. Te Fout ode Ruge-Kutta metod wit embedded eoia mea fo step +1 is give by R.Poalagusamy ad S.Setilkuma [22]. y( t ) y( t ) k 2( k 9 k ) k k k k k k k Wee k1 f ( t, y( t )) Coside k f ( t a, y( t ) a k ) (4.5) (4.3) (4.4) Copyigt to IJIRSET DOI: /IJIRSET

5 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 k3 f ( t ( a2 a3), y( t) a2k1 a3k2 ) (4.6) k4 f ( t ( a4 a5 a6 ), y( t ) a4k1 a5k2 a6k3 ) (4.7) ad te paametes a 1, a 2, a 3, a 4, a 5, a 6 ae cose to make y 1 close to y( t 1).Te value of paametes ae a1, a 2, a3, a4, a5, a V. FOURTH ORDER RUNGE- KUTTA EMBEDDED HERONIAN MEAN FOR SOLVING FUZZY DIFFERENTIAL EQUATIONS Let te exact solutio[ Y( t) ] [ Y( t; ), Y ( t; ) ],is appoximated by some [ y( t)] [ y( t; ), y( t; ) ] poits at wic te solutios is calculated ae Fom 4.3 to 4.7 we defie T t 0 N, ti t0 i ; 0 i N.Te gid k1( t, y( t, )) 2( k2 ( t, y( t, )) k3( t, y( t, ))) y( t 1, ) y( t, ) k4( t, y( t, )) k1( t, y( t, )) k2( t, y( t, )) 9 k2( t, y( t, )) k3( t, y( t, )) k3( t, y( t, )) k4( t, y( t, )) (5.1) wee k1 F[ t, y( t, ), y( t, )] (5.2) 1 1 k2 F[ t, y( t, ) k1( t, y( t, )), y( t, ) k1( t, y( t, ))] (5.3) k3 F[ t, y( t, ) k1( t, y( t, )) k2 ( t, y( t, )), (5.4) y( t, ) k1( t, y( t, )) k 2 ( t, y( t, ))] k4 F[ t, y( t, ) k1( t, y( t, )) k2( t, y( t, )) k3( t, y( t, )), y( t, ) k1( t, y( t, )) k2( t, y( t, )) k3( t, y( t, ))] (5.5) ad k1( t, y( t, )) 2( k2 ( t, y( t, )) k3( t, y( t, ))) y( t 1, ) y( t, ) k4( t, y( t, )) k1( t, y( t, )) k2( t, y( t, )) 9 k2( t, y( t, )) k3( t, y( t, )) k3( t, y( t, )) k4( t, y( t, )) Copyigt to IJIRSET DOI: /IJIRSET

6 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 (5.6) wee k1 G[ t, y( t, ), y( t, )] (5.7) 1 1 k2 G[ t, y( t, ) k1( t, y( t, )), y( t, ) k1( t, y( t, ))] (5.8) k3 G[ t, y( t, ) k1( t, y( t, )) k2( t, y( t, )), y( t, ) k1( t, y( t, )) k2( t, y( t, ))] (5.9) k4 G[ t, y( t, ) k1( t, y( t, )) k2( t, y( t, )) k3( t, y( t, )), y( t, ) k1( t, y( t, )) k2( t, y( t, )) k3( t, y( t, ))] (5.10) We defie k1 ( t, y( t, )) 2( k2( t, y( t, )) k3( t, y( t, ))) F[ t, y( t, )] k4 ( t, y( t, )) k1( t, y( t, )) k2( t, y( t, )) 9 k2( t, y( t, )) k3( t, y( t, )) k3( t, y( t, )) k4( t, y( t, )) k1( t, y( t, )) 2( k2( t, y( t, )) k3( t, y( t, ))) G[ t, y( t, )] k4( t, y( t, )) k1( t, y( t, )) k2( t, y( t, )) 9 k2( t, y( t, )) k3( t, y( t, )) k3( t, y( t, )) k4( t, y( t, )) (5.11) (5.12) Teefoe we ave Y ( t, ) (, ) [, (, )] 1 Y t F t Y t Y ( t, ) (, ) [, (, )] 1 Y t G t Y t (5.13) Ad y ( t, ) (, ) [, (, )] 1 y t F t y t (5.14) y ( t, ) (, ) [, (, )] 1 y t G t y t Clealy y( t; ) ad y( t; ) covege to Y( t; ) ad Y ( t; ) weeve 0 VI. NUMERICAL EXAMPLE Coside fuzzy iitial value poblem y( t) y( t), t 0 y(0) ( , ) Te exact solutio is give by (6.1) Copyigt to IJIRSET DOI: /IJIRSET

7 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 t t Y ( t, ) [( ) e,( ) e ] At t=1 we get Y(1, ) [( ) e,( ) e], 0 1 Te values of exact ad appoximate solutio wit = 0.1 is give i Table :1.Te appoximate solutios obtaied by te poposed metod is plotted i Fig :I ad te eo estimatio of te exact ad appoximate solutio is give i Fig II. Exact Solutio t=1 Table:1 Appoximate Solutio (=0.1) Y( t, ) Y ( t, ) y( t, ) y( t, ) Eo 1 Eo , , e e , , e e , , e e , , e e , , e e , , e e , , e e , , e e , , e e , , e e , , e e-002 Fig-I Fig II. Copyigt to IJIRSET DOI: /IJIRSET

8 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 VI. CONCLUSION I tis pape te fout ode uge-kutta metod fo embedded eoia mea as bee applied fo fidig te umeical solutio of fist ode fuzzy diffeetial equatio usig tapezoidal fuzzy umbe.te accuacy ad applicability of te poposed metod ave bee illustated by a suitable example. Fom te umeical example it as bee obseved tat by miimizig te step size te exact solutios at diffeet poits of te solutio cuve ad te coespodig appoximate solutio coicides. ACKNOWLEDGEMENT I umbly ackowledge ad ecod my sicee gatitude to te Uivesity Gat Commissio (UGC) fo avig sactioed a mio eseac poject o te title Fuzzy Diffeetial Equatios.Tis study as eabled me to big out tis pape. I also tak te maagemet of Bisop Hebe College fo tei suppot ad ecouagemet. REFERENCES [1]S.Abbasbady,T AllaVialoo,.(2002), Numeical Solutio of fuzzy diffeetial equatios by Taylo metod,joual of ComputatioalMetods i Applied Matematics2(2),pp [2] S.Abbasbady,T.Alla Vialoo,(2004), Numeical solutio of fuzzy diffeetial equatios by Ruge-Kutta metod, Noliea studies.11(1), pp [3] J.J.Buckley ad E.Eslami, Itoductio to Fuzzy Logic ad Fuzzy Sets, Pysica-Velag, Heidelbeg,Gemay [4] J.J.Buckley ad E.Eslami ad T.Feuig, Fuzzy Matematics i Ecoomics ad Egieeig,Pysica-Velag, Heidelbeg, Gemay [5] J.C Butce, (1987), Te Numeical Aalysis of Odiay Diffeetial equatios Ruge-Kutta ad Geeal Liea Metods, New Yok: Wiley. [6] S.L.Cag ad L.A.Zade, O Fuzzy Mappig ad Cotol, IEEE Tas. Systems Ma Cybeet., 2 (1972) [7] D.Dubois,H.Pade,.(1982), Towads fuzzy diffeetial calculus:pat3, Diffeetiatio,Fuzzy sets ad systems 8, pp [8] C.Duaisamy ad B.Usa Aote appoac to solutio of Fuzzy Diffeetial Equatios Applied Matematical scieces Vol.4,2010,o.16, [9] D.Goeke, Joso.(2000), Ruge Kutta wit ige ode deivative Appoximatios Applied.Numeical Matematics 34, pp [10] R Goetscel ad W.Voxma, Elemetay Calculus,Fuzzy sets ad systems,18 (1986) [11] R,Getsi samila & E.C,Hey Amitaaj, Numeical Solutios of N t ode fuzzy iitial value poblems by No-liea Tapezoidal metod based o logatimic mea wit step size cotol Iteatioal Joual of applied Matematics & Statistical Scieces Vol 3,Issue 3 july 2014,11-24 [12] R,Getsi samila & E.C,Hey Amitaaj Numeical Solutios of N t ode fuzzy iitial value poblems by fout ode Ruge-kutta Metod based o Cetoidal mea IOSR joual of Matematics Vol 6,Issue 3(May-ju 2013),pp [13] R,Getsi samila & E.C,Hey Amitaaj, Numeical Solutios of fist ode fuzzy iitial value poblems by No-liea Tapezoidal fomulae based o vaiety of Meas Idia joual of Reseac,Vol 3,Issue-5 May Copyigt to IJIRSET DOI: /IJIRSET

9 ISSN (Pit) : Iteatioal Joual of Iovative Reseac i Sciece, Egieeig ad Tecology (A ISO 3297: 2007 Cetified Ogaizatio) Vol. 5, Issue 3, Mac 2016 [14] R,Getsi samila & E.C,Hey Amitaaj Numeical Solutios of N t ode fuzzy iitial value poblems by fout ode Ruge-kutta Metod based o Cota-amoic Mea Iteatioal joual o ecet ad iovatio teds i computig ad commuicatios,vol 2 Issue:8,ISSN: [15] O.Kaleva, (1987), Fuzzy diffeetial equatios, Fuzzy sets ad systems 24 pp [16] O.Kaleva, (1990), Te Caucy poblem fo Fuzzy diffeetial equatios, Fuzzy sets ad systems 35,pp [17] K.Kaagaaj ad M.Sambat Numeical solutio of Fuzzy Diffeetial equatios by Tid ode Ruge-Kutta Metod, Iteatioal joual of Applied Matematics ad Computatio Volume.2(4),pp 1-8,2010. [18] M.Ma,M. Fiedma, M., Kadel, A (1999), Numeical solutios of fuzzy diffeetial equatios, Fuzzy sets ad System 105, pp [19] V.Nimala ad S.Cetupadia, New Multi-Step Ruge Kutta Metod fo solvig Fuzzy Diffeetial equatios,matematical Teoy ad Modelig ISSN (Pape),ISSN (olie)vol.1, No.3,2011. [20] V.Nimala,N.Saveeta,S.Cetupadiya,(2010) Numeical Solutio of Fuzzy Diffeetial Equatios by Ruge-Kutta Metod wit Hige ode Deivative Appoximatios,Poceedigs of te Iteatioal cofeece o Emegig Teds i Matematics ad Compute Applicatios,Idia:MEPCO sclek Egieeig College,Sivakasi Tamiladu,pp (ISBN: ) [21]Palligkiis, S.C.,G.Papageogiou,Famelis,I.TH.(2009), Ruge-Kutta metods fo fuzzy diffeetial equatios, Applied Matematics Computatio, 209,pp [22] R.Poalagusamy ad S.Setilkuma, A New Fout Ode Embedded RKAHeM(4,4) metod wit Eo Cotol o Multilaye Raste Cellula Neual Netwok, Sigal Image ad Video Pocessig, 2008[Accepted i Pess]. [23] M. L. Pui ad D. A. Ralescu, Diffeetials of Fuzzy Fuctios, J. Mat. Aal. Appl., 91 (1983) [24] S.Seikkala,(1987), O te Fuzzy iitial value poblem, Fuzzy sets ad systems 24, pp Copyigt to IJIRSET DOI: /IJIRSET