Numerical solution of fuzzy differential equations by Milne s predictor-corrector method and the dependency problem

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1 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 Numeical soluion o uzz dieenial equaions b Milne s pedico-coeco meod and e dependenc poblem Kanagaajan K Indakuma S Muukuma S Depamen o MaemaicsSi Ramakisna mission Vidalaa College o As & Science Coimbaoe 6400 ABSTRACT Te sud o is pape suggess on dependenc poblem in uzz compuaional meod b using e numeical soluion o Fuzz dieenial equaions(fdes) in Milne s pedico-coeco meod. Tis meod is adoped o solve e dependenc poblem in uzz compuaion. We solve some uzz iniial value poblems o illusae e eo. KEYWORDS Fuzz iniial value poblem Dependenc poblem in uzz compuaion Milnes pedico-coeco meod.. INTRODUCTION Fuzz Dieenial Equaions (FDEs) ae used in modeling poblems in science and engineeing. Mos o e poblems in science and engineeing equie e soluions o FDEs wic ae saisied b uzz iniial condiions eeoe a Fuzz Iniial Value Poblem(FIVP) occus and sould be solved. Fuzz se was is inoduced b Zade[]. Since en e eo as been developed and i is now emeged as an independen banc o Applied Maemaics. Te elemena uzz calculus based on e exension pinciple was sudied b Dubois and Pade [4]. Seikkala[] and Kaleva[6] ave discussed FIVP. Buckle and Feuing[] compaed e soluions o FIVPs wic wee obained using dieen deivaives. Te numeical soluions o FIVP b Eule's meod was sudied b Ma e al.[8]. Abbasband and Allvianloo [ ] poposed e Talo meod and e ou ode Runge-Kua meod o solving FIVPs. Palligkinis e al.[0] applied e Runge-Kua meod o moe geneal poblems and poved e convegence o n-sage Runge-Kua meod. Allavianloo e. al.[8] and Banabas Bed [0] o solve e numeical soluion o FDEs b pedico-coeco meod. Te dependenc poblem in uzz compuaion was discussed b Amad and Hasan[4] and e used Eule's meod based on Zade's exension pinciple o inding e numeical soluion o FIVPs. Oma and Hasan[7] adoped e same compuaion meod o deive e ou ode Runge-Kua meod o FIVP. Lael Amad and Hasan[4] invesigae e dependenc poblem in uzz compuaion based on Zade exension pinciple. In is pape we sud e dependenc poblem in uzz compuaions b using Milne's pedico-coeco meod. 65

2 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04. PRELIMINARY CONCEPTS In is secion we give some basic deiniions. Deiniion. Subse à o a univesal se Y is said o be a uzz se i a membesip uncion µ à () akes eac objec in Y ono e ineval [0]. Te uncion µ à () is e possibili degees o wic eac objec is compaible wi e popeies a caaceized e goup. A uzz se can also be pesened as a se o odeed pais () Te suppo e coe and e eig o A ae especivel :> () : () sup. (4) Deiniion. A uzz numbe is a convex uzz subse A o R o wic e ollowing condiions ae saisied: (i) is nomalized. i.e. ; (ii) ae uppe semiconinuous; (iii) : ae compac ses o 0< and (iv) : ae also compac ses o 0<. Deiniion. I is e se o all uzz numbes and we can caaceize b is α-levels b e ollowing closed-bounded inevals: [] : [ ] 0< (5) [] : [ ] 0< (6) Opeaions on uzz numbes can be descibed as ollows: I en o 0<. [ ];. [] [ ];. [ ]; 4. ee 0 [] ; 5. [] [] wee s is scala and 6. o 0<. Deiniion.4 A uzz pocess is a mapping : wee I is a eal ineval [7]. Tis pocess can be denoed as: [] [ ] 0<. (7) 66

3 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 Te uzz deivaive o a uzz pocess x() is deined b [] 0<. (8) Deiniion.5 Tiangula uzz numbe ae ose uzz ses in F(R) in wic ae caaceized l c l c l b an odeed iple ( ) R wi U 0 and c [ U ] [ ] o an en α R. FUZZY INITIAL VALUE PROBLEM Te FIVP can be consideed as ollows suc a [ ] [ ] [ ] α c c l c c [ U ] ( α )( ) ( α )( ) d( ) ~ ( ( ) ) (0) Y 0 (0) d Wee : is a coninuous mapping and wi -level ineval [ ] [ ] 0<. () Wen is a uzz numbe e exension pinciple o Zade leads o e ollowing deiniion: I ollows a () [] [ ] 0< () Wee [ ] 0< (4) [ ] 0<. (5) Teoem. Le sais 0 (6) Wee : is a coninuous mapping suc a is non deceasing e IVP 0 (7) (9) 67

4 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus Has a soluion on o >0 and a 0 is e onl soluion o equaion (7) o 0. Ten e FIVP (0) as a unique uzz soluion. Poo.See [7] In e uzz compuaion e dependenc poblem aises wen we appl e saigowad uzz ineval aimeic and Zade's exension pinciple b compuing e ineval sepaael. Fo e dependenc poblem we ee [7]. 4. THE MILNE S PREDICTOR-CORRECTOR METHOD IN DEPENDENCY PROBLEM We conside e IVP in equaion (0) bu wi cisp iniial condiion and [ ]. Te omula o Milne s pedico-coeco meod is ollows: [ ] [ ] ) ( ) ( ) ( ) ( 4 4 P C P (8) Wee. 0 0 N N T K We conside e ig-and side o equaion (8) we modi e Milne s pedico-coeco meod b using dependenc poblem in uzz compuaion as one uncion [ ] 4 ) ( P V (9) B e equivalen omula [ ]. 4 4 C (0) Now le e omula ; 0 () Can exend equaion (0) in e uzz seing. Le [ ] epesen e -level o e uzz numbe deined in equaion (). We ewie equaion () using e -level as ollows:

5 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 [ ] () B appling equaion () in (8) we ge [ ] () Wee Teeoe [ ] (4) [ ]. (5) α α [ ( ( )) 4 ( ( )) ( ( ))] [ ] α min ( ) P (6) α α [ ( ( )) 4 ( ( )) ( ( ))] [ ] max ( ) P α (7) B using e compuaional meod poposed in [5] we compue e minimum and maximum in equaions (6) (7) as ollows (8) 9 5. NUMERICAL EXAMPLES In is secion we pesen some numeical examples including linea and nonlinea FIVPs. Example 5. Conside e ollowing FIVP. [0]; 0 <0.5; ; (0) 0 >0.5; 69

6 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 Te exac soluion o equaion (0) is given b []. () Te absolue esuls o e numeical uzz Milne's pedico-coeco meod appoximaed soluions a 0. See Table and Figue and. TABLE Te eo o e obained esuls wi e exac soluion a. Figue : Te appoximaion o uzz soluion b Milne's pedico-coeco (0.) 70

7 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 Figue : Compaison beween e exac Milne's pedico-coeco pedico-coeco In is example e compaison o e absolue local eo beween Milne's pedico-coeco meod wi e uzz exac soluion is given in Table o vaious values o α -level α 00. K0.9 and ixed value o ( 0 ). Te esuls sows a Milne's pedicocoeco meod is moe accuae an pedico-coeco meod sows e gapical compaison o a uzz soluion beween exac Milne's pedico-coeco pedico-coeco a ixed ( 0 ). Te beaviou soluions o e end poins o e uzz inevals o a uzz exac soluion Milne's pedico-coeco and pedico-coeco uzz appoximaed soluions ae ploed and compaed in Figue a α 0. Figue cleal sow a Milne's pedicocoeco povides a moe accuae esuls an pedico-coeco meod. Example 5. Conside e ollowing FIVP 4 [0]; 0 <0.5; ; 0 >0.5; () Te exac soluion o equaion () is given b []. () Te absolue esuls o e numeical uzz Milne's pedico-coeco meod appoximaed soluions a 0. See Table and Figue and 4. 7

8 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 TABLE Te eo o e obained esuls wi e exac soluion a. Figue : Te appoximaion o uzz soluion b Milne's pedico-coeco (0.) 7

9 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 Figue 4: Compaison beween e exac Milne's pedico-coeco pedico-coeco In is example we compae e soluion obained b Milne's pedico-coeco meod wi e exac soluion and pedico-coeco. We ave given e numeical values in Table ixed value o 0. and o dieen values o α. 6. CONCLUSION In is pape we used e Milne's pedico-coeco meod o solving FIVP b consideing e dependenc poblem in uzz compuaion. We compaed e soluions obained in wo numeical examples. ACKNOWLEDGEMENTS Tis wok as been suppoed b Tamilnadu Sae Council o Science and Tecnolog Tamilnadu India. REFERENCES [] S. Abbasband T. Allavianloo Numeical soluions o uzz dieenial equaions b Talo Meod Compuaional Meods in Applied Maemaics (00) -4. [] S. Abbasband T. Allavianloo Numeical soluion o Fuzz dieenial equaion b Runge- Kua meod Nonlinea Sudies (004) 7-9. [] M. Aamad M. Hasen A new appoac o incopoae unceaini ino Eule meod Applied Maemaical Sciences 4(5) (00) [4] M. Aamed M. Hasan A new uzz vesion o Eule's meod o solving dienial equaions wi uzz iniial values Sians Malasiana 40 (0) [5] M. Amad M. Hasan Incopoaing opimizaion ecnique ino Zade's exension pinciple o compuing non-monoone uncions wi uzz vaiable Sains Malasiana 40 (0) [6] N. Z. Amad H. K. Hasan B. De Baes A new meod o compuing coninuous uncion wi uzz vaiable Jounal o Applied Sciences (7) (0) [7] A. H. Alsonosi Oma Y. Abu Hasan Numeical soluion o uzz dieenial equaions and e dependenc poblem Applied Maemaics and Compuaion 9 (0) 6-7. [8] T. Allavianloo N. Amad E. Amad Numeical soluions o uzz dieenial equaions b pedico-coeco meod Inomaion Sciences 77(7) (007)

10 Applied Maemaics and Sciences: An Inenaional Jounal (MaSJ ) Vol. No. Augus 04 [9] T. Allavianloo N. Amad E. Amad Eaum o Numeical soluions o uzz dieenial equaions b pedico-coeco meod Inomaion Sciences 77(7) (007) Inomaion Sciences 78 (008) [0] Banabas Bede Noe on Numeical soluions o uzz dieenial equaions b pedico-coeco meod Inomaion Sciences 78 (008) [] A. Bonaini G. Bonempi A Qualiaive simulaion appoac o uzz dnamical modelsacm Tans. Model. Compu. Simula 4 (994) 85-. [] R. en Algoims o Minimizaion wiou Deivaives Dove Pubns 00. [] J. J. Buckle T. Feuing Fuzz dieenial equaions Fuzz Ses and Ssems0 (000) [4] D. Dubois H. Pade Towads uzz dieenial calculus pa : dieeniaion Fuzz Ses and Ssems 8 (98) 5-. [5] E. Isaacson H. B. Kelle Analsis o Numeical Meods Wile New Yok 966. [6] Kaleva Osmo Fuzz dieenial equaions Fuzz Ses and Ssems 4 (987) 0-7. [7] O. Kaleva Inepolaion o uzz daa Fuzz Ses and Ssems 60 (994) [8] M. Ma M. Fiedman A. Kandel Numeical soluions o uzz dieenial equaions Fuzz Ses and Ssems 05 (999) -8. [9] R. E. Mooe Ineval Analsis Paenice-Hall Englewood clis N. J 966. [0] S. Palligkinis G. Papageogiou I. Famelis Runge-Kua meods o uzz dieenial equaions Applied Maemaics and Compuaion 09 (009) [] S. Seikkala On e uzz iniial value poblem Fuzz Ses and Ssems 4() (987) 9-0. [] L. A. Zade Fuzz Ses Inomaion and Conol 8(965)