Abstct nd Applied Anlysis Volume 23, Aticle ID 968464, 8 pges http://dx.doi.og/.55/23/968464 Resech Aticle Modeling of Theml Distibutions ound Bie t the Intefce of Coting nd Substte Ali Shin Deptment of Mthemtics, Ftih Univesity, Buyukcekmece, 345 Istnbul, Tukey Coespondence should be ddessed to Ali Shin; shin@ftih.edu.t Received 7 Mch 23; Accepted 2 August 23 Acdemic Edito: Adem Kılıçmn Copyight 23 Ali Shin. This is n open ccess ticle distibuted unde the Cetive Commons Attibution License, which pemits unesticted use, distibution, nd epoduction in ny medium, povided the oiginl wok is popely cited. Due to constnt het flux, the theml distibution ound n insulted bie t the intefce of substte nd functionlly gded mteil (FGM) which e essentilly two-phse pticulte composites is exmined in such wy tht the volume fctions of the constituents vy continuously in the thickness diection. Using integl tnsfom method, two-dimensionl stedy-stte diffusion eqution with vible conductivity is tuned into constnt coefficient diffeentil eqution. Reducing tht eqution to singul integl eqution with Cuchy type, the tempetue distibution ound the bie is obtined by defining n unknown function, which is clled density function, s seies expnsion of othogonl polynomils. Results e shown fo diffeent thickness nd nonhomogeneity pmetes of FGM.. Intoduction Thee e mny engineeing pplictions unde sevee theml loding tht equie high tempetue esistnt mteils in vious foms of composites nd bonded mteils such s powe genetion, tnspottion, eospce, nd theml bie cotings. ew developments in science nd technology ely on the developments of new mteils. Composites ppe to povide the necessy flexibility in the design of these new mteils, which e essentils tht evey pt of the mteil in use exhibits unifom popeties. In the mid-98s, new composite mteil, which ws initilly designed s kind of theml bie coting used in eospce stuctul pplictions nd fusion ectos, ws found by goup of scientists in Jpn. Becuse of the mteil s stuctue, it is nmed s functionlly gded mteil (FGM). FGMs wee used in moden technologies s dvnced stuctues whee the composition o the micostuctue is loclly vied so tht cetin vition of the locl mteil popeties is chieved []. FGMs e lso developed fo genel use s stuctul components in extemely high-tempetue envionments. The concept is to mke composite mteil by vying the micostuctue fom one mteil to nothe mteil with specific gdient. The tnsition between the two mteils cn usully be ppoximted by mens of powe seies o n exponentil function [2 5]. The icft nd eospce industy nd the compute cicuit industy e vey inteested in the possibility of mteils tht cn withstnd vey high theml gdients. Thisisnomllychievedbyusingcemiclyeconnected with metllic lye. The composition pofile which vies fom % cemic t the intefce to % cemic ne the sufce, in tun, is selected in such wy tht the esulting nonhomogeneous mteil exhibits the desied themomechnicl popeties. The concept of FGMs could povide get flexibility in mteil design by contolling both composition pofile nd micostuctue [6]. A numbe of eviews deling with vious spects of FGMs hve been published in the pst few decdes. They show tht most of ely esech studies in FGMs hd focused moe on theml stess nlysis nd fctue mechnics. Fctue mechnics of FGMs hve been studied nlyticlly by Edogn nd cowokes [7 9]. Edogn identified numbe of typicl poblem es elting to the fctue of FGMs by consideing minly the investigtion of the ntue of stess singulitynethetipofcckindiffeentgeomety [].Edognlsoinvestigtedthentueofthecck-tip stess field in nonhomogeneous medium hving she
2 Abstct nd Applied Anlysis modulus with discontinuous deivtive. The poblem ws consideed fo the simplest possible loding nd geomety, nmely, the ntiplne she loding of two bonded hlf spces, in which the cck is pependicul to the intefce. It ws shown tht the sque-oot singulity of the cck-tip stess field is unffected by the discontinuity in the deivtive of the she modulus []. Relted to the fctue poblems in composite mteils, the solution of integl equtions with stongly singul kenels is exmined by Ky nd Edogn [2, 3]. In n xisymmetic coodinte system, n embedded xisymmetic cck in nonhomogeneous infinite medium ws studied by Oztuk nd Edogn [4]. They showed the effect of the mteil nonhomogeneity on the stess intensity fctos unde constnt Poisson s tio. Due to the mteil mismtch t the intefce of substte nd coting, the theml distibutions nd theml stesses on the cck o insulted bie t the intefce e exmined by eseches. A genel nlysis of one-dimensionl stedystte theml stesses in hollow thick cylinde mde of functionlly gded mteil is developed by Jbbi et l. [5]. The mteil popeties wee ssumed to be nonline with powe lw distibution. The mechnicl nd theml stesses wee obtined though the diect method of solution of the vie eqution. The poblem of genel solution fo the mechnicl nd theml stesses in shot length functionlly gded hollow cylinde due to the two-dimensionl xisymmetic stedy-stte lods ws solved using the Bessel functions by Jbbi et l. [6].Astnddmethodws used to solve nonhomogeneous system of ptil diffeentil vie equtions with nonconstnts coefficients, using Fouie seies. Jin nd od [7] exmined n intenl cck poblem in nonhomogeneous hlf-plne unde theml loding using the iy stess function method nd Fouie tnsfom. They educed the poblem to system of singul integl equtions nd solved it by numeicl methods. They used supeposition method by defining the poblem in two cses. One is the line one-dimensionl het conduction poblem unde constnt het flux without cck, nd the othe one is the two-dimensionl het conduction poblem with n insulted cck subject to constnt het flux which is cted in the opposite diection. It ws lso consideed the poblem of n xisymmetic penny-shped cck embedded in n isotopic gded coting bonded to semi-infinite homogeneous medium by Rekik et l. [8]. The coting s mteil gdient is pllel to the xisymmetic diection nd is othogonl to the cck plne. They used Hnkel tnsfom to convet the equtions into coupled singul integl equtions long with the density function, nd they solved it numeiclly. In this study, the Hnkel integl tnsfom method will be used to solve the het eqution in xisymmetic coodinte system. Poblem will be exmined s one-dimensionl nd two-dimensionl het conduction poblem tht is mixed boundy vlue poblem ove the el line. Using mixed boundy conditions Fedholm integl eqution will be obtined with Cuchy type singulity nd then it will be solved by using some known numeicl techniques [9, 2]. h k k(z) z Q Coting Substte Figue : Geomety of the het conduction poblem. 2. Definition of the Poblem The theml distibution ound penny-shped bie t the intefce of gded composite coting nd substte is given by the following stedy-stte het eqution in xisymmetic coodinte system: T (k (z) )+ z T (k (z) )=, () z nd the conductivities of the substte nd the gded composite coting e, espectively, given s k (z) =k, z, k (z) =k e δz, z, whee k is constnt nd δ is the nonhomogeneity pmete elted to the gded coting. ote tht the conductivity is continuous t the intefce of substte nd the gded composite coting. As shown in Figue, it is consideed penny-shped bie with dius, centeed t the oigin of the xisymmetic coodinte system. It is ssumed tht unifom het flux is pplied ove the stess fee boundy, nd the bie fces emin insulted. Thesolutioncnbeobtinedusingsupepositionmethod which is n ddition of one- nd two-dimensionl het conduction poblems, T (z) nd T 2 (, z), espectively, s shown in Figues 2() nd 2(b). As in Figue 2(), it will be ssumed tht thee will be no bie nd in flux cuses theml distibution only z diection. On the othe hnd, in Figue 2(b),fluxwillbessumedinnoppositediectionon the bie tht cuses theml distibution in (, z) plne. Rewiting () ssuming tht no chnging in diection: δ dt dz + d2 T =, dz2 z h; d 2 T =, dz2 z, (2) (3)
Abstct nd Applied Anlysis 3 z Q with stndd boundy conditions: k (z) z T 2 (, z) finite, z, k (z) z T 2 (, z) =, z=h, <, (8) z T 2 (, + )= z T 2 (, ), <<, z () nd mixed boundy conditions: k (z) z T 2 (, z) =Q, z, <, (9) Q T 2 (, + )=T 2 (, ), <<. () (b) Figue 2: () One-dimensionl het conduction without n insulted bie, (b) Two-dimensionl het conduction with n insulted bie. long with suitble boundy conditions: k (z) d dz T (z) finite, z, k (z) d dz T (z) = Q, z = h, T ( + )=T ( ), the solution of the one-dimensionl het conduction is obtined stightfowd s Q { k T (z) = δ e δz, z h, { Q, z, { k δ wheethecontinuitycnbeseensz.fotwodimensionl het conduction poblem, () cn be simplified fo <z hs nd fo z<, 2 2 T 2 (, z) + T 2 (, z) + 2 z 2 T 2 (, z) +δ z T 2 (, z) =, (4) (5) (6) 2 2 T 2 (, z) + T 2 (, z) + 2 z 2 T 2 (, z) = (7) Eqution (6) will be solved using Hnkel integl tnsfom such tht T 2 (, z) denote the Hnkel tnsfom of zeo ode nd τ(ρ, z) denote invese Hnkel tnsfom of zeo ode [2] shown s below, espectively, T 2 (, z) = τ(ρ,z)j (ρ) ρ, τ(ρ,z)= T 2 (, z) J (ρ) d. () Using Hnkel tnsfom, the solution of (6) long with boundy conditions (8)cnbeobtineds A(ρ)e (m 2+m )z +A(ρ) m 2 +m m 2 m { e 2m 2h+(m 2 m )z, <z h, τ(ρ,z)= A(ρ) { ρ (e 2m2h )(m 2 m ) { e ρz, < z <, (2) whee m = δ/2, m 2 = ρ (δ/2ρ) 2 +nd obseving tht m 2 +m >nd m 2 m >. 3. Evlution of Integl Eqution The unknown vlue A(ρ) cnbeobtinedbydefiningnew function, which is clled density function [], such s ψ () = (T 2 (, + ) T 2 (, )), (3) whee ψ() stisfies the following conditions: ψ () d =, ψ () =, t <<, ψ () = ψ( ). (4) Substituting (2)into(3) long with the conditions given in (4), the unknown function A(ρ) cnbeobtined: A (ρ) = F(ρ) ψ (s) J (sρ) sds, (5)
4 Abstct nd Applied Anlysis whee F(ρ)= ρ ρ( m 2 +m m 2 m )e 2m 2h +(e 2m 2h )(m 2 m ). (6) Using the boundy condition (9) in tnsfomed domin s z,itcnbeobtineds (e 2m2h )(m 2 m )A(ρ)J (ρ) ρ = Q, (7) k nd substituting the vlue of A(ρ) into (7), the integl eqution to be solved fo unknown ψ(s) cnbeobtineds whee ( (e 2m2h )η(ρ)j (ρ) J (sρ) ρ ) ψ (s) sds = Q k, (8) η(ρ)= m 2 m F(ρ). (9) Defining new nomlized vibles nd pmetes such s s = s, =, ρ =ρ, h = h, δ =δ, ψ(s) =ψ(s )=ψ(s ), (2) the integl eqution in (8) cn be expessed in the fom of ψ (s) sds J (ρ) J (sρ) ρ + =q, ψ (s) sds 2e 2m2h η(ρ)j (ρ) J (sρ) ρ ψ (s) sds (2η (ρ) + ) J (ρ) J (sρ) ρ (2) whee q =2Q /k ndthepimesignisemovedfotheske ofsimplicity.thefistdoubleinteglin(2) cn be expessed in tems of the fist nd second kinds of elliptic integls, K nd E, espectively.ask, the second kind of elliptic integl E(k) hsfinitevluewhilethefistkindofelliptic integl K(k) hs logithmic singulity such s 4 K (k) = log ( ), E(k) =. (22) k2 Defining new function M(s, ), given in the ppendix, nd using some lgebic mnipultions, Cuchy type singulity cn obtined by using the popeties of ψ(s) in (4). Hence, the fist integl of (2)becomes ψ (s) sds J (ρ) J (sρ) ρ = π + π ψ (s) s ds M (s, ) ( + s M (s, ) )ψ(s) ds. s+ (23) The second double integl in (2) hs n exponentil integnd with negtive exponent so tht s ρ,the integnd symptoticlly ppoches to zeo. The symptotic expnsion of the integnd cn be expessed s e 2hρ ρ( + hδ2 δ 2 2 ρ h2 δ 4 2hδ 3 2 5 ρ 2 + (/3) h3 δ 6 h 2 δ 5 2hδ 4 +2δ 3 2 7 ρ 3 (/3) h4 δ 8 + (4/3) h 3 δ 7 8h 2 δ 6 4hδ 5 2 ρ 4 +O(ρ 5 )). (24) Since thee is no singulity nd ny discontinuity ove the intevl,ndduetothesymptoticexpnsionsρ, the infinite integl cn be ppoximted ove the intevl [, B]. Depending on pmetes h nd δ, thevlueofb cn be chosen lge enough to obtin smll enough vlue of the integl (less thn 25 )ove[b, ).Hence,thesecond double integl in (2) my be expessed s ψ (s) sds 2e 2m2h η(ρ)j (ρ) J (sρ) ρ B(h,δ) = ψ (s) sds 2e 2m2h η(ρ)j (ρ) J (sρ) ρ. (25) Finlly, the lst integl in (2) cnbeevlutedbyusing seies G A (ρ) = δ 2 2 + δ3 2 6 ρ 2 δ5 2 9 ρ 4 + 5δ7 2 4 ρ 6 + 429δ5 2 3 ρ 4 O(ρ 6 ), (26) n symptotic expnsion of the integnd G(ρ) = ρ(2η(ρ) + ) s ρ. Let us define vlue C tht depends on
Abstct nd Applied Anlysis 5 pmetes h nd δ nd supeposes the infinite integl s follows: nd the othe integls in (29) cn be evluted itetively fo k=,2,3,...s follows: G(ρ)J (ρ) J (sρ) = G(ρ)J (ρ) J (sρ) G A (ρ) J (ρ) J (sρ) + [G (ρ) G A (ρ)] J (ρ) J (sρ). (27) H k (, s) = whee J (ρ) J (sρ) ρ k = J (C) J (sc) + s kc k k L k (, s) + k k (, s), (3) Fo pticul vlue of C, G(C) G A (C) cnbeobtined; then the lst infinite integl cn be expessed s G(ρ)J (ρ) J (sρ) = G(ρ)J (ρ) J (sρ) + G A (ρ) J (ρ) J (sρ). (28) Since the function G(ρ) is smooth function ove the intevl [, C],itcnbeevlutednumeicllyusingGussin qudtue long with Chebyshev othogonl polynomils. On the othe hnd, the evlution of the integl ove the intevl [C, ) cnbeevlutedseptelyfoechtemin G A (ρ) like G A (ρ) J (ρ) J (sρ) whee = δ 2 2 J (ρ) J (sρ) + ( δ3 64 ρ 2 δ5 52 ρ 4 + J (ρ) J (sρ), H (, s) = J (ρ) J (sρ) 5δ7 6384 ρ 6 ) (29) L k (, s) = J (ρ) J (sρ) ρ k = J (C) J (sc) (k ) C k k (, s) = H k (s, ) k J (ρ) J (sρ) ρ k = J (C) J (sc) (k+) C k + H k (, s) k+ sh k (, s), k (32) + sh k (s, ). k+ (33) The initil vlues of the integl H (, s), L 2 (, s),nd 3 (, s) e shown in the ppendix. 4. umeicl Evlution of Integl Eqution The integl eqution in (2) cnbesolvedusinggussin qudtue method. Using the condition given in (4), the unknown function ψ(s) cnbedefinedintemsofthetuncted th tem seies expnsion of Chebyshev othogonl polynomils of the fist kind, T n (s),s ψ (s) = n= T A 2n (s) 2n, < s <, (34) s2 whee A 2n e coefficients. Substituting the seies expnsion of the unknown function ψ(s) into the fist integl in (23), it cn be found tht s J (ρ) J (sρ), { = 2s J (ρ) J (sρ), { J (ρ) J (sρ), { s >, s =, s <, (3) π ψ (s) s ds = A 2n π = n= n= A 2n U 2n 2 (), T 2n (s) (s ) s 2 ds (35)
6 Abstct nd Applied Anlysis whee U 2n 2 () is the Chebyshev polynomils of the second kind, nd eliminting the logithmic singulities, shown in ppendix, the second integl in (23)cn be witten s π M (s, ) ( + s = n= A 2n π ( M (, s) + s M (s, ) )ψ(s) ds s+ M (, s) s+ + 2 log s s+ ) T 2n (s) s 2 ds. (36) Finlly, substituting the tuncted seies epesenttion of ψ(s) into (25) nd(29) system of lgebic eqution cn be obtined the to be solved fo A 2n, n= A 2n (U 2n 2 ( i )+ T 2n ( i ) 2 i 2n +Z ( i ) +Z 2 ( i ) Z 3 ( i ))=q, whee i, i =, 2, 3,...,ecolloctionpointsnd Z ( i )= π Z 2 ( i )= Z 3 ( i )= ( M( i,s) s i + M( i,s) s+ i + s log i 2 i s+ i ) T 2n (s) s ds, 2 B(h,δ) ( 2e 2m2h η(ρ)j ( i ρ) J (sρ) ρ) st 2n (s) s 2 ds, ( G(ρ)J ( i ρ) J (sρ) + G A (ρ) J ( i ρ) J (sρ) ) (37) st 2n (s) s 2 ds. (38) Then, we get ( )system of equtions whose solution gives the coefficients A 2n+. With known coefficient vlues, the tempetue distibution ound the insulted bie my be obtined by integting (3)such s T 2 (, + ) T 2 (, ) = n= Defining new vible like / A 2n T 2n (s) s 2 ds. (4) s=cos θ, π θ ccos ( ), (4) theinteglin(4)cnbeevlutedusingtheeltion U n (t) = sin {(n+) ccos t}, (42) sin (ccos t) nd the diffeence in tempetue distibution on the plne of theinsultedbiecnbeobtineds Appendix T () = T 2 (, + ) T 2 (, ) = ( U )2 A 2n 2 (/) 2n. 2n n= The function M(s, ) in (23) cn be defined [4]s (43) { M (s, ) = s E(s 2 )+s2 K( s s ), s<, { E( (A.) { s ), s>, 5. Results Becuse of the ntue of the poblem it is necessy to incese the density of the colloction points ne the ends =±. Thus, these points my be selected s follows: (2i ) π T n ( i )=, i = cos ( 2 ), i=,2,...,. (39) in tems of the complete elliptic integls of the fist nd second kinds, espectively, π/2 dθ K (k) = k 2 sin 2 θ, E (k) = π/2 k 2 sin 2 θdθ. (A.2)
Abstct nd Applied Anlysis 7 Fo k 2, the vlues of the integls in (3), (32), nd (33), espectively, cn be obtined by solving the initil vlue of echinteglfok =such tht H (, s) = = π π L (, s) = J (ρ) J (sρ) ρ s cos φ R J (ρ) J (sρ) ρ ( + J (RC (h, δ)) = π π RC (h, δ) ( γ log ( ) 2 (, s) = R + J (ρ) J (sρ) ρ R J (V) dv)dφ, J (u) du) dφ, u = s π π sin 2 φ R ( J (Rρ) ) dφ, (A.3) whee R 2 = 2 +s 2 2scos φ. Due to the logithmic singulity s s in (23), the tio ((M(s, ) )/(s )) (/) hs undetemined limiting cse. Using (22)long with L Hospitl sule,we hve M (s, ) lim = s s 2 log s + (log 8 ). (A.4) ow, by dding nd subtcting the leding pt of the logithmic function, π M (, s) M (, s) ( + + log s )ψ(s) ds s s+ 2 2π (log s )ψ(s) ds, ndusingthesymmetypopetyofψ(s) like (log s )ψ(s) ds = (log s ) ψ(s) ds (A.5) (log s+ )ψ(s) ds, (A.6) we hve π whee M (, s) M (, s) ( + s s+ π (log s )ψ(s) ds, 2 (A.7) + 2 log s )ψ(s) ds s+ π log s T 2n (s) s ds = T 2n () 2 2n, (A.8) using the seies expnsion of unknown function ψ(s) in (34). Acknowledgment ThiswokissuppotedbytheScientificResechFundof Ftih Univesity unde Poject numbe P548-2. Refeences [] H.-S. Shen, Functionlly Gded Mteils online Anlysis of Pltes nd Shells, CRC Pess, ew Yok, Y, USA, 29. [2] S. Suesh nd A. Motensen, Fundmentls of Functionlly Gded Mteils, Institute of Mteils Communictions Limited, London, UK, 998. [3] K. Tumble, Functionlly Gded Mteils 2, Cemic Tnsctions 4, The Ameicn Cemics Society, Westeville, Ohio,USA,2. [4] W. Pn, Functionlly Gded Mteils VII,TnsTechPublictions LTD, Züich, Switzelnd, 23. [5] O. vn de Biest, FGM, 24, Functionlly Gded Mteils VIII, TnsTech. PublictionsLTD, Züich, Switzelnd, 25. [6] I. Shiot nd Y. Miymoto, Functionlly Gded Mteils 996, Elsevie,ewYok,Y,USA,997. [7] Y. F. Chen nd F. Edogn, The intefce cck poblem fo nonhomogeneous coting bonded to homogeneous substte, Jounl of the Mechnics nd Physics of Solids, vol. 44,no.5,pp.77 787,996. [8] A. Şhin, An intefce cck between gded coting nd homogeneous substte, Tukish Jounl of Engineeing nd Envionmentl Sciences,vol.28,no.2,pp.35 48,24. [9] A. Shin nd F. Edogn, On debonding of gded theml bie cotings, Intentionl Jounl of Fctue,vol.29,no. 4, pp. 34 359, 24. [] F. Edogn, Fctue mechnics of functionlly gded mteils, Composites Engineeing,vol.5,no.7,pp.753 77,995. [] F. Edogn, The cck poblem fo bonded nonhomogeneous mteils unde ntiplne she loding, ASME Jounl of Applied Mechnics,vol.52,pp.823 828,995. [2] A. Ky nd F. Edogn, On the solution of integl equtions with stongly singul kenels, Qutely of Applied Mthemtics,vol.45,pp.5 22,987. [3] A. Ky nd F. Edogn, On the solution of integl equtions with genelized cuchy kenel, Qutely of Applied Mthemtics,vol.45,pp.455 469,987. [4] M. Oztuk nd F. Edogn, Axisymmetic cck poblem in nonhomogeneous medium, Jounl of Applied Mechnics, Tnsctions ASME, vol. 6, no. 2, pp. 46 43, 993.
8 Abstct nd Applied Anlysis [5] M.Jbbi,S.Sohbpou,ndM.R.Eslmi, Mechniclnd theml stesses in functionlly gded hollow cylinde due to dilly symmetic lods, Intentionl Jounl of Pessue Vessels nd Piping,vol.79,no.7,pp.493 497,22. [6] M. Jbbi, A. Bhtui, nd M. R. Eslmi, Axisymmetic mechnicl nd theml stesses in thick shot length FGM cylindes, Intentionl Jounl of Pessue Vessels nd Piping, vol. 86, no. 5, pp. 296 36, 29. [7] Z. H. Jin nd. od, An intenl cck pllel to the boundy of nonhomogeneous hlf plne unde theml loding, Intentionl Jounl of Engineeing Science,vol.3,no. 5, pp. 793 86, 993. [8] M. Rekik, M. eif, nd S. El-Bogi, An xisymmetic poblem of n embedded cck in gded lye bonded to homogeneous hlf-spce, Intentionl Jounl of Solids nd Stuctues, vol. 47, no. 6, pp. 243 255, 2. [9] F. Edogn nd G. D. Gupt, umeicl solution of singul integl equtions, in MethodsofAnlysisndSolutionof Cck Poblems, G.C.Sih,Ed.,MtinusijhoffPublishes, Dodecht, The ethelnds, 973. [2] F. Edogn, Mixed Boundy Vlue Poblems in Mechnics, in Mechnics Tody,.sse,Ed.,chpte,PegmonPess, Oxfod, UK, 975. [2] G.. Wtson, ATetiseontheTheoyofBesselFunctions, Cmbidge Univesity Pess, Cmbidge, UK, 966.
Advnces in Opetions Resech Volume 24 Advnces in Decision Sciences Volume 24 Jounl of Applied Mthemtics Algeb Volume 24 Jounl of Pobbility nd Sttistics Volume 24 The Scientific Wold Jounl Volume 24 Intentionl Jounl of Diffeentil Equtions Volume 24 Volume 24 Submit you mnuscipts t Intentionl Jounl of Advnces in Combintoics Mthemticl Physics Volume 24 Jounl of Complex Anlysis Volume 24 Intentionl Jounl of Mthemtics nd Mthemticl Sciences Mthemticl Poblems in Engineeing Jounl of Mthemtics Volume 24 Volume 24 Volume 24 Volume 24 Discete Mthemtics Jounl of Volume 24 Discete Dynmics in tue nd Society Jounl of Function Spces Abstct nd Applied Anlysis Volume 24 Volume 24 Volume 24 Intentionl Jounl of Jounl of Stochstic Anlysis Optimiztion Volume 24 Volume 24