Fourth Order Runge-Kutta Method Based On Geometric Mean for Hybrid Fuzzy Initial Value Problems

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1 IOSR Joural of Mahemacs (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 2 Ver. II (Mar. - Apr. 27), PP Fourh Order Ruge-Kua Mehod Based O Geomerc Mea for Hybrd Fuzzy Ial Value Problems R. Gehs Sharmla Asssa Professor Deparme of Mahemacs Bshop Heber College (Auoomous), Truchrappall -62 7, Tamladu, Ida. Absrac: I hs paper, a umercal algorhm for solvg hybrd fuzzy al value problem usg he Fourh Order Ruge-Kua mehod based o Geomerc Mea (RK4GM) s proposed. The algorhm s llusraed by solvg hybrd fuzzy al value problems usg ragular fuzzy umber. I s compared wh he classcal fourh order Ruge Kua mehod. The resuls show ha he proposed fourh order Ruge-Kua mehod based o Geomerc Mea wors well for solvg hybrd fuzzy al value problems. Keywords: Numercal soluo, Hybrd Fuzzy Ial Value Problems, Tragular fuzzy umber, Fourh order Ruge-Kua mehod, Geomerc Mea I. Iroduco Fuzzy se heory s a ool ha maes possble o descrbe vague ad ucera oos. Fuzzy Dffereal Equao (FDE) models have wde rage of applcaos may braches of egeerg ad he feld of medce. The cocep of a fuzzy dervave was frs roduced by Chag ad Zadeh [5], laer Dubos ad Prade [6] defed he fuzzy dervave by usg Zadeh s exeso prcple ad he followed by Pur ad Ralescu [2]. Fuzzy dffereal equaos have bee suggesed as a way of modellg ucera ad compleely specfed sysems ad were suded by may researchers [9,]. The exsece of soluos of fuzzy dffereal equaos has bee suded by several auhors [, 4]. I s dffcul o oba a exac soluo for fuzzy dffereal equaos ad hece several umercal mehods where proposed [, 2, 4, 8]. Abbasbady ad Allahvraloo [2] developed umercal algorhms for solvg fuzzy dffereal equaos based o Seala s dervave of fuzzy process [25]. Ruge-Kua mehod for fuzzy dffereal equao has bee suded by may auhors [, 9]. The umercal soluo o hybrd fuzzy sysems has bee suded by he auhors [8, 2,2,22 ad 24]. Abdul-Majd Wazwaz [] roduced a comparso of modfed Ruge-Kua formulas based o a varey of meas. Evas ad Yaacub [7] compared he umercal o.d.e. solvers based o arhmec ad geomerc meas for frs order IVPs. Murugesa e al. [6] compared fourh order RK mehods based o varey of meas ad cocluded ha RK4CeM wors very well o solve sysem of IVPs ad hey also developed [7] a ew embedded RK mehod based o AM ad CeM. The applcably of he RK4CeM : Dvso by zero, Error RK4CeM formulae, ad Sably aalyss are dscussed by Murugesa e al. [5]. Hybrd sysems are devoed o modelg, desg, ad valdao of eracve sysems ha are capable of corollg complex sysems whch have dscree eve dyamcs as well as couous me dyamcs ca be modeled by hybrd sysems. The dffereal sysems coag fuzzy valued fucos ad eraco wh a dscree me coroller are amed hybrd fuzzy dffereal sysems. The umercal soluo o hybrd fuzzy sysems has bee suded by he auhors [8, 2, 2, 22 ad 24]. I hs paper, he RK4GM s appled o solve hybrd fuzzy al value problems. The srucure of he paper s orgazed as follows: I Seco 2, some ecessary oaos ad defos of fuzzy se heory, hybrd fuzzy dffereal equaos, he fourh order Ruge-Kua formula based o Geomerc Mea o solve IVPs are gve. I Seco Hybrd Fuzzy Ial Value Problem s defed ad he umercal algorhm for solvg he fuzzy al value problems by he fourh order Ruge-Kua mehod based o Geomerc Mea are gve. Seco 4 coas he umercal example. The cocluso s gve seco 5. II. Prelmares I hs seco, some of he basc defos of fuzzy umbers ad Ruge-Kua mehods are revewed. 2. Defos ad oaos Deoe by E he se of all fucos u : R [,] such ha () u s ormal, ha s, here exs a x R such ha ux ( ),() u s a fuzzy covex, ha s, for x, y R ad, DOI:.979/ Page

2 u ( x ( ) y) m { ( x), u( y)} { x R : u( x) } s compac. For r, u E s gve by u,() u s upper sem-couous, ad (v) we defe u r { x R : u( x) r}. 4x, f x(.75,], u( x) 2x, f x(,.5],, f x (.75,.5], The r - level ses of u (2.) are gve by u r r For laer purpose, we defe seala dervave [25] of I ad r, problem (IVP) r [.75.25,.5.5 ]., he u he closure of A example of a DOI:.979/ Page (2.) (2.2) E as ( x), f x = ad ( x), f x. Nex we revew he x : I E where R x ( ) x ( ; r), x ( ; r) r I s a erval. If x ( ) x( ; r), x( ; r) r f x () E. Nex cosder he al value y( ) f (, y( )), uy ( ) y() y, (2.) for all where f : [, ) R R s couous. We would le o erpre (2.) usg he Seala dervave ad x x ( r), x ( r) E. Le r ge f :[, ) E E ad Where x( ) x( ; r), x( ; r) r by he zadeh exeso prcple we [ f (, y)] [m{ f (, u) : u [ x( ; r), x( ; r)]},max{ f (, u) : u [ x( ; r), x( ; r)]}]. The r x :[, E ) s a soluo of (2.) usg he seala dervave ad x ( ; r) m{ f (, u) : u [ x( ; r), x( ; r)]}, x(; r) x ( r) ( ; ) max{ (, ) : [ ( ; ), ( ; )]}, (; ) ( ) x r f u u x r x r x r x r [, ad r [,]. f [,) R R R For all ) Cosder a fuco (2.4) : whch s couous ad he IVP y ( ) f (, y( ), ), y y o erpre(2.4) usg he seala dervave ad f :[, E ) E E where y, E, x E f by he zadeh exeo prcple we use [ f (, y, )] [m{ f (, u, u ) : u [ y( ; r), y( ; r)], u [ ( r ), ( r )]}, where r ( r), ( r) y, E f r max{ f (, u, u ) : u [ y( ; r), y( ; r)], u [ ( r), ( r)]}, he y:[, ) E s a soluo of (2.4) usg he seala dervave ad y(, y; r) m{ f (, u, u ) \ u [ y( ; r), y( ; r)], u [ ( r), ( r)]}, y(; r) y ( r) y(, y; r) max{ f (, u, u ) \ u [ y( ; r), y( ; r)], u [ ( r), ( r)]}, y(; r) y ( r) for all [, ) ad r [,]. x

3 2.2 Ruge Kua Mehod for Ial Value Problem Cosder he al value problem y( ) f (, y( )) ; a b y(a) r, The bass of all Ruge-Kua mehod s o express he dfferece bewee he value of y a ad as y y m w (2.5) (2.6) where for =, 2,, m, w s are cosas ad h Equaos (2.6) s o be exac for powers of h hrough order m. f c h, y j a j m h, because s o be cocde wh Taylor seres of j 2. The Fourh Order Ruge-Kua mehod Based O Geomerc Mea For Solvg Ial Value Problem Evas [7] developed a ew fourh order RK mehod for solvg IVPs s derved by replacg he arhmec meas he formula y ( 2 2 ) where f y h 6 y 2 4 h - j a j j by her geomerc meas.e. ( 2)/2 2 ec. o yeld ally a low order accuracy formula. For y f (, y), he fourh order Ruge-Kua mehods usg Geomerc Mea ca be wre he form h y y Meas where meas cludes Geomerc Mea(GM) whch volves, 4, where, f (, y), 2 f ( a h, y a h ), f ( ( a2 a) h, y a2 h a h 2) f ( ( a a a ) h, y ( a h a h a h )) (2.8) where he parameers for Geomerc Mea : a, a, a, a, a, a The fourh order formulae based o Ruge-Kua scheme usg Geomerc Mea s as follows: h Geomerc Mea : y y Wh he grd pos a... (2.9) h ( ba) N b ad N The LTE for hs formula s gve as follows: LTE GM = 5 h 842 (2.) (2.7) f f yyyy - 28 f fy fyyy - 744f f yy f fy f yy 56 ff y (h ) Theorem 2.. Le f (, y ) belogs o C 4 [a, b] ad le s paral dervaves are bouded ad assume ha here exss L, M, posve cosas such ha f (, y) j j f L M,, j m. j j y M DOI:.979/ Page

4 he erms of he error boud due o Lo (see Lamber [], we have a src upper boud (wh respec o y oly) as he fourh order Ruge-Kua mehod based o Geomerc Mea : Theorem 2.2 Le f sasfy 2429 y y LTE h ML O h ( ) GM ( ) f (, v) f (, v) g(, v v ),, v, v R. (2.) where g : R + R + R + s a couous mappg such ha r g(, r) s o-decreasg ad he al value problem. (2.2) u ( ) g(, u( )), u() u has a soluo o R + for u > ad ha u() = s he oly soluo of (.4) for u =. The he fuzzy al value problem (2.) has a uque fuzzy soluo. Refer [25]. III. The Hybrd Fuzzy Ial Value Problem Cosder he hybrd fuzzy al value problem y ( ) f (, y( ), ( y )),,, y( ) y, where deoes seala dffereao......, f C[ R E E, E ], C[ E, E ] To be specfc he sysem loo le y ( ) f (, y( ), ( y)), y( ) y,, y ( ) f (, y( ), ( y)), y( ) y, 2, y( )... y ( ) f (, y ( ), ( y )), y ( ) y,,... Assumg ha he exsece ad uqueess of soluo of (.) hold for each we mea he followg fuco: y( ),, y( ), 2, y( ) y(,, y )... y ( ),,... (.),, by he soluo of (.) [, ] We oe ha he soluo of (.) are pecewse dffereable each each erval for for fxed x E,,,2,... Therefore we may replace (.) by a equvale sysem y( ) f (, y, ( y ) F [, y, y], y( ) y, y( ) f (, y, ( y ) G [, y, y], y( ) y, whch possesses a uque soluo ( yy, ) whch s a fuzzy fuco. Tha s for each, he par ( y( ; r), y( ; r )) s a fuzzy umber, where, y( ; r), y( ; r) are respecvely he soluos of he paramerc form gve by (.2) DOI:.979/ Page

5 y( ; r) F [, y( ; r), y( ; r)], y( ; r) y ( r), y( ; r) G [, y( ; r), y( ; r)], y( ; r) y ( r), for r, (.).. The Fourh Order Ruge-Kua mehod based o Geomerc Mea For Hybrd Fuzzy Ial Value Problem I hs seco, for a hybrd fuzzy al value problem (.) he fourh order Ruge Kua mehod based o Geomerc Mea s developed. Here he assumpo s ha he exxece ad uqueess of he soluos of (.) [, ]. hold for each For a fxed r, o egrae he sysem (.) of,,,,...,,,..., we replace each erval by a se 2 N dscree equally spaced grd pos a whch he exac soluo Y( ; r) ( Y( ; r), Y( ; r)) approxmaed by some ( y ( ; r), y ( ; r)) For each chose grd pos o Le, a h, h, N,. N s ( ; ), Y r Y ( ; r) ( y( ; r), y( ; r)) ( Y ( ; r), Y ( ; )), r ad ( y ( ; r), y ( ; r)) may be deoed ( Y, ( r), Y ( r)) ad ( y ( r ), y ( )), r. We allow he N s o vary over he, s respecvely by, so ha he h s may be comparable., The fourh order Ruge Kua mehod based o Geomerc mea s a a approxmao of DOI:.979/ Page Y ( ; r) ad Y ( ; r). To develop he fourh order Ruge Kua mehod based o Geomerc Mea for (2.), defe (, ; y, ( r )) m f (,, u, ( u )) \ u [ y ( r ), y,, ( r )], u y ( r ), y,,( r ) (, ; y, ( r )) max f (,, u, ( u )) / u [ y ( r ), y,, ( r )], u y ( r ), y,,( r ) h 2 (, ; y, ( r)) m f (,, u, ( u )) \ u [ z(, ; y, ( r)), z(, ; y, ( r))], u y ( r), y,,( r) 2 h (, ; y, ( r)) m f (,, u, ( u )) \ u [ z 2(, ; y, ( r)), z 2(, ; y, ( r))], u y ( r), y,,( r) 2 h (, ; y, ( r)) max f (,, u, ( u )) \ u [ z 2(, ; y, ( r)), z 2(, ; y, ( r))], u y ( r), y,,( r) 2 h 2(, ; y, ( r)) max f (,, u, ( u )) \ u [ z(, ; y, ( r)), z(, ; y, ( r))], u y ( r), y,,( r) 2 4 (, ; y, ( r)) m f (, h, u, ( u )) \ u [ z(, ; y, ( r)), z(, ; y, ( r))], u y ( r), y,,( r) 4(, ; y, ( r)) max f (, h, u, ( u )) \ u [ z(, ; y, ( r)), z(, ; y, ( r))], u y ( r), y,,( r) where z ( ; y ( r)) y ( r) h ( ; y ( r)), z( ; y ( r)) y ( r) h ( ; y ( r)),,, 2,,,,, 2,, z ( ; y ( r)) y ( r) h ( ; y ( r)) h ( ; y ( r)) 9 2,,, 6,, 6 2,, z y r y r) h ( ; y ( r)) h ( ; y ( r)) 9 2 (, ;, ( )), ( 6,, 6 2,, z ( ; y ( r)) y ( r) h ( ; y ( r)) h ( ; y ( r)) h ( ; y ( r)) 5,,, 8,, 24 2,, 2,, z ( ; y ( r)) y ( r) h ( ; y ( r)) h ( ; y ( r)) h ( ; y ( r)) 5,,, 8,, 24 2,, 2,, Nex defe

6 S, ; y ( r), y,, ( r) (, ; y, ( r)) 2(, ; y, ( r)) 2(, ; y, ( r)) (, ; y, ( r)) ( ; y ( r)) ( ; y ( r)),, 4,, T, ; y ( r), y,, ( r) ( ; y ( r)) ( ; y ( r)) ( ; y ( r)) ( ; y ( r)),, 2,, 2,,,, ( ; y ( r)) ( ; y ( r)),, 4,, The exac soluos a,+ s gve by Y, ( r) Y, (r) h S, ; Y, ( r), Y, ( ),, ( ) Y, (r), ;, ( ),, ( ) r Y r h T Y r Y r The approxmae soluos a,+ s gve by (.4) y ( r ) y (r),,, ; ( ),,, ( ),, ( ), (r), ; ( ),,, ( ) h S y r y r y r y h T y r y r (.5) Theorem.. Cosder he sysems (.2) ad (.5). For a fxed ϵ Z + ad rϵ[,], h,..., h lm y ( r) x( ; r), N, h,..., h lm y ( r) x( ; r), N, IV. Numercal Example Before llusrag he umercal soluo of Hybrd Fuzzy Ial Value Problem (HFIVP), frs we recall he fuzzy IVP; y( ) y( ), [,] y(; r) [ r, r], r The exac soluo of (4.) s gve by (4.) A = we ge Y( ; r) [( r) e,( r) e ], r r. (4.2) Y r r e r e (; ) [( ),( ) ], r By he fourh order Ruge Kua mehod based o Geomerc Mea wh N=2, (4.) gves he approxmae soluo as follows: y(.; r) [( r)( c ),( r)( c ) ], r where c, 2 2,, h h h h h h h h h h Example 4.. Nex cosder he followg hybrd fuzzy al value problem, y ( ) y( ) m( ) ( y( )), [, ],,,,2,,..., y( ; r) [( r) e,( r) e ], r, where 2( (mod)) f (mod).5, m () 2( (mod)) f (mod).5, (4.) (4.4) DOI:.979/ Page

7 , f = ( ), f,2,..., The hybrd fuzzy IVP (4.4) s equvale o he followg sysems of fuzzy IVPs. y( ) y( ), [,] y(; r) [( r) e,( r) e], r y ( ) y ( ) m( ) y ( ),,, y ( ) y ( ),,2,..., I (4.4) y( ) m( ) ( y( )) s couous fuco of, x ad ( y ( )) fuzzy IVP y ( ) y( ) m( ) ( y( )), [, ], y( ) y,. For each =,, 2,, he (4.5) has a uque soluo o [, + ]. To umercally solve he hybrd fuzzy IVP (4.4)we wll apply he fourh order Ruge Kua mehod based o Geomerc Mea for hybrd fuzzy dffereal equao wh N=2 o oba y,2 (r)approxmag Y(2.; r). Le f :[, ) R R R be gve by where : R Rs gve by f (, y, ( y( ))) y( ) m( ) ( y( )),,,,2,...,, f = ( y) y, f,2,..., Sce he exac soluo of (4.4) for ϵ[,.5] s Y r Y r e r e ( ; ) (; )( 2 ),, Y(.5;r)=Y(;r)( -), r. The Y(.5;r) s approxmaely 5.29 ad y, s approxmaely Sce he exac soluo for (4.4) for [.5, 2] s Therefore.5 Y( ; r) Y(; r)(2 2 e ( e 4)), r. Y(2.;r)=Y(;r)(2+e-4 e ). The Y(2.;r) s approxmaely ad y (2.;) s approxmaely The absolue errors by RK4GM ad he classcal fourh order RK mehod for he r-level se wh h=.5 ad for =.5, 2 of example 4. s gve Table 4.. The graphcal represeao for he approxmae values bewee RK4GM ad he classcal fourh order RK mehod of example 4. s gve fgure.4.. Table 4.. The absolue error able for example 4. for r- level se wh h=.5 ad =.5, 2. r Absolue Error Value (h=.5 ad =.5) Absolue Error Value (h=.5 ad =2) HFRK4GM HFRK4AM HFRK4GM HFRK4AM 2.57E-2.85E E- 4.4E- 4.E E-2 5.2E- 7.97E E-2.8E E- 4.29E- 4.E-2 5.9E E- 7.88E E-2.77E-2.9E- 4.24E- 4.26E E E- 7.8E-. 2.8E-2.7E-2.8E- 4.9E- 4.4E E E- 7.7E E-2.68E-2.28E- 4.5E- 4.5E-2 5.7E-2 6.2E- 7.62E-.5.E-2.64E-2.7E- 4.E- 4.66E E-2 6.2E- 7.5E-.6.8E-2.6E-2.47E- 4.5E- 4.8E E-2 6.8E- 7.44E-.7.7E-2.55E-2.57E- 4.E- 4.9E-2 5.5E E- 7.5E-.8.25E-2.5E-2.66E-.95E- 5.6E E-2 6.7E- 7.26E-.9.4E-2.47E-2.76E-.9E- 5.2E-2 5.9E-2 6.9E- 7.8E-.4E-2.4E-2.86E-.86E- 5.E-2 5.E-2 7.9E- 7.9E- DOI:.979/ Page

8 Fgure 4. Comparso of RK4GM wh RK4AM ad he Exac Soluo whe N=2 ad =2 V. Cocluso I hs wor, he proposed fourh order Ruge-Kua mehod based o Geomerc Mea for fdg he umercal soluo of hybrd fuzzy al value problem wors very well. From he Tables 4. fgure 4. of example 4., s show ha he fourh order Ruge- Kua mehod based o Geomerc Mea s suable for solvg hybrd fuzzy al value problem. Acowledgemes Ths wor has bee suppored by Uversy Gras Commsso, MRP 584/5, SERO, Hyderabad, Ida. Refereces []. Abbasbady. S, Allah Vraloo. T, Numercal soluo of fuzzy dffereal equaos by Ruge-Kua mehod, Nolear sudes.(24)n.,7-29. [2]. Abbasbady. S, Allahvraloo. T, Numercal soluo of fuzzy dffereal equaos by Taylor mehod, Joural of Compuaoal Mehods ad Appled Mahemacs, 2(22)-24. []. Abdul-Majd Wazwaz, A comparso of modfed Ruge-Kua formulas based o a varey of meas, Ieraoal Joural of Compuer Mahemacs, vol.5,pp-5-2. [4]. Balachadra. K, Praash. P, Exsece of soluos of fuzzy delay dffereal equaos wh olocal codo, Joural of he Korea Socey for Idusral ad Appled Mahemacs, 6(22)8-89. [5]. Balachadra. K, Kaagaraja. K, Exsece of soluos of fuzzy delay egrodffereal equaos wh olocal codo, Joural of he Korea Socey for Idusral ad Appled Mahemacs, 9(25) [6]. Chag. S.L, Zadeh. L.A, O fuzzy mappg ad corol, IEEE Tras, Sysems Ma Cybere. 2(972)-4. [7]. Dubos. D, Prade. H, Towards fuzzy dffereal calculus par : Dffereao, Fuzzy Ses ad Sysems, 8(982) [8]. Evas. D.J & Saug. B.B. (99). A comparso of umercal o.d.e. solvers based o arhmec ad geomerc meas. Ieraoal Joural of Compuer Mahemacs, 2-5. [9]. Jayaumar. T ad Kaagaraja. K, Numercal soluo for hybrd fuzzy sysem by Ruge-Kua mehod of order fve, Appled Mahemacal Sceces, Vol. 6,22, o.72, []. Kaleva. O, Fuzzy dffereal equaos, Fuzzy Ses ad Sysems, 24(987)-7. []. Kaleva. O, The Cauchy problem for fuzzy dffereal equaos, Fuzzy Ses ad Sysems, 5(99) [2]. Kaagaraja. K, Sambah. M, Ruge- Kua Nysrom mehod of order hree for solvg fuzzy dffereal equaos, Compuaoal mehods Appled Mahemacs, Vol.(2), No.2, pp []. Kaagaraja. K, Sambah. M,Numercal soluo of fuzzy dffereal equaos by hrd order Ruge-Kua mehod, Ieraoal joural of Appled Mahemacs ad Compuao, 2(4) (2) pp.-8. [4]. Lamber. J.D, Numercal Mehods for ordary dffereal sysems, Joh Wley &Sos, 99. [5]. Ma. M, Fredma. M, Kadel. A, Numercal soluos of fuzzy dffereal equaos, Fuzzy Ses ad Sysems, 5(999)-8. [6]. Murugesa. K, Paul Dhayabara. D, Hery Amrharaj. E.C ad Davd J. Evas, Numercal Sraegy for he sysem of secod order IVPs usg RK mehod based o Cerodal Mea.,Ier.J.Comp.Mah.8(2), (2), pp [7]. Murugesa. K, Paul Dhayabara. D, Hery Amrharaj. E.C ad Davd J. Evas, A Comparso of exeded Ruge Kua formulae based o Varey of meas o solve sysem of IVPs, Ier.J.Comp.Mah.78 (2), pp [8]. Murugesa. K, Paul Dhayabara. D, Hery Amrharaj. E.C ad Davd J. Evas, A fourh order embedded Ruge Kua RKACeM (4,4) mehod based o Arhmec ad Cerodal meas wh error corol, Ier.J.Comp.Mah.79(2) (22), pp [9]. Nrmala. V, Chehur Pada. S, Numercal Soluo of Fuzzy Dffereal Equao by Fourh Order Ruge-Kua Mehod wh Hgher Order Dervave Approxmaos, Europea Joural of Scefc Research, Vol.62 N.2 (2),pp [2]. Pallgs. S.ch., Papageorgou. G, Famels,I.TH., Ruge-Kua mehods for fuzzy dffereal equaos, App.Mah.Comp 29(29),97-5. DOI:.979/ Page

9 [2]. Pederso.S ad Sambadham, Numercal soluo o hybrd fuzzy sysems,mahemacal ad Compug Modellg, vol.45, o. 9-,pp.-44, 27. [22]. Pederso.S ad Sambadham.M, The Ruge-Kua mehod for hybrd fuzzy dffereal equaos, Nolear Aalyss: hybrd sysems 2,pp ,26. [2]. Praash.P ad Kalaselv. V, Numercal soluo of hybrd fuzzy dffereal equaos by predcor-correcor mehod, Ieraoal Joural of Compuer Mahemacs, vol.86, o..pp.2-4, 29. [24]. Pur. M.L, Ralescu. D.A, Dffereals of fuzzy fucos, Joural of Mahemacal Aalyss ad Applcaos, 9(98) [25]. Saveeha.N ad Chedur Pada.S, Numercal soluo of fuzzy hybrd dffereal equao by hrd order ysrom mehod, Mahemacal Theory ad Modelg, vol.2,no.4,22 [26]. Seala. S, O he fuzzy al value problem, Fuzzy Ses ad Sysems, 24(987)9-. DOI:.979/ Page