Solving fuzzy differential equations by Runge-Kutta method

Transcription

1 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- The Joural of Mathematics ad Computer Sciece Available olie at The Joural of Mathematics ad Computer Sciece Vol No (0) 08- Solvig fuzzy differetial equatios by Ruge-Kutta method Z Akbarzadeh Ghaaie,*, M Mohsei Moghadam Mathematics Departmet, Kerma Uiversity ad Youg Research s Society of Shahid Bahoar Uiversity of Kerma, Kerma, Ira akbarzadehghaaie@yahoocom Ceter of Excellece of Liear Algebra ad Optimizatio, Shahid Bahoar Uiversity of Kerma, Kerma, Ira mohsei@mailukacir Received: September 00, Revised: November 00 Olie Publicatio: Jauary 0 Abstract I this paper, we iterpret a fuzzy differetial equatio (FDE) by usig the strogly geeralized differetiability cocept The we show that by this cocept ay FDE ca be trasformed to a system of ordiary differetial equatios (ODEs) Next by solvig the associate ODEs we will fid two solutios for FDE Here we express the geeralized Ruge-Kutta approximatio method of order two ad aalyze its error Fially oe example i the uclear decay equatio show the rich behavior of the method Keywords: fuzzy differetial equatio, geeralized differetiability, geeralized Ruge-Kutta method Itroductio Kowledge about the behavior of differetial equatio is ofte icomplete or vague For example, values of parameter, fuctioal relatioship or iitial coditios, may ot be kow,* Correspodig author: Graduate studet ad Numerical aalysis Uiversity professor ad Numerical aalysis 08

2 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- preciselythe cocept of fuzzy derivative was first itroduced by Chag ad Zadeh i [9] It was followed up by Dubois ad Prade i [], who defied ad used the extesio priciple Other methods have bee discussed by Puri ad Ralescu i [9] ad Goetshel ad Voxma i [] The iitial value problem for fuzzy differetial equatio (FIVP)has bee studied by Kaleva i [,5] ad by Seikkala i [0] There are differet approaches to the study of fuzzy differetial equatios First approach uses H-derivative or its geeralizatio, the Hukuhara derivative This approach has the disadvatage that it leads to solutios with icreasig support, fact which is solved by iterpretig a FDE as a system of differetial iclusios (see eg [,0]) The strogly geeralized differetiability was itroduced i [5] ad studied i [,8] This cocept allows us to resolve the above metioed shortcomigideed, the strogly geeralized derivative is defied for a larger class of fuzzy umber valued fuctio tha the H-derivative So, we use this differetiability cocept i the preset paper Uder appropriate coditios, the fuzzy iitial value problem cosidered uder this iterpretatio has locally two solutios [] The topics of umerical methods for solvig FDE have bee rapidly growig i recet years Hu llermeier i [] obtaied umerical solutio of a FDE, by extedig the existig classical methods to the fuzzy case Some umerical methods for FDE uder Hukuhara differetiability cocept such as the fuzzy Euler method, predictor-corrector method, Taylor method ad Nystro m method are preseted i [-,] The local existece of two solutios of a FDE uder geeralized differetiability implies that we preset ew umerical methods I [], Bede proved a characterizatio theorem which states that uder certai coditios a FDE uder Hukuhara differetiability is equivalet to a system of ODEs Bede also remarked that this theorem ca help to solve FDEs umerically by covertig them to a system of ODEs, which ca the be solved by ay umerical method suitable for ODEs I [8], the authors exteded Bede s characterizatio theorem to geeralized derivatives ad the used that result to solve FDE umerically by Euler method for ODEs uder strogly geeralized differetiability I this paper, after prelimiary sectio, we study FDE uder this cocept of differetiability ad preset the geeralized characterizatio theorem I sectio, we exted Ruge-Kutta method expressed o [] for solvig ODEs uder strogly geeralized differetiability ad the use for solvig FDE umerically Prelimiaries I this sectio, we give some defiitios ad itroduce the ecessary otatio which will be used throughout the paper Defiitio Let X be a oempty set A fuzzy set u i Xis characterized by its membership fuctiou: X [0,]The u(x) is iterpreted as the degree of membership of a elemet x i the fuzzy set u for each x X Let us deote by R F the class of fuzzy subsets of the real axes (ie u: R [0,]) satisfyig the followig properties: (i) u R F, u is ormal, ie x 0 R with u x 0 = ; (ii) u R F, u is covex fuzzy set(ie u tx + t y mi u x, u y, t 0,, x, y R); (iii) u R F, u is upper semicotiuous o R; (iv) cl{x R; u x > 0} is compact, where cl(a) deotes the closure of subset A 09

3 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- The R F is called the space of fuzzy umbers Obviously R R F For 0 < deote [u] = {x R; u(x) } ad[u] 0 = cl{x R; u x > 0} The it is well-kow that for each [0,], [u] is a bouded closed iterval For u, v R F, λ R, the sum u + v ad λ u are defied by [u + v] = [u] + [v], [λ u] = λ[u], [0,], where [u] + [v] meas the usual additio of two itervals of R ad λ[u] meas the usual product betwee a scalar ad a subset of R The metric structure is give by the Hausdorff distace D: R F R F R + 0 D u, v = sup [0,] max { u v, u v } where [u] = u, u, v = v, v, (R F, D) is a complete space ad the followig properties are wellkow: D u + w, v + w = D u, v, u, v, w R F, D k u, k v = k D u, v, k R, u, v R F, D u + v, w + e D u, w + D v, e, u, v, w R F Defiitio Let x, y R F If there exists z R F such that x = y + z, the z is called the H-differece of x, y ad it is deoted by x y Note that x y x + y = x y I what follows,we fix I = (a, b), for a, b R Bede i [] itroduced a more geeral defiitio of a derivative for a fuzzy-umber-valued fuctio I this paper we cosider the followig defiitio [8]: Defiitio Let f: I R F be give Fix t 0 I We say f is ()-differetiable at t 0 ad its derivative deoted by D f, if there exists a elemet f (t 0 ) R F such that for all > 0 sufficietly small, there exist f(t 0 + ) f(t 0 ), f t 0 f(t 0 ) ad the followig limits ( i metric Hausdorff): f t 0 + f t 0 lim 0 + f t 0 f(t 0 ) = lim = f t Similarly a fuctio f is ()-differetiable at t 0 ad its derivative deoted by D f, if there exists a elemet f (t 0 ) R F such that for all > 0 sufficietly small, there exist f t 0 + f(t 0 ), f t 0 f(t 0 ) ad the followig limits: f t 0 + f(t 0 ) f t 0 f(t 0 ) lim = lim = f t Theorem Let F: I R F ad put [F(t)] = [f (t), g (t)] for each [0,] (i) If F is ()-differetiable the f ad g are differetiable fuctios ad [D F t ] = [f (t), g (t)] (ii) If F is ()-differetiable the f ad g are differetiable fuctios ad [D F(t)] = [g t, f (t)] Proof See [8] Geeralized characterizatio theorem 0

4 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- Let us cosider the FDE with iitial value coditio: x t = f t, x, x t 0 = x 0 () where f: [t 0, T] R F R F is a cotiuous fuzzy mappig adx 0 R F ad T is positive umber or ifiity Theorem Let f: [t 0, T] R F R F is a cotiuous fuzzy fuctio If there exists k > 0 such that D f t, x, f t, z kd x, z, t I, x, y R F The the problem () has two solutios o I Oe is ()- differetiable solutio ad the other oe is ()-differetiable solutio Proof See[8] Defiitio Let y: I R F be a fuzzy fuctio such that D yord y exists If y ad D y satisfy problem (), we say y is a ()-solutio of problem () Similarly, if y ad D y satisfy problem (), we say y is a ()- solutio of problem () By usig theorem we ca state useful approach for solvig FIVP: Let us suppose -cut of fuctios x t, x 0, f(t, x) are the followig form: [x(t)] = [x t, x t ], [x 0 ] = [x 0, x 0 ], [f(t, x(t))] = [f t, x, x, f t, x, x ], The we have two followig cases: Case (I): if x(t) is ()-differetiable the solvigfivp () traslates ito the followig algorithm: step (i) solvig the followig system of ODEs: x t = f t, x, x = F t, x, x, x t 0 = x 0 x t = f t, x, x = G t, x, x, x t 0 = x 0 () step (ii) esure that the solutio [x t, x t ] ad [x t, x t ] are valid level sets step (iii) by usig the represetatio theorem agai, we costruct a ()-solutio x(t) such that [x(t)] = [x t, x t ], for all [0,] Case (II): if x(t) is ()-differetiable the solvigfivp () traslates ito the followig algorithm: step (i) solvig the followig system of ODEs: x t = f t, x, x = G t, x, x, x(t 0 ) = x 0 x t = f t, x, x = F t, x, x, x(t 0 ) = x 0 () step (ii) esure that the solutio [x t, x t ] ad [x t, x t ] are valid level sets step (iii) by usig the represetatio theorem agai, we costruct a ()-solutio x(t) such that [x(t)] = [x t, x t ], for all [0,]

5 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- Now we exted Bede s characterizatio theorem [] to fuzzy differetial equatio uder geeralized differetiability: Theorem Let us cosider the FIVP () where f: I R F R F is such that (i) [f(t, x)] = [f t, x, x, f (t, x, x )] (ii) (iii) f, f are equicotiuous there exists L > 0 such that: f t, x, y f t, x, y Lmax x x, y y, 0, ; f t, x, y f t, x, y Lmax x x, y y, 0, ; The for()-differetiability, the FIVP () ad the system of ODEs() are equivalet ad i ()- differetiability, the FIVP () ad the system of ODEs() are equivalet Proof I the paper [], the authors proved for ()-differetiability The result for ()-differetiability is obtaied aalogously by usig theorem Ruge-Kutta method for FDE I this sectio we preset Ruge-Kutta method for solvig () by the geeralized characterizatio theorem Here we state the existece theorem for FDE: Theorem Uder appropriate coditios, the FIVP () cosidered uder geeralized differetiability has locally two solutios, ad the successive iteratios x 0 = x 0, x + t = x 0 + f(s, x (s))ds ad T x 0 = x 0, x + t = x 0 ( ) f(s, x (s))ds t 0 coverge to the ()-solutio ad the ()-solutio, respectively Proof The authors of [] proved for ()-differetiability The result for ()-differetiability is obtaied i [] Based o the geeralized characterizatio theorem, we replace the fuzzy differetial equatio with its equivalet system ad the, for approximatig the two fuzzy solutios, we solve umerically two ODE systems which cosist of four classic ordiary differetial equatios with iitial coditios Now we exted Ruge-Kutta method i [] for fidig two fuzzy solutios of FDEs uder geeralized differetiabilitywe cosider the partitio P for iterval [t 0, T] P t 0 = a 0 < a < < a N = T, a i = a 0 + i, = T t 0 N Suppose two exact solutios [Y (t)] = Y (t, ), Y (t, ) ad [Y (t)] = [Y t,, Y (t, )]are approximated by some[y (t)] = y t,, y t,, [y t ] = [y (t, ), y (t, )], respectively T t 0

6 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- The exact ad approximate solutio at grid poit a i, 0 i Nare deoted by Y, Y, y ( ) ad ( ), respectively y The geeralized Ruge-Kutta method based o the secod order approximatio of Y t,, Y t,, Y t,, Y (t, ) ad equatios () ad () is obtaied as follows: y + = y + θ F t, y, y + y + = y + θ G t, y, y + y 0 ( ) = y 0 ( ) y 0 = y 0( ) θ F t + θ, z + θ G t + θ, z +, z +, z + () z + z + = y + θf t, y, y = y + θg t, y, y (5) y + = y + θ G t, y, y + y + = y + θ F t, y, y + y 0 ( ) = y 0 ( ) y 0 = y 0( ) θ G t + θ, z + θ F t + θ, z +, z +, z + () z + z + = y + θg t, y, y = y + θf t, y, y () Lemma [] let the sequeces of umbers W 0 W W Amax W, V B, V V Amax W, V B V, 0 satisfy for some give positive costats A ad B, ad deote U W V,0 N, the ( A) U ( A) U0 B, N ( A) The followig theorem shows that the geeralized Ruge-Kutta approximatio poitwise coverge to the exact solutios Let F(t, u, v) ad G(t, u, v) be the fuctios F ad G of equatios () ad (), where u ad v are costats ad u vthe domai where F ad G are defied is therefore:

7 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- K = t, u, v 0 t A, < v <, < u v Theorem Let F t, u, v ad G(t, u, v) belog to C (K) ad let the partial derivatives of F ad G be bouded over K The for arbitrary fixed : [0,] the geeralized Ruge-Kutta approximatio of Eqs () ad () coverge to the exact solutio Y (t, ), Y (t, ) uiformly i it ProofIf we cosider ()-differetiability, the for covergece of Eq () similar to [] is sufficiet to show: lim y = Y t,, lim y 0 N 0 N = Y t, by usig the Taylor theorem, we have: Y ( ) Y ( ) hf t, Y ( ), Y ( ) hf t h, Z, Z ad h Y Y ( ) Y ( ) hg t, Y ( ), Y ( ) hg t h, Z, Z where t, t The we have: h Y ad Y ( ) y ( ) Y ( ) y ( ) h Y h F t, Y ( ), Y ( ) F t, y ( ), y ( ) hf t h, Z, Z F t, h z, z

8 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- Y ( ) y ( ) Y ( ) y ( ) h Y h G t, Y ( ), Y ( ) G t, y ( ), y ( ) hg t h, Z, Z G t, h z, z Similarly we have: Z ( ) z ( ) Y ( ) y ( ) ad where t, t h F t, Y ( ), Y ( ) F t, y ( ), y ( ) h Y Z ( ) z ( ) Y ( ) y ( ) h Y Now, we defie W, V, P, T by the followig terms: h G t, Y ( ), Y ( ) G t, y ( ), y ( ) The we have: W Y ( ) y ( ), V Y ( ) y ( ), P Z ( ) z ( ), T Z ( ) z ( ) h W W Lh max W, V Lh max P, T N h V V Lh max W, V Lh max P, T N h P W Lh max W, V M h T V Lh max W, V M N sup Y ( t, ), N sup Y ( t, ), M sup Y ( t, ), M sup Y ( t, ) ad L 0 is a boud where for the partial derivatives of FG, 5

9 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- By substitute ad P, T i W, V, we have: W W Lh max W, V h N h Lh max max W, V Lh max W, V K V V Lh max W, V h N h Lh max max W, V Lh max W, V K where K max M, M ad Now the above term ca abbreviate to the followig: h L W W N K max W, V Lh Lh( Lh), h L V V N K Lh max W, V max W, V Lh Lh( Lh) The by Lemma we have: h L Lh( Lh) W Lh( Lh) U0 N K, Lh( Lh) h L Lh( Lh) V Lh( Lh) U0 N K Lh( Lh) where U W V I particular 0 0 0

10 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- Sice W V ( T t0 )/ h N h L Lh( Lh) W Lh( Lh) U0 N K, N Lh( Lh) ( T t0 )/ h N h L Lh( Lh) V Lh( Lh) U0 N K N Lh( Lh) ad kow for, relatioship k k e ( ) satisfy, the by assumptio T t k 0, h Lh( Lh) we have: L( Lh)( T t0 ) h L e N, W N K Lh( Lh) L( Lh)( T t0 ) h L e N V N K Lh( Lh) ad if h 0 we get W 0, V 0 which cocludes the proof N N Now we will preset a example to show that our method works Example Let us cosider the uclear decay equatio x t = λ x t, x t 0 = x 0, where x(t) is the umber of radiouclides preset i a give radioactive material, λ is the decay costat ad x 0 is the iitial umber of radiouclides I the model, ucertaity is itroduced if we have ucertai iformatio o the iitial value x 0 of radiouclides preset i the material Note that the pheomeo of uclear disitegratio is cosidered a stochastic process, ucertaity beig itroducedby the lack of iformatio o the radioactive material uder study I order to take ito accout the ucertaity we cosider x 0 to be a fuzzy umber Let, I [0,0] ad x0 [, ] the we have: F t, y, y = y, G t, y, y = y By usig the formulatio () we get exact solutio t t Y ( t, ) [( ) e,( ) e ] That is a ()-differetiable solutio of the problem () Usig the formulatio (), t t Y ( t, ) [( ) e,( ) e ],

11 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- is a ()-differetiable solutio of the problem () To get the geeralized Ruge-Kutta approximatio we devide I ito N 0 equally spaced subitervals ad calculate y + = y θ y θ z + y + = y θ y θ z + y 0 ( ) = y 0 ( ) y 0 = y 0( ) z + z + = y θy = y θy for fidig the ()-solutio ad compute for fidig ()-solutio y + = y θ y θ z + y + = y θ y θ z + y 0 ( ) = y 0 ( ) y 0 = y 0( ) z + z + = y θy = ( θ)y = y θy = ( θ)y By substitutig z +, z, z + + adz i y + +, y +, y ad y + +, we have: y + = + y y, y + = y + + y y + = + y, y + = + y 8

12 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- A compariso betwee the exact ad the approximate solutios at t = 0ad the error of geeralized Ruge-Kutta ad Euler method is show i the followig tables ad figures ad Table y Y Ruge-Kutta Euler Error Ruge-Kutta Euler Error y Error Y Error e- -589e e- 589e e- -99e e- 99e e- -90e e- 90e e- -859e e- 859e e- -95e e- 95e e- -9e e- 9e e- -95e e- 95e e- -8e e- 8e e e e e e- -589e e- 589e Table y Y Ruge-Kutta Euler Error Ruge-Kutta Euler Error y Error Y Error e- -550e e- 550e e e e e e- -e e- e e- -80e e- 80e e- -058e e- 058e e- -5e e- 5e e- -8e e- 8e e- -09e e- 09e e- -908e e- 908e e- -550e e- 550e

13 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- Figure (-) exact ()-solutio, (+) approximated poits usig Hukuhara differetiability Figure (-) exact ()-solutio, (+) approximated poits usig ()-differetiability Now, if cosider the same differetial equatio uder Hukuhara differetiability, the the ()- solutio(it exists ad uique by theorems i []) has a icreasig legth of its support, which leads us to the coclusio that there is a possibility that the radioactivity of the system icreases as time goes o ad eve a o-zero possibility that it is egative! Fortuately, the real situatio is differet, ad the radioactivity of a material always decreases with time ad it caot be egative The we coclude that the secod solutio is more efficiet tha the first oe ad ()-solutio models the radioactive decay better This is a advatage of the geeralized differetiability that allows us to select better solutio Also,by compariso the errors of geeralized Ruge-Kutta ad Euler methods i tables ad we observe that the error of geeralized Ruge-Kutta method less tha the geeralized Euler method That is the geeralized Ruge-Kutta method is better tha geeralized Euler method 5 Ackowledgemets This work has partially supported by the Mahai Mathematical Research Ceter ad Ceter of Excellece of Liear Algebra ad Optimizatio, Shahid Bahoar Uiversity of Kerma, Kerma, Ira 0

14 Z Akbarzadeh Ghaaie, M Mohsei Moghadam/ TJMCS Vol No (0) 08- Refereces [] S Abbasbady, T Allahviloo, Numerical solutios of fuzzy differetial equatios by Taylor method, Joural of Computatioal Methods i applied Mathematics, -, 00 [] S Abbasbady, T Allahviloo, O Lopez-Pouso, JJ Nieto, Numerical methods for fuzzy differetial iclusios, Joural of Computer ad Mathematics with Applicatios 8, -, 00 [] T Allahviloo, N Ahmadi, E Ahmadi, Numerical solutio of fuzzy differetial equatios by predictor-corrector method, Iformatio Scieces, -, 00 [] B Bede, Note o Numerical solutios of fuzzy differetial equatios by predictor-corrector method, Iformatio Scieces, 8, 9-9, 008 [5] B Bede, SG Gal, Almost periodic fuzzy-umber-valued fuctios,fuzzy Sets ad Systems, 85-0, 00 [] B Bede, SG Gal,Geeralizatios of the differetiability of fuzzy-umber-valued fuctios with applicatios to fuzzy differetial equatios, Fuzzy Sets ad Systems 5, , 005 []JC Butcher, Numerical methods for ordiary differetial equatios, Joh Wiley & Sos, Great Britai, 00 [8] Y Chalco-Cao,HRoma-Flores, O ew solutios of fuzzy differetial equatios,chaos, Solitos ad Fractals 8, -9, 008 [9] SL Chag, LA Zadeh, O fuzzy mappig ad cotrol, IEEE Tras, Systems Ma Cyberet, 0-, 9 [0] P Diamod, Stability ad periodicity i fuzzy differetial equatios, IEEE Tras Fuzzy Systems 8, , 000 [] D Dubois, H Prade, Towards fuzzy differetial calculus: part, differetiatio, Fuzzy Sets ad Systems 8, 5-, 98 [] R Goetshel, W Voxma, Elemetary fuzzy calculus, Fuzzy Sets ad Systems 8, -, 98 [] E Hu llermeier, A approach to modelig ad simulatio of ucertai dyamical systems, Iterat J Ucertaity Fuzzyess Kowledge-Based Systems 5, -, 99 [] O Kaleva, Fuzzy differetial equatios, Fuzzy Sets ad Systems, 0-, 98 [5] O Kaleva, The Cauchy problem for fuzzy differetial equatios, Fuzzy Sets ad Systems 5, 89-9, 990 [] A Khasta, K Ivaz, Numerical solutio of fuzzy differetial equatios by Nystrom method, Chaos, Solitos & Fractals, , 009 []M Ma, M Friedma, A Kadel, Numerical solutios of fuzzy differetial equatios, Fuzzy Sets ad Systems, 05, -8, 999 [8]JJ Nieto, A Khasta, K Ivaz, Numerical solutio of fuzzy differetial equatios uder geeralized differetiability, Noliear Aalysis:Hybrid System, 00-0, 009 [9] ML Puri, D Ralescu, Differetials of fuzzy fuctios, J Math Aal Appl, 9, -5, 98 [0] S Seikkala, O the fuzzy iitial value problem, Fuzzy Sets ad Systems, 9-0, 98 [] S Sog, C Wu, Existece ad uiqueess of solutios to the Cauchy problem of fuzzy differetial equatios, Fuzzy Sets ad Systems 0, 55-5, 000