Numerical Solution of Linear Emden Fowler Boundary Value Problem in Fuzzy Environment

Transcription

1 Jounal of mathematics and compute Science 15 (2015) Numeical Solution of Linea Emden Fowle Bounday Value Poblem in Fuzzy Envionment A F Jameel 1, Samad A Altaie 2 1 School of Mathematical Sciences, USM, Univesity Sains Malaysia, Penang, Malaysia homotopy33@gmailcom 2 Compute Engineeing Depatment, Univesity of Technology, Baghdad, Iaq altaiesamad@yahoocom Aticle histoy: Received Januay 2015 Accepted Apil 2015 Available online Apil 2015 Abstact In this pape a numeical method fo solving Tow Point Fuzzy Bounday Value Poblems '(TPFBVP) involving linea Emden Folwe equation is consideed The finite diffeence method (FDM) fo solving TPFBVP is intoduced and the poof of convegence of appoximate solutions is bought in detail Finally a numeical example is solved fo illustating the capability of method Keywods: Fuzzy numbes, fuzzy diffeential equations, two point fuzzy bounday value poblems, Finite diffeence method 1 Intoduction Nowadays, fuzzy diffeential equations (FDEs) [1, 2] ae a popula topic studied by many eseaches since it is utilized widely fo the pupose of modeling poblems in science and engineeing Most of the pactical poblems equie the solution of a fuzzy diffeential equation (FDE) which satisfies fuzzy initial o bounday conditions, theefoe a fuzzy initial o bounday poblem should be solved Howeve, many fuzzy initial o bounday value poblems could not be solved exactly; sometimes it is even impossible to find thei analytical solutions Thus, consideing thei appoximate solutions is becoming moe impotant The two-point bounday value poblem (TPBVP) occus in a wide vaiety of poblems in engineeing [3, 4] and science, including the modeling of chemical eactions [5, 6], heat tansfe [7,8], and diffusion [9,10], and the solution of optimal contol poblems [11,12] Fuzzy two point bounday value poblems (FTPBVP) appeas when the modeling of these poblems cannot be sue is pefect and its natue is unde uncetainty Fuzzy odinay diffeential equations ae suitable mathematical models to model dynamical systems in which thee exist uncetainties o vagueness These models ae used in vaious applications including population models [13], quantum optics gavity [14], and medicine [15, 16] Fuzzy two point bounday value poblems have been solved using FDM in [17] fo linea poblems with fuzzy bounday conditions, also a initial value method in [18] ae applied to get the numeical solution fo linea TPFBVP The Finite Diffeence Method (FDM) is one of the simplest and of the oldest methods to linea TPFBVP 179

2 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) solve diffeential equations It consists in appoximating the diffeential opeato by eplacing the deivatives in the equation using diffeential quotients The domains is patitioned in space and in time and appoximations of the solution ae computed at the space o time To the best of ou knowledge, this is the fist attempt fo solving linea fuzzy Emden Folwe The stuctue of this pape is as follows In section 2, some basic definitions and notations ae given which will be used in othe sections In section 3, the defuzzification of TPFBVP is given in details In section 4, stuctue of FDM is fomulated fo solving TPFBVP In section 5, the definitions and themos of consistency, stability and convegence elated with FDM in section 4 ae pesented in details In section 6, we intoduced linea Emden Folwe in fuzzy envionment with the poof the uniqueness of the diffeence appoximation In section 7 a numeical example is solved fo illustating of the method and finally, in section 8, we give the conclusion of this study 2 Peliminaies The definitions eviewed in this section ae equied in ou wok Definition 1 [19, 20]: Let E be the set of all uppe semi-continuous nomal convexfuzzy numbes with - level bounded intevals such that: [μ] = {t R: μ } An abitay fuzzy numbe is epesented by an odeed pai of functions [μ (t)] = [μ(t), μ(t)] fo all [0,1] which satisfies: 1 μ(t)is nomal, ie t 0 Rwith μ(t 0 ) = 1 2 μ(t)is convex fuzzy set, ieμ(λt + (1 λ)s) min {μ(t), μ(s)}, t, s R, λ [0,1] 3 [μ (t)] E, μ is uppe semi continuous onr 4 {t R: µ( t ) >0 } is compact 5 μ(t)is a bounded left continuous non-deceasing function ove[0,1] 6 μ(t) is a bounded left continuous non-inceasing function ove[0,1] 7 μ(t) μ(t), fo all [0, 1] The -level sets of a fuzzy numbe ae much moe effective as epesentation foms of fuzzy set than the above Fuzzy sets can be defined by the families of thei -level sets based on the esolution identity theoem [21] Definition 21 [22]: A mapping f : T E (o P (E)) fo some inteval T E iscalledafuzzy pocess o fuzzy function with cisp vaiable, and we denote -level set by: [f (t)] = [f(t; ), f(t; )], t T, [0,1] whee E be the set of all uppe semi-continuous nomal convexfuzzy numbes Definition 22 [23, 24]: Each function f: X Y induces anothe function f : F(X) F(Y) defined fo each fuzzy inteval U in X by: This is called the Zadeh extension pinciple f (U)(y) = { Sup x f 1 (y)u(x), if y ange (f) 0, if y ange (f) 180

3 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) Definition 23 [25]: A fuzzy matix of ode m s is defined [A ] = [a ij, μ a ij ] as, whee μ a ij is the membeship function of the element a ij in[a ], a ij E, fo i=1,2,m,j=1,2,s Thus fo all [0,1] [A ] = [A, A], and[a ij ] = [a ij, a ij ] 3 Defuzzification of FTPBVP Conside the following second ode FTPBVP { y (x) = f (x, y (x), y (x)), x [x 0, x n ] y (x 0 ) = α, y (x n ) = β (1) Accoding to Section 2, It s not difficult fo defuzzification of Eq (1), whee y (x) is a fuzzy function of the cisp vaiable x and f is the fuzzy function of the cisp vaiable x and the fuzzy vaiable y Hee y (x) is the second ode fuzzy deivative [26] of y (x), y (x), with α, β ae the fuzzy numbes that efe to the fuzzy bounday conditions y (x 0 ), y (x n ) of Eq(1) We denote the fuzzy function y (x) by [y ] = [y, y], fo x [x 0, x n ] and [0,1] it means that the -level set of y (x) can be defined as: [y (x)] = [y(x; ), y(x; )] [y (x)] = [y (x; ), y (x; )] (2) { [y (x)] = [y (x; ), y (x; )] [y (x 0 )] = [y(x 0 ; ), y(x 0 ; )] = [α, α] { (3) [y (x n )] = [y(x n ; ), y(x n ; )] = [β, β] Now if we let Y(x) = y(x), y (x), and fo defuzzification we have whee Also we can wite and by using the extension fuzzy pinciple we have Y (x; ) = [Y(x; ), Y(x; )] Y(x; ) = [y(x; ), y (x; ), y (x; )] Y(x; ) = [y(x; ), y (x; ), y (x; )] [f (x, Y )] = [f(x, Y ; ), f(x, Y ; )] (4) { f(x, Y ; ) = F [x, Y, Y], f(x, Y ; ) = G [x, Y, Y] (5) 181

4 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) Since y (x) = f(x, Y(x)), we can define the following membeship function { f(x, Y (x; ); ) = min{f (x, μ ()) μ () Y (t; )} f(x, Y (x; ); ) = max{f (x, μ ()) μ () Y (t; )} (6) whee f(x, Y (x; ); ) = F (x, Y(x; ), Y(x; )) { f(x, Y (x; ); ) = G (x, Y(x; ), Y(x; )) (7) Afte this defuzzification can ewite Eq (1) fo the lowe and the uppe bound of espectively fo Eq (1) and fo all [0,1], we have y (x; ) = F (x, Y(x; ), Y(x; )), x [x 0, x n ] { (8) y(x 0 ; ) = [α], y(x n ; ) = [β] { y (x; ) = G (x, Y(x; ), Y(x; )), x [x 0, x n ] y(x 0 ; ) = [α], y(x n ; ) = [β] (9) Moe details about the existence and uniqueness ae in [17] 4 Finite Diffeence Scheme fo Second Ode Linea FTPBVP It consists in appoximating the diffeential opeato by eplacing the deivatives in the equation using diffeential quotients The domain is patitioned in space and in time and appoximations of the solution ae computed at the space o time points The eo between the numeical solution and the exact solution is detemined by the eo that is committed by going fom a diffeential opeato to a diffeence opeato This eo is called the called the tuncation eo [9] The tem tuncation eo eflects the fact that a finite pat of a Taylo seies is used in the appoximation We fomulate this scheme in ode to solve second ode TPBVP in fuzzy envionment We conside fuzzy bounday value Eq (1) can be witten as: y (x) = p(x)y (x) + q(x)y (x) + w (x), x [x 0, x n ] { (10) y(x 0 ) = α, y(x n ) = β Whee p(x), q(x) and w(x) ae eal continuous functions on J = [x 0, x n ] with α, β E, then accoding to Section 2 and fo all [0,1], we have the same defuzzification of Eq (1) of y (x), w (x), y (x), y (x), y (x 0 ), y (x n ) 41 Appoximation of the Fist Ode Deivative In this section we deive a finite diffeence fomula fo the fuzzy deivative [y (x)] with its accuacy analysis Suppose the function [y (x)] C 2 (E ) continuous fo all x J Using Taylo seies fo any h>0 we have the fowad diffeence fomula: 182

5 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) y (x + h; ) = y (x; ) + hy (x; ) + h2 2 y (x; ) + h3 6 y (x; ) + h4 24 y (4) (ζ + ; ) (11) whee ζ + [x + h, x],and backwad diffeence fomula: y (x h; ) = y (x; ) hy (x; ) + h2 y (x; ) h3 y (x; ) + h4 y (4) (ζ ; ) (12) whee ζ [x h, x]by subtacting (34) fom (35) we have fuzzy cental diffeence fomula : y (x; ) = y (x+h;) y (x h;) + O(h 2 ) (13) 2h whee O(h 2 ) = h3 y (ζ; ), and ζ [x h, x + h] Hence, fo evey h [0, h 3 0 ], we have the following bound on the appoximation tuncation eo: whee y (x + h; ) y (x h; ) C h 2 2h C = C = {min ( h3 ζ [x h,x+h] 3 y (ζ; μ), h3 3 y (ζ; μ) μ [y(ζ), y(ζ)] )} {max ( h3 ζ [x h,x+h] 3 y (ζ; μ), h3 3 y (ζ; μ) μ [y(ζ), y(ζ)] )} Hee μ epesent the membeship function of the FTPBVP (10) fo all [0,1] and h is the step size given byh = x n x 0 N, whee N is the numbe of iteative 42 Appoximation of the Second Ode Deivative In this section we deive a finite diffeence fomula fo y (x; )with its accuacy analysis Suppose the function [y (x)] C 4 (E ) continuous fo all x J Using the Taylo seies fo any h>0 and by adding fomula (34) fom (35) we have cental fuzzy diffeence fomula: y (x; ) = y (x+h;) 2y (x;)+y (x h;) h 2 + O(h 2 ) (14) whee O(h 2 ) = h4 12 y (4) (ζ; ), ζ [x h, x + h] Hence, fo evey h [0, h 0 ], we have the following bound on the appoximation tuncation eo: whee C = C = y (x + h; ) 2y (x; ) + y (x h; ) h 2 C h 2 {min ( h4 ζ [x h,x+h] 12 y(4) (ζ; μ), h4 12 y(4) (ζ; μ) μ [y(ζ), y(ζ)] )} {max ( h4 ζ [x h,x+h] 12 y(4) (ζ; μ), h4 12 y(4) (ζ; μ) μ [y(ζ), y(ζ)] )} Hee μ epesent the membeship function of the FTPBVP (10) fo all [0,1] and h is the step size given by h = x n x 0 N, whee N is the numbe of iteations In ou wok we set 183

6 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) y (x + h; ) = [y i+1 ], y (x h; ) = [y i 1 ], and y (x; ) = [y i] fo i=1, 2 n fo abitay positive intege n we subdivide the inteval J = [x 0, x n ] as x 0 < x 1 < < x n 1 < x n Let π = {x i } n+1 i=0 denote a unifom patition of the inteval J such that x i = x 0 + ih, i=0,1, n+1, with h = x n x 0 n + 1, and fo all [0,1] we can ewite the fuzzy cental diffeence fomula as: [y i ] = [y i+1 ] [y i 1 ] 2h { [y i ] = [y i+1 ] (15) 2[y i] +[y i 1 ] h 2 The Uniqueness of the Diffeence Appoximation of Eq (10) is given in [15] 5 Consistency, Stability and Convegence To study the accuacy and the computability of the diffeence appoximation in fuzzy envionment, such that {y i+1 } n i=0, we intoduce the concepts of consistency, stability and convegence of finite diffeence methods The basic esult poved in this section is that, fo a consistent method, stability implies convegence Definition 51 (Consistency): Let [T i,π ] = L hν (x i ; ) L ν (x i ; ), i = 1,2,, n whee ν is smooth and continuous fuzzy function on E, x J, L h is the diffeence opeato and L is the linea opeato of Eq (10)Then the diffeence poblem (15) is consistent with the diffeential poblem (10) if: [T i,π (ν )] 0 as h 0 Whee [T i,π (ν )] efe to local tuncation (o local discetization) eos Definition 52: The diffeence poblem (15) is locally p th ode accuate if sufficiently smooth data, thee exists a positive constant[c ] = [C, C], independent of h, such that: {min [T i,π (ν)] 1 i n, [T i,π (ν)] } [C] h p {max [T i,π (ν)] 1 i n, [T i,π (ν)] } [C] h p The following lemma demonstates that the diffeence poblem (15) is consistent with (10) and is locally second ode accuate Lemma 51: If ν C 4 (J) then [T i,π (ν )] = h2 12 (ν (4) (τ i ; ) 2p(x i )ν (θ i ; )) [0,1], i = 1,2,, n τ i, θ i lie in (x i 1, x i+1 ) 184

7 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) Poof: Accoding to definition (52) one can wite [T i,π (ν )] = [ ν (x i+1;) 2ν (x i ;)+ν (x i 1 ;) (16) It is easy to show using Taylo s the theoem that: h 2 ν (x i ; )] + p(x i ) [ ν (x i+1;) ν (x i 1 ;) ν (x 2h i ; )] θ i (x i 1, x i+1 ) ν (x i+1 ;) ν (x i 1 ;) 2h ν (x i ; ) = h2 3 ν (θ i ; ) (17) ν (x i+1 ;) 2ν (x i ;)+ν (x i 1 ;) ν (x h 2 i ; ) = h2 ν (4) (τ 12 i ; ) (18) τ i (x i 1, x i+1 ) The desied esult now follows on substituting (17) and (18) in (16) Definition 53 (Stability): The linea diffeence opeato L his stable if, fo sufficiently small h, thee exists a constant K, independent of h, such that [ν i ] K { {min ((( [ν 0 ], [ν n+1 ] ), [ν 0 ], [ν n+1 ] ))}} + {min ([L h ν j ], [L h ν j ] )} [ν i ] K { {max ((( [ν 0 ], [ν n+1 ] ), [ν 0 ], [ν n+1 ] ))}} + {max ([L h ν j ], [L h ν j ] )} [0,1], i = 0,1,, n + 1 We now pove that, fo h sufficiently small, the diffeence opeato L h of (16) is stable Theoem 51: If the functions p and q satisfy ( ), then the diffeence opeato L h of Eq (41) is stable fo, h < min ( 2 p, p q ) with K = {1, 1 q } Poof: If [0,1], i = 0,1,, n + 1 then fom (41) we obtain [ν i ] {min ( [ν j ], [ν j ] [ν i ] {min ( [ν j ], [ν j ] d i [ν i ] ( e i + c i ) [ν j ] + h 2 {min ( [L h ν j ], [L h ν j ] 185

8 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) then Thus if h < min ( 2 p, p q ), then d i [ν i ] ( e i + c i ) [ν j ] + h 2 {max ( [L h ν j ], [L h ν j ] d i [ν i] = e i [ν i+1 ] c i [ν i 1 ] + h 2 [L hν i+1 ] and it follows that o h 2 q i [ν i ] h 2 d i = c i + e i + h 2 q i {min ( [L h ν j ], [L h ν j ] h 2 q i [ν i ] h 2 {max ( [L h ν j ], [L h ν j ] [ν i ] 1 q [ν i ] 1 q {min ( [L h ν j ], [L h ν j ] {max ( [L h ν j ], [L h ν j ] Thus if max x 0 i x n+1 [ν i+1 ] occus fo 1 j n, then 0 i n+1 {min ( [ν i ], [ν i ] 1 q {min ( [L h ν j ], [L h hν j ] and clealy 1 i n+1 {min ( [ν i ] 0 i n+1, [ν i ] {max ( [ν i ], [ν i ] 1 q 186 {max ( [L h ν j ], [L h ν j ] K { {min (( [ν 0 ], [ν n+1 ] ), ( [ν 0 ], [ν n+1 ] ))}} + {min ( [L h ν j ], [L h ν j ] {max ( [ν i ] 1 i n+1, [ν i ] K { {max (( [ν 0 ], [ν n+1 ] ), ( [ν 0 ], [ν n+1 ] ))}} + {max ( [L h ν j ], [L h ν j ] Thus if max x 0 i x n+1 [ν i] = { [ν 0], [ν n+1 ] } such that [ν i ] = {min (( [ν 0 ], [ν n+1 ] ), ( [ν 0 ], [ν n+1 ] ))}

9 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) [ν i ] = {max (( [ν 0 ], [ν n+1 ] ), ( [ν 0 ], [ν n+1 ] ))} then the above equation follows immediately An immediate consequence of stability is the uniqueness (and hence existence since the poblem is linea) of the diffeence appoximation{[y i] } n i=0, fo, if they wee two solutions, thei diffeence{[ν i] } n i=0, say, would satisfy: [L hν j] = 0, j = 1,2,, n [ν 0] = [ν n+1 ] = 0 Stability then implies that [ν i] = 0 fo i = 0,1,2,, n + 1 Definition 54(Convegence): Let Y (x; ) be the exact solution of the bounday value Eq (10), and {[y i] } n i=0 be the diffeence appoximation defined by Eq (15) The diffeence appoximation conveges to Y (x; )if max Y (x j ; ) y (x j ; ) 0 o [E] = {min ( Y(x j ; ) y(x j ; ), Y(x j ; ) y(x j ; ) 0 [E] = {max ( Y(x j ; ) y(x j ; ), Y(x j ; ) y(x j ; ) 0 [0,1], as h 0 The diffeence Y (x j ; ) y (x j ; ) is the global tuncation (o discetization) eo [E ] = [E, E] at the point x j, j = 1,2,, n Definition 59: The diffeence appoximation{[y i] } n i=0 is a p th appoximation to the solution Y (x; ) if fo h sufficiently small; thee exists a constant [C ] independent of h such that max Y (x j ; ) y (x j ; ) [C ] h p The basic esult connecting consistency, stability and convegence is given in the following theoem Theoem 52: SupposeY (x; ) C 4 (J), andh < 2, Then the diffeence solution {[y i] p } n i=0 of (21) is convegent to the Y (x; ) solution of Eq (15) Moeove, whee C = max Y (x j ; ) y (x j ; ) [C ] h 2 {min ( h4 ζ [x h,x+h] 12 y(4) (ζ; μ), h4 12 y(4) (ζ; μ) μ [y(ζ), y(ζ)] )} C = {max ( h4 ζ [x h,x+h] 12 y(4) (ζ; μ), h4 12 y(4) (ζ; μ) μ [y(ζ), y(ζ)] )} Poof Unde the given conditions, the diffeence poblem (15) is consistent with the bounday value poblem (10) and the opeato [L h] is stable Since 187

10 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) and [L h] (Y (x j ; ) y (x j ; )) = W (x j ; ) y (x j ; ) = [L h] Y (x j ; ) [L h] y (x j ; ) = [T i,π (y )] the stability of [L h] implies that Y (x 0 ; ) y (x 0 ; ) = Y (x n+1 ; ) y (x n+1 ; ) = 0 such that Y(x j ; ) y(x j ; ) 1 q Y(x j ; ) y(x j ; ) 1 q Y (x j ; ) y (x j ; ) 1 q max [T i,π (y )] {min ([T i,π (y)], [T i,π (y)])} {max ([T i,π (y)], [T i,π (y)])} The desied esult follows fom the Lemma in [ ] It follows fom this theoem that {[y i] } n i=0 is a second ode appoximation to the solution Y (x; ) of Eq (10) 6 FTPBVP Linea Emden Fowle Equation Conside the following linea Emden Fowle FTPBVP y (x) = p (x)y (x) + q(x)y (x) + w (x), x [x 0, x n ] { y(x 0 ) = α, y(x n ) = β Whee q(x) ae eal continuous functions on J = [x 0, x n ]with α, β E Hee p (x) =, and w (x)ae η x continuous fuzzy function, η is fuzzy numbe such that η E (17) 61 The Uniqueness of the Diffeence Appoximation Accoding to Eq (17) p (x) E such that p (x; ) = [p(x; ), p(x; )], If thei exist positive constant [p, p ], q, and [p, p ], q such that 0 [p ] p(x; ) [p ], 0 [p ] p(x; ) [p ], 0 q q(x) q (18) Substituting and simplify fomula (38) in Eq (61) we have 188

11 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) h 2 [L i y i ] : (1 + h 2 [p i] ) [y i 1 ] + (2 + h 2 q i ) [y i ] (1 h 2 p i) [y i+1 ] = h 2 [w i ] y(x 0 ; ) = [α], y(x n ; ) = [β] h 2 [L i y i ] : (1 + h 2 [p i ] ) [y i 1 ] + (2 + h2 q i )[y i ] (1 h 2 p i) [y i+1 ] = h 2 [w i ] { y(x 0 ; ) = [α], y(x n ; ) = [β] i = 1,2,, n Now fom the above equations we can constuct the following linea system in matix fom AY=B as follows: (19) [Y] = [ [y 0 ] [y 1 ] [y n 1 ] [y n ] ], [Y] = [A] = [ [y 0 ] [y 1 ] [y n 1 ] [y n ] ], [A] = [ d 1 [e 1 ] [c 2 ] d 1 [e 1 ] [c 2 ] [e n 1 ] [ [c n ] d n ] [e n 1 ] [c n ] d n ] [w 0 ] [c 2 α] [w 0 ] [c 2 α] [w 1 ] 0 [w 1 ] 0 [B] = h 2, [B] = h 2 [w n 1 ] 0 [w n 1 ] 0 [ [w n ] ] [ [e n β] ] [ [w n ] ] [[e n β] ] whee [A, A] ae m s fuzzy matices such that fo all [0,1] we have [c i ] = (1 + h 2 [p i] ), [c i ] = (1 + h 2 [p i ] ), d i = (2 + h 2 q i ), and [e i ] = (1 h [p 2 i] ), [e i ] = (1 h [p 2 i ] ) fo i = 1,2,, n We pove that thee is a unique {y i+1 } n i=0 by showing that the ti-diagonal matix [A ] is stictly diagonally dominant and hence nonsingula Theoem 61: Suppose that the functions p (x; ) and q(x) satisfy (18) and the step size, 189

12 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) h 1 < inf {min {( 2, [p ] [p ] q ), ( 2, [p ] [p ] q )}} h 2 < inf {max {( 2, [p ] [p ] q ), ( 2, [p ] [p ] q )}} then the matix [A ] is stictly diagonally dominant and hence nonsingula Poof If h 1 < inf {min {( 2, [p ] [p ] q ), ( 2, [p ] [p ] q )}}, then [c i ] = (1 + h 2 [p i] ), [e i ] = (1 h 2 [p i] ) and [c i ] + [e i ] = 2 < d i i = 1,2,, n then [e 1 ] < d 1,, [c n ] < d n Similaly if, then h 2 < inf {max {( 2, [p ] [p ] q ), ( 2, [p ] [p ] q )}} and [c i ] = (1 + h 2 [p i] ), [e i ] = (1 h 2 [p i] ) [c i ] + [e i ] = 2 < d i i = 1,2,, n then [e 1 ] < d 1,, [c n ] < d n Since h 1 = h 2 = h, we conclude that [c i] + [e i] = 2 < d i i = 1,2,, n then [e 1] < d 1,, [c n] < d n which completes the poof Thus Eq (17) has unique solution 190

13 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) Numeical Examples Conside the following line Emden- Fowle FTPBVP y (x) η x y (x) = 2 x2 y (x), x [1,2] (20) y(1) = σ, y(2) = σ Whee σ is a tiangula fuzzy numbe [18] having -level set [08+02, 12-02] and η = [ + 1, 3 ] fo all [0,1] Using Maple16 package to obtain the exact solution of this poblem such that: Y(x; ) = Y(x; ) = [0,1] As in the pevious section it s easy to apply finite diffeence fomula (15) on Eq (20) Fo simplicity we only need one point of 1 x i 2 to shows the exact and the numeical esults fo the appoximate diffeence solution fo the lowe and uppe bounds of Eq (20) with step size h=1/300 and x=17 ae given in the following table below: Table 1: Diffeence appoximate solution y(17; ) and y(17; ) of Eq (20) at h=1/100 y(17; ) y(17; )

14 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) Fom Table (1) one can see that the numeical esults ae satisfies the convex tiangula fuzzy numbe, and fo plotting these solutions with exact one we need to plot fo each point x such that 0 1 Fo simplest illustation we plot the numeical and exact solution when x=17 as shows in the following figue: -level Y (17; ), y (17; ) Figue 1: Exact and FD solution y (17; ) of Eq (20) when h=1/300 Fo simplest illustation of finite diffeence method in fuzzy envionment of Eq (20) we solved this poblem at = 05 with step size h = 1/300 fo 1 x i 2, i=0, 2 n as shows in next figues and table: Figue 2: Diffeence appoximate solution y(x; 05)and y(x; 05) Eq (20) when h=1/300 The next table shows the absolute eos [E h] = [E h, E h ] fo Eq (20) fo all x [0,1] as such that 192

15 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) Table 2: Absolute eo of Eq (20) at h=1/300 and =05 x [E 1/300 ] 05 [E 1/300 ] Conclusions In this pape, Finite Diffeence Method (FDM) has been successfully intoduced and applied to solve 'Tow Point Fuzzy Bounday Value Poblems involving linea Emden Folwe equation Numeical examples including linea and nonlinea fuzzy initial value poblems show the efficiency of implemented numeical method The convegence and uniqueness of the diffeence appoximation have been pesented and poved Refeences [1] Z Akbazadeh Ghanaie, M Mohseni Moghadam, Solving fuzzy diffeential equations by Runge- Kutta method, The Jounal of mathematics and compute science, 2(2) (2011), [2] M Rostami, M Kianpou, E bashadoust, A numeical algoithm fo solving nonlinea fuzzy diffeential equations, The Jounal of mathematics and compute science, 2(4) (2011), [3] RA Usmani, Discete methods fo a bounday value poblem with engineeing applications, Math Comput 32(1978), [4] S Ghosh, D Roy, Numeic-Analytic Fom of the Adomian Decomposition Method fo Two- Point Bounday Value Poblems in Nonlinea Mechanics, J Eng Mech 133(10) (2007), [5] T Ray Mahapata, A S Gupta, Heat tansfe in stagnation-point flow towads a stetching sheet, Heat and Mass Tansfe 38(6) (2002), [6] R Shama, A Ishak, I Pop, Patial Slip Flow and Heat Tansfe ove a Stetching Sheet in a Nano fluid, (2013): [7] N Kopteva, N Madden, M Stynes, Gid equidistibution fo eaction-diffusion poblems in one dimension, Numeical Algoithms 40(30) (2005), [8] N Kopteva, M Stynes, A obust adaptive method fo a quasilinea one-dimensional convectiondiffusion poblem, SIAM J Nume Anal 39 (2001),

16 A F Jameel, Samad A Altaie / J Math Compute Sci 15 (2015) [9] F Schlögl, Chemical eaction models fo non-equilibium phase tansitions, Z Physik 253(1972), [10] K Fukui, The path of chemical eactions - the IRC appoach, Chem Res 14 (12) (1981), [11] M Goebel, U Raitums, Optimal contol of two point bounday value poblems, Contol and Infomation Sciences (143) (1990), [12] M Popescu, Two-point bounday value poblem of contol systems with paamete, CCCA 29 (2011), 1-6 [13] A Ome and O Ome, A Pay and Petdou Model with Fuzzy Intial Values, Hacettepe Jounal of Mathematics and Statistics, 41(3) (2013), [14] MS El Naschie, Fom Expeimental Quantum Optics to Quantum Gavity Via a Fuzzy Kahle Manifold, Chaos Solution and Factals, 25(2005), , 2005 [15] M F Abbod, D G Von Keyselingk, DA Linkens, M Mahfouf, Suvey of Utilization of Fuzzy Technology in Medicine and Healthcae Fuzzy sets and system, 120(2001), [16] Bao, R Main, Fuzzy Logic in Medicine, Heidelbeg: Physica - Velag, (2002) [17] T Allahvianloo, K Khalilpou, A Numeical Method fo Two-Point Fuzzy Bounday Value Poblems, Wold Applied Sciences Jounal 13 (10) (2011): [18] T Allahvianloo, K Khalilpou, An Initial-value Method fo Two-Point Fuzzy Bounday Value Poblems, Wold Applied Sciences Jounal 13 (10) (2011): , 2011 [19] D Dubois, H Pade, Towads fuzzy diffeential calculus Pat 3: Diffeentiation, Fuzzy Sets and Systems, 8 (1982), [20] S Seikkala, On the Fuzzy Initial Value Poblem, Fuzzy Sets and Systems 24(3), (1987), [21] M Ghanbai, Numeical Solution of Fuzzy Initial Value Poblems Unde Genealization Diffeentiability by HPM, IntJIndustial Mathematics 1(1) (2009), [22] O S Fad, An Iteative Scheme fo the Solution of Genealized System of Linea Fuzzy Diffeential Equations, Wold Applied Sciences Jounal 7(2009) [23] H Sabei Najafi, F Ramezani Sasemasi, S Saboui Roudkoli, S Fazeli Nodehi, Compaison of two methods fo solving fuzzy diffeential equations based on Eule method and Zadeh s extension, 2(2) (2011), [24] L A Zadeh, The Concept of A linguistic Tuth Vaiable and Its Application to Appoximate Reasoning-I, II, III, Infom Sci 8 (1975), [25] S Siam, P Muugadas, On Semiing of Intuitionstic Fuzzy Matices, Applied Mathematical Sciences 4 (23) (2010), [26] S Siah Mansoui, N Ahmady, Fuzzy Diffeential Equation by using Chaacteization Theoem, communication in numeical analysis (2012), doi:105899/2012/cna