Stochastic Finite Element Based on Stochastic Linearization for Stochastic Nonlinear Ordinary Differential Equations with Random coefficients

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1 Proc. of the 5th WSEAS It. Cof. o No-Lear Aalyss, No-Lear Systems ad Chaos, Bucharest, Romaa, October 6-8, 6 Stochastc Fte Elemet Based o Stochastc Learzato for Stochastc Nolear Ordary Dfferetal Equatos wth Radom coeffcets M.M. SALEH, I.L. EL-KALLA ad EHAB M.M. Departmet of Egeerg Mathematcs ad Physcs Faculty of Egeerg, Masoura Uersty, Egypt Abstract: We propose a algorthm for solg stochastc olear ordary dfferetal equatos wth radom coeffcets. Ths algorthm s based o stochastc learzato ad soluto of the learzed equato by stochastc fte elemet method. The determstc coeffcet of the olear term s dded to leels ad the coeffcet of the learzato at each leel depeds o the momets of the soluto at the preous leel. Numercal examples are llustrated to choose the optmum leels for each problem. Key-Words: Stochastc learzato; stochastc fte elemet; Karhue- Loee expaso; chaos polyomals. Itroducto The stochastc problem ca be modeled as a dfferetal or tegral equato, lear or olear []. The objecte of solg a stochastc dfferetal equato s to obta the dfferet momets of the soluto process ad to detfy ts trasto probablty desty fucto. For olear problems, approxmate methods should be troduced to hae a approxmate form for the momets of the soluto process. For example: the stochastc aeragg [-], stochastc learzato [5-7], Adoma's decomposto method [8,9] ad stochastc fte elemet method [-]. I ths paper, the stochastc olear ordary dfferetal wth stochastc coeffcet s replaced by mult-leels learzed equatos. The learzed equato s soled usg stochastc fte elemet method based o dscretzg the stochastc coeffcet by Karhue- Loee expaso ad projectg the soluto o two dmeso-frst order chaos polyomals. Radom Feld. Defto A radom feld ca be ewed as the spatal exteso of a radom arable. I fact, t descrbes the spatal correlato of structural parameters that V x, θ s radomly fluctuate. The radom feld defed by ts mea fucto E ( x ) ad ts coarace fucto (, ) = ( (, θ ) [ ] )( (, θ ) [ ] ) C x x E V x E x V x E x () The θ depedece suggests the radomess. For stace, the frst order autoregresse feld (Marko process) s ofte used umercal applcatos. Its coarace fucto s ge by (, x ) d x C ( x, x ) = σ exp, x, x L l cor where d (.,.) s a approprate dstace measure, s the pot arace of the feld ad correlato legth. cor () σ l s ts. Karhue-Loee Expaso The use of Karhue-Loee (K-L) expaso wth orthogoal determstc bass fuctos ad ucorrelated radom coeffcets gaed terest because of ts b-orthogoal property that s both the determstc bass fuctos ad the correspodg radom coeffcets are orthogoal. Let ω ( x ) deotes the mea alue of ω ( x, θ ) ad C ( x, x ) deotes ts coarace fucto whch s bouded ad poste defte. It has spectral decomposto as, λ C x, x = f x f x, () where = λ ad f x are the egealues ad the egeectors of the coarace kerel, respectely.

2 Proc. of the 5th WSEAS It. Cof. o No-Lear Aalyss, No-Lear Systems ad Chaos, Bucharest, Romaa, October 6-8, 6 5 They are the solutos of the homogeeous Fredholm tegral equato of secod kd ge by, C ( x, x ) f ( x ) dx = λ f ( x ). () D Clearly, ω ( x, θ ) ca be wrtte as, ω ( x, θ ) ω ( x ) α ( x, θ ) where α ( x, ) coarace fucto (, ) decomposto of the feld α ( x, ) = + (5) θ s a process wth zero mea ad ( x ) f ( x ) = C x x. Fally, the K-L θ s ge by, α, θ = ζ θ λ, (6) where ζ θ s a set of ucorrelated radom arables. For example, the egefuctos of the expoetal coarace descrbed () wll be the form: ω Cos ( x ) S( x ) f ( x ) ω + ω =, (7) ω S( ω) S( ω) S ( ω) ω ω where ω s the soluto of olear equato ωcos ( ω) + ω S( ω) =, (8) The, the egealues becomes λ ω = σ (9) λ. The Chaos Polyomals The chaos polyomals are a partcular bass of the radom arables space based o Hermte polyomals of depedet stadard radom arables ζ, ζ,, ζ. Classcally, the oe dmeso Hermte polyomals are defed by x x d ( e ) h ( x ) = ( ) e. () dx The multarable Hermte polyomals ca be defed as tesor product of Hermte polyomals. Cosder the mult-dex, α = { α,..., αm } α, = : m The multarable Hermte polyomals assocated wth ths sequece are: H M = h α α ζ = Fally, ay radom arable k ( θ ) wth fte arace ca be expressed as k ( θ ) = a H ( ζ ) () = where a are determstc costats ad H are eumerato of the H α. The expaso s coerget the mea square sese. I the applcato of chaos polyomals, oe problem s to select the umber of radom arables ad the order of chaos polyomals. The umber of radom arables ca be selected accordg to the K-L expaso, but o formula ca be utlzed for selectg the order []. Stochastc Nolear Ordary Dfferetal Equato Cosder the followg stochastc olear ordary dfferetal equato m LU + εu = f ( x ), x L, () where L s a stochastc lear dfferetal operator, ε s a arbtrary parameter ad m s a poste teger. The correspodg equalet learzato equato s [] + α =, () LU equ. U f x, x L where equ. α s the equalet learzato coeffcet whch ca be foud by mmzg the expectato of the mea square error. The error caused by the aboe replacemet s m Φ = εu α U, equ. Mmzg the mea square error, we get m E + U αequ. = ε. () E U The the stochastc fte elemet method s employed to calculate the momets of the soluto process [5]. Let the soluto be projected o - dmeso-frst order chaos polyomals, so the respose at eery ode of the fte elemet mesh wll be the form: U = a H + a H + a H, f m =, the we eed to calculate U ad U ; = U a H a H a H a a H H a a H H + a a H H

3 Proc. of the 5th WSEAS It. Cof. o No-Lear Aalyss, No-Lear Systems ad Chaos, Bucharest, Romaa, October 6-8, 6 6 = U a H a H a H a a H H a a H H + 6a a H H + a a H H + a a H H + a a H H + a a H H + a a H H a a a H HH + a a H H + + a a a H H H + a a a H H H So the requred momets at each ode of the mesh are: E U = a + a + a, (5) E U = a + a + a + a a + a a + a a (6) At ths stage we kow the secod ad fourth momets at eery ode of the mesh. The α equ. ca be ealuated wth the ht of the followg proposed approach.. A Proposed Approach ) Dde the ε alue to leels such that ε = δ δ = ε (7) ) Sole the stochastc lear dfferetal equato LU = f x, (8) ad ealuate the secod ad fourth momets of the soluto process usg equatos (5) ad (6). ) Trasform the olear equato LU + δu = f x, (9) to the correspodg lear oe, LU + α U = f x, () where α ca be ealuated usg the obtaed momets step () from relato, E U α = δ. () E U We expect that the dfferece the soluto momets betwee the equatos (8) ad (9) s ery small for a suffcetly small alue of δ ad therefore a accurate result ca be obtaed. The proposed way to get the learzato coeffcet s to dscretze ths coeffcet oer the doma. So the coeffcet of learzato α ca be calculated oer the th elemet equ. by usg lear terpolato betwee the alues of momets as fg.() ( ) ( ) x x x x E IP ( ) = E U + E U () h h x x ( ) x x ( ) E IP ( ) = E ( U ) + E ( U ) () h h E IP are the lear where E IP ( ) ad terpolatg fuctos of secod ad fourth momets oer th elemet respectely, so E IP ( ) α E U ( ) = δ. () E IP ) Sole the learzed dfferetal equato () ad calculate the secod ad fourth momets. 5) Repeat steps () ad trasform the olear equato LU + δu = f x, (5) to the correspodg lear oe, LU + α U = f x, (6) E IP E U E U th elemet where α ca be ealuated by usg the momets obtaed step () from relato, E IP ( ) α ( ) = δ, (7) E IP 6) Sole the learzed dfferetal equato (6) ad calculate the secod ad fourth momets. 7) Repeat steps (5) ad (6) utl we reach the leels where E IP ( ) α ( ) = δ. (8) E IP Fg.() E U The ma problem ths approach s to determe the leels at whch the problem coerges. Ths problem s umercally studed the followg examples. E IP

4 Proc. of the 5th WSEAS It. Cof. o No-Lear Aalyss, No-Lear Systems ad Chaos, Bucharest, Romaa, October 6-8, 6 7 A proposed Approach the ed of doma U from ts lear soluto to the olear soluto at fxed.ths study s llustrated fgure (). LU = f ( x ) E U, E U, α = = LU + δu = f x st leel LU + αu = f x = =,, E U E U α LU + δu = f x d leel LU + αu = f x = 5 = 6,, E U E U α Fg.() The trasto of mea of soluto at ed pot oer the leels of olearty LU + δu = f x th leel Fg.() LU + α U = f x The optmum alue of makes the trasto of soluto to be creasg or decreasg mootoously. So the requred alue s 5, the mea ad stadard deato of soluto are llustrated at ths leel the ext fgures.. Numercal Examples Example () d du A.5U +.5U =, x (9) dx dx subjected to the followg boudary codtos U ( ) =, A du =., where A s a stochastc dx x = process wth mea oe ad expoetal coarace as (). I the suggested algorthm we must choose a sutable alue of δ or umber of leels. Ths choce was studed umercally by drawg the trasto alues of the mea of soluto process at Mea =5 =6 Det. soluto, A= Mea of the soluto Dstace alog x Fg.()

5 Proc. of the 5th WSEAS It. Cof. o No-Lear Aalyss, No-Lear Systems ad Chaos, Bucharest, Romaa, October 6-8, 6 8 stadard deato. x S.D. of the soluto at =6 The requred s, the mea ad stadard deato of soluto are llustrated below at ths leel..8 Mea of the soluto x-dstace M ea. Fg.(5). = = Det.soluto, A= If A = the equato (9) trasformed to the correspodg determstc oe whch takes the form d U.5U +.5U =, () dx x whch has exact determstc soluto U = tah. Ths determstc soluto was llustrated fg.() ad cocde wth the mea of stochastc soluto at ery small σ. Example () d d U A U s ( x ) s ( x ), x + = + π dx dx subjected to the followg boudary codtos U ( π ) U =, = d U d U A =, A = dx dx x = x = π () Where A s the same as example ().The trasto of the mea of soluto process at the mddle pot of the doma from lear to the olear soluto at fxed s llustrated fgure (6). = 5 = Fg.(6) The trasto of mea of soluto at the mddle pot oer the leels of olearty If A = the equato () trasformed to correspodg determstc oe whch takes the form d U + U = s ( x ) + s ( x ), () dx whch has exact determstc soluto U = s ( x ). The determstc soluto of () was llustrated fg.(7) ad cocde wth the mea of stochastc soluto at ery small σ. We ote from the preous examples that the mea of soluto process of the ge stochastc dfferetal equatos approach the exact soluto of the correspodg determstc equato ( A = ) usg ery small pot arace ( σ =. ). stadard deato Dstace alog x.5 x Fg.(7) S.D. of the soluto at = x-dstace Fg.(8) 5

6 Proc. of the 5th WSEAS It. Cof. o No-Lear Aalyss, No-Lear Systems ad Chaos, Bucharest, Romaa, October 6-8, 6 9 Cocluso The suggested algorthm s a effecte tool for solg stochastc olear ordary dfferetal equatos wth radom coeffcet. The choce of the optmum leels s umercally studed for the selected examples. Ths choce s based o the trasto of mea of soluto process at ay pot from lear to olear case. Ths trasto must be creasg or decreasg mootoously. We ca obta the exact determstc soluto from the stochastc problem by assumg a small alue of pot arace σ of the stochastc process. Refereces [] Bert Øksedal, Stochastc dfferetal equatos: a troducto wth applcatos, Sprger-erlage, (998). [] G.Q. Ca ad Y.K. L, Strogly olear system uder o-whte radom exctato, stochastc structural dyamc, -6, (999). [] Peter H. Baxedale, Stochastc aeragg ad asymptotc behaor of the stochastc Duffg- Va der Pol equato, stochastc processes ad ther applcatos, 5-7, (). [] H. He ad U. Lepk, O stochastc respose of olear oscllators, dyamcal systems ad applcatos, 95-7, (). [5] L. Dual ad M.N. Noor, Drect stochastc equalet learzato of SDOF smart mechacal systems, 8 th ASCE specalty coferece o probablstc mechacs ad structural relablty, (). [6] Carste Proppe, stochastc learzato of dyamcal systems uder parametrc posso whte ose exctato, Iteratoal joural of o-lear mechacs 8, 5-555, (). [7] Goa Falsoe, A exteso of Kazako relatoshp for o-gaussa radom arables ad ts use the olear stochastc dyamcs, probablstc egeerg mechacs, 5-56, (5). [8] M. El-Tawl, M.M. Saleh, M.M. Elsady ad I.L. El-kalla, Decomposto soluto of stochastc olear oscllator, Iteratoal joural of dfferetal equatos ad applcatos, ol. 6, o., -, (). [9] M.M. Saleh ad I.L. El-Kalla, O the geeralty ad coergece of Adoma's method for solg partal dfferetal equatos, Iteratoal joural of dfferetal equatos ad applcatos, ol. 6, (). [] Marc Kamsk, Stochastc secod order perturbato approach computatoal mechacs, umercal methods cotuum mechacs, (). [] M. El-Tawl, W. El-Taha ad A. Husse, A proposed techque of SFEM o solg ordary radom dfferetal equato, Appled mathematcs ad computatos 6, 5-7, (5). [] Stephe A. Rzz ad Alexader A. Murayo, Comparso of olear radom respose usg equalet learzato ad umercal smulato, the 7 th teratoal coferece structural dyamcs, ol. Π, 8-86, (). [] P.D. Spaos ad N.P. Polts, olear stochastc drll-strg bratos, joural of brato ad acoustcs, (). [] Jorge E. Hurtado, Aalyss of oedmesoal stochastc fte elemet usg eural etworks, probablstc egeerg mechacs, 5-, (). [5] Roger G. Ghaem ad Pol D. Spaos, Stochastc fte elemet: a spectral approach, sprger, Verlage, (99). 6