Research Article Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique

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1 Hndaw Publshng Corporaon Journal of Appled Mahemacs Volume 3, Arcle ID 68537, pages hp://dx.do.org/.55/3/68537 Research Arcle Numercal Approxmaon of Hgher-Order Soluons of he Quadrac Nonlnear Sochasc Oscllaory Equaon Usng WHEP Technque Mohamed A. El-Belagy and Amnah S. Al-Johan Engneerng Mahemacs & Physcs Deparmen, Faculy of Engneerng, Caro Unversy, Gza, Egyp Mahemacs Deparmen, Faculy of Scence, Norhern Borders Unversy, Arar, Saud Araba Correspondence should be addressed o Mohamed A. El-Belagy; zbelagy@homal.com Receved March 3; Acceped 4 July 3 Academc Edor: Turgu Özş Copyrgh 3 M. A. El-Belagy and A. S. Al-Johan. Ths s an open access arcle dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced. Ths paper nroduces hgher-order soluons of he sochasc nonlnear dfferenal equaons wh he Wener-Herme expanson and perurbaon (WHEP) echnque. The echnque s used o sudy he quadrac nonlnear sochasc oscllaory equaon wh dfferen orders, dfferen number of correcons, and dfferen srenghs of he nonlnear erm. The equvalen deermnsc equaons are derved up o hrd order and fourh correcon. A model numercal negral solver s developed o solve he resulng se of equaons. The numercal solver s esed and valdaed and hen used n smulang he sochasc quadrac nonlnear oscllaory moon wh dfferen parameers. The soluon ensemble average and varance are compued and compared n all cases. The curren work exends he use of WHEP echnque n solvng sochasc nonlnear dfferenal equaons.. Inroducon Analyss of he response of lnear and nonlnear sysems subjeced o random excaons s of consderable neres o he felds of mechancal and srucural engneerng []. Sochasc dfferenal equaons based on he whe nose process provde a powerful ool for dynamcally modelng complex and unceran aspecs. In many praccal suaons, may be approprae o assume ha he nonlnear erm affecng he phenomena under sudy s small enough; hen s nensy s conrolled by means of a small parameer, say ε []. Accordng o [3], he soluon of sochasc paral dfferenal equaons (SPDEs) usng Wener-Herme expanson (WHE) has he advanage of converng he problem o a sysem of deermnsc equaons ha can be solved effcenly usng he sandard deermnsc numercal mehods. The man sascs, such as he mean, covarance, and hgher-order sascal momens, can be calculaed by smple formulae nvolvng only he deermnsc Wener-Herme coeffcens. In WHE approach, here s no randomness drecly nvolved n he compuaons. One does no have o rely on pseudorandom number generaors, and here s no need o solve he sochasc PDEs repeaedly for many realzaons. Insead, he deermnsc sysem s solved only once. The applcaon of he WHE [4 ] ams a fndng a runcaed seres soluon o he soluon process of a sochasc dfferenal equaon. The runcaed seres s composed of wo majorpars:hefrsshegaussanparwhchconsssof he frs wo erms, whle he res of he seres consues he non-gaussan par. In nonlnear cases, here exs always dffcules n solvng he resulan se of deermnsc negrodfferenal equaons go from he applcaons of a se of comprehensve averages on he sochasc negrodfferenal equaon obaned afer he drec applcaon of WHE. Many auhors nroduced dfferen mehods o face hese obsacles. Among hem, he WHE wh perurbaon (WHEP) echnque [4] was nroduced usng he perurbaon echnque o solve perurbed nonlnear problems. The WHE was orgnally sared and developed by Wener n 938 and 958 []. Wener consruced orhonormal

2 Journal of Appled Mahemacs random bases for expandng homogeneous chaos dependng on whe nose and used o sudy problems n sascal mechancs. Cameron and Marn [] developed a more explc and nuve formulaon for WHE (now known as Wener chaos expanson, WCE). Ther developmen s based on an explc dscrezaon of he whe nose process hrough s Fourer expanson, whch was mssed n Wener s orgnal formalsm. Ths approach s much easer o undersand and more convenen o use and hence replaced Wener s orgnal formulaon. Snce Cameron and Marn s work, WHE has become a useful ool n sochasc analyss nvolvng whe nose (Brownan moon) [3]. We wll denoe by Imamura formulaon [3]. Anoher formulaon was suggesed and appled by Imamura and hs coworkers [3, 4]. They have developed a heory of urbulence nvolvng a runcaed WHE of he velocy feld. The randomness s aken up by a whe-nose funcon assocaed, n he orgnal verson of he heory, wh he nal sae of he flow. The mechancal problem hen reduces o a se of coupled negrodfferenal equaons for deermnsc kernels. In [], he WHE (Imamura formulaon [3]) was used o compue he nonsaonary random vbraon of a Duffng oscllaor whch has cubc nonlneary under whe-nose excaon. Soluons up o second order are obaned by solvng he equvalen deermnsc sysem by an erave scheme. El- Tawl and hs coworkers [4, 5, 6] used he WHEP echnque o solve a perurbed nonlnear sochasc dffuson equaon and he quadrac and cubc nonlnear sochasc oscllaory equaon wh frs-order approxmaons. In [7], he analyss of nonlnear random vbraon has been suded usng several mehods, such as equvalen lnearzaon mehod [8], sochasc averagng mehod [9], he WHE approach wh nonsaonary excaons [], he WHEP echnque [5], egenfuncon expansons [], and he mehod of dealed balance []. All of he above mehods are appled and used for nonlnear random oscllaons of real sysems subjeced o random nonsaonary (or saonary) excaons. Accordng o [5, 6], quadrae oscllaon arses hrough many appled models n appled scences and engneerng when sudyng oscllaory sysems []. These sysems can be exposed o a lo of unceranes hrough he exernal forces, he dampng coeffcen, he frequency, and/or he nal or boundary condons. These npu unceranes cause he oupu soluon process o be also unceran. For mos of he cases, geng he probably densy funcon (p.d.f.) of he soluon process may be mpossble. So, developng approxmae echnques (hrough whch approxmae sascal momens can be obaned) s an mporan and necessary work. There are many echnques whch can be used o oban sascal momens of such problems. The man goal of hs paper s o nroduce hgher-order WHEP soluons and o sugges a numercal solver suable o handle he equvalen deermnsc sysem. In [6], he WHEP echnque s generalzed o nh nonlneary, general order of WHE, and general number of correcons. Also, he exenson o handle whe nose n more han one varable and general nonlneares are oulned. The generalzed algorhm s mplemened and lnked o MahML [3] scrplanguageoprnouheresulngequvalen deermnsc sysem. In he curren work, he WHE formulaon suggesed by Meecham and hs coworkers (Imamura formulaon) s used o solve he sochasc nonlnear dfferenal models of he form L (x ()) = εx n +f() +g() N (), (, T] wh he proper se of nal condons whch wll be assumed o be deermnsc. The dfferenal operaor L s a general lnear operaor. The nonlneary s nroduced as losses of degree n>srenghened by a deermnsc small parameer (ε). For he quadrac nonlneary, n wll be equal. The uncerany s nroduced hrough whe nose N() scaled by a deermnsc envelope funcon g().the funconf() s a deermnsc forcng funcon. Theorem wll be used n he dervaon of he WHEP echnque. Theorem. The soluon of (),fexss,sapowerseresnε; ha s, x() = = ε x (). Proof. Usng Pcard s mehod whch generaes a sequence of approxmaons ha converge o he soluon; ha s, he soluon s obaned as x() = lm k x (k) () where he kh approxmaon s compued as L(x (k) ()) = ε[x (k ) ()] n +f() +g() N (), k, apply he nverse operaor, L, on boh sdes o ge x (k) () = εl ([x (k ) ()] n ) +L (f () +g() N ()). Le x () () = L (f() + g()n()) whch s he soluon a ε=. Then he Pcard s kh approxmaon s compued as () () (3) x (k) () =x () () εl ([x (k ) ()] n ). (4) Now, we need o prove ha he kh approxmaon s a power seres n ε;has,canbewrenas x (k) () = M k = ε x (k) (), (5) where M k +s he number of erms n he seres. Usng he mahemacal nducon, we need o prove ha a power seres soluon wll be obaned a k=and a k+provded ha he kh approxmaon s a power seres soluon.

3 Journal of Appled Mahemacs 3 A k=, he Pcard s s approxmaon wll be x () () = x () () εl ([x () ()] n ), whch s a power seres n ε. Now we need o prove ha x (k+) () s a power seres n ε gven ha x (k) () s a power seres n ε. ThePcard s(k + )h approxmaon s compued as x (k+) () =x () () εl ([x (k) ()] n ), (6) or,afersubsunghepowerseresofx (k) (),wege M k x (k+) () =x () () εl ([ = ε x (k) ()] ). (7) The second erm n he rgh hand sde can be expanded usng he mulnomal heorem o ge where c h ( n+m k n M k [ = ε x (k) ()] n = h M k c h = [ε x (k) ()] j h, (8) = n!/ M k = j h! and he couner h runs over all he ) combnaons of he posve negers j h,j h,...,jm k h =n. In a more smplfed form, we can wre such ha M k = j h M k [ = ε x (k) ()] n = h where w h = M k = j h.le M k expanson akes he form M k [ = ε x (k) ()] n c h ε w h = [x(k) M k [x (k) = n ()] j h, (9) ()] j h = V h (); hen he = c h ε w h V h (). () h Subsue n Pcard s (k + )h approxmaon o ge or x (k+) () =x () () εl ( c h ε w h V h ()) () h x (k+) () =x () () c h ε +w h L (V h ()), () h whch s a power seres n ε. Thscompleesheproof.We noe ha he prevous heorem s appled also n case of deermnsc force erm; ha s, g() =.Also,heheorem s appled f he unknown funcon x s a funcon of more han one varable [6]. Ths paper s organzed as follows. In Secon, he WHEP echnque s revewed and he generalzed WHEP dervaon seps are oulned. The deermnsc se of equaons equvalen o he sochasc nonlnear oscllaory equaon are abulaed n Secon 3. The analycal and numercal soluons of he oscllaory equaon are descrbed n Exac Numercal (d =.) Fgure : Comparson beween he exac and he numercal soluons a Δ =.. Error E+ E E E 3 E 4 E 5 E 6 E 7 E 8 E 9 E E E. Acual Quadrac. Tme sep. Lnear.5 order.. Fgure : The convergence order of he developed numercal solver. Secon 4. The smulaons up o hrd order and fourh correcon are shown n Secon 5.. WHEP Technque As a consequence of he compleeness of he Wener-Herme se [], any arbrary sochasc process can be expanded n erms of he Wener-Herme polynomal se, and hs expanson converges o he orgnal sochasc process wh probably one.

4 4 Journal of Appled Mahemacs (a) (b) Zero correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon (c) Fgure 3: Frs-order response mean and varance for (a) ε =.,(b)ε =.3,and(c)ε =.5.

5 Journal of Appled Mahemacs (a) Zero correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon (b) Fgure 4: Frs-order response mean and varance for (a) ε =.7 and (b) ε =.. The soluon funcon x(; w) can be expanded n erms of Wener-Herme funconals as [4] x (; w) =x () () + x () (; )H () ( ;w)d + x () (;, )H () (, ;w)d d + x (3) (;,, 3 ) H (3) (,, 3 ;w)d d d 3 + (3) or afer elmnang he parameers, for he sake of brevy, we ge x (; w) =x () () + x k= R (k) H (k) dτ k, (4) k where dτ k = d d d k and R k s a k-dmensonal negral over he varables,,..., k.thefrsermnhe expanson (4) s he nonrandom par or ensemble mean of he funcon. The frs wo erms represen he normally dsrbued (Gaussan) par of he soluon. Hgher-order erms n he expanson depar more and more from he Gaussan form [6]. The Gaussan approxmaon s usually a bad approxmaon for nonlnear problems, especally when hgh-order sascs are concerned [8].

6 6 Journal of Appled Mahemacs (a) Var [x] (b) Zero correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon (c) Fgure 5: The frs-order response mean and varance for (a) ε =.,(b)ε =.3,and(c)ε =.5. Case of exponenal funcon e.5 mulpled by he whe nose.

7 Journal of Appled Mahemacs (a) Zero correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon s correcon nd correcon 3rd correcon 4h correcon 5h correcon (b) Fgure 6: The frs-order response mean and varance for (a) ε =.7 and (b) ε =.. Case of exponenal funcon e.5 mulpled by he whe nose. The componens x (j) (;,,..., j ) are called he deermnsc kernels of he WHE of x(). Theyarefuncons of me and space varables and fully accoun for he me dependence of x() as well as for s sascal properes [3]. w s a random oupu of a rple probably space (Ω, B, P), where Ω s a sample space, B s a σ-algebra assocaed wh Ω, andp s a probably measure. For smplcy, w wll be dropped laer on. The funconal H (n) (,..., n ) s he nh order Wener- Herme me-ndependen funconal. The WH funconals form a complee se [], and hey sasfy he followng recurrence relaon for n : H (n) (,,..., n )=H (n ) (,,..., n )H () ( n ) n H (n ) ( n, n,..., nn ) m= (5) δ( n m n ), wh H () = and H () ( ) = N( ):hewhenose.by consrucon, he Wener-Herme funconals are symmerc

8 8 Journal of Appled Mahemacs (a) (b) Zero correcon s correcon nd correcon 3rd correcon 4h correcon s correcon nd correcon 3rd correcon 4h correcon (c) Fgure 7: The second-order response mean and varance for (a) ε =.,(b)ε =.3,and(c)ε =.5.

9 Journal of Appled Mahemacs (a) Zero correcon s correcon nd correcon 3rd correcon 4h correcon s correcon nd correcon 3rd correcon 4h correcon (b) Fgure 8: The second-order response mean and varance for (a) ε =.7 and (b) ε =.. n her argumens and are sascally orhonormal wh respec o he weghng funcon e (/) n = ξ ( ) ;has, Var [x ()] = m k= (k!) R k (x (k) ) dτ k. (8) E[H () H (j) ]= =j. (6) The average of almos all Wener-Herme funconals vanshes, parcularly, E [H () ] =. (7) The expecaon and varance of he soluon wll be E [x ()] =x () (), The WHE mehod can be elemenarly used n solvng sochasc dfferenal equaons by expandng he soluon as well as he sochasc npu processes va he WHE. The resulan equaon s more complex han he orgnal one due o beng a sochasc negrodfferenal equaon. Takng a se of ensemble averages ogeher wh usng he sascal properes of he WHE funconals, a se of deermnsc negrodfferenal equaons are obaned n he deermnsc kernels x () (;,,..., ), =,,,.... Tooban approxmae soluons of hese deermnsc kernels, one can

10 Journal of Appled Mahemacs Zero correcon s correcon nd correcon 3rd correcon 4h correcon 5 5 s correcon nd correcon 3rd correcon 4h correcon (a) (b) Fgure 9: The second-order response mean and varance for ε =.5 andmenervalofseconds. use perurbaon heory n he case of havng a perurbed sysem dependng on a small parameer ε. Expandng he kernels as a power seres of ε, anoher se of smpler erave equaons n he kernel seres componens are obaned. ThsshemanalgorhmofheWHEPalgorhm[4]. The echnque was successfully appled o several nonlnear sochasc equaons; see for example [4, 5, ]. The WHEP echnque for general nonlnear exponen (n), general order (m), and general number of correcons (NC) follows he followng seps [6]. () Truncae he expanson (4) o conan only m+ (m ) kernelsx (j), j m;has,x(; w) = x () () + m k= R k x (k) H (k) dτ k. () Subsue no he sochasc oscllaory equaon (). (3) Use he mulnomal heorem o expand he quadrac nonlnear erm n (), x n, n=. (4) Mulply by H (j), j m, and hen apply he ensemble average. Ths wll lead o (m + ) equaons n he kernels x (j), j m. (5) For each kernel x (j), j m, apply he perurbaon echnque up o NC correcons; ha s, x (j) = NC = ε x (j). (6) Equae he coeffcens of ε k, k NC, n boh sdes o ge NC+equaons for each kernel x (j), j m. where Ths wll lead o he followng (m + )(NC +)equaons: (j!)l(x (j) )=δ jf () +δ j g () δ( ), (j!)l(x (j) b )= c f D (j) f,b Ej f, f D (j) j m, b NC, f,b = R z ( Var c m NC g [x () = p= And he expecaons E j f are compued as j m, (9) () p ]hp g)dτ z. () E j m f = H(j) (H () ) kj f. () = I was explaned n [6] howogee j f n erms of he Drac dela funcons and hen use hem o reduce he negrals ha appear n D (j) f,b.thesummaon Var means ha all varaons h p q, q m, p NC, ha sasfy he equaly d = m q= NC p= php q are seleced. Ths can be done be a searchng echnque. For hese varaons, he facors c g = k f!/ NC p= hp g! wll be mulpled by each oher o ge

11 Journal of Appled Mahemacs (a) (b) Zero correcon s correcon nd correcon 3rd correcon 4h correcon s correcon nd correcon 3rd correcon 4h correcon (c) Fgure : The hrd-order response mean and varance for (a) ε =.,(b)ε =.3,and(c)ε =.5.

12 Journal of Appled Mahemacs (a) Zero correcon s correcon nd correcon 3rd correcon 4h correcon s correcon nd correcon (b) Fgure : The hrd-order response mean and varance for (a) ε =.7 and (b) ε =.. 3rd correcon 4h correcon c g ;has,c g = Var c g. The Kronecker dela funcons are defned as δ jl ={, j= l (3), j =l. The couner f, nhesummaonnherghhandsdeof (), runs over all of he ( n+m n ) combnaons of he posve negers k f,k f,...,km f such ha m = k f =n. Equaons (9) and() canalwaysbesolvedusnghe proper sequence. The frs m+equaons of (9)aresolved ndependenly o ge x (j), j m; hen hey are used o compue he oher componens n (). For j =,he componen u () s obaned by solvng L(x () ) = f() wh he orgnal nal and boundary condons. For j=, he componen x () s obaned by solvng L(x () )=g()δ( ) wh zero nal and boundary condons. The oher componens u (j), j, wll be zeros due o zero rgh hand sde and zero nal and boundary condons. Equaon () specfes hesoluonsequenceobefollowed.thecomponenx (j) s evaluaed n erms of he prevously compued componens x (p) k, p j, k <.Thsmeanshahescorreconsfor all kernels, x (j), j m, are solved frsly and hen he nd correcons for all kernels, x (j), j m,...up o he (NC)h correcons for all kernels x (j) NC, j m.

13 Journal of Appled Mahemacs (a) s correcon (mean of s, nd, and 3rd orders s concden) (varance of nd and 3rd orders s concden) s order nd order 3rd order s order nd order 3rd order (b) nd correcon (mean of s, nd, and 3rd orders s concden) Fgure : Comparson beween he response mean and varance for he frs, second, and hrd orders wh number of correcons and, ε =.3. These resuls are conssen wh he known resuls obaned usng WHE. In WHE, hgher-order kernels are drven by lower-order kernels, and a he boom, he Gaussan kernels are drven by he random forcng drecly. So, he lower-order kernels are usually domnan n magnude [3]. The sascal properes of he soluon wll be calculaed as Var [x ()] = E [x ()] = m k= NC = ε x (), NC (k!) ( R k = ) ε x (k) dτ k. (4) If x (j) = = ε x (j),henwllbeconvergenf[6] ε x (j) x (j) + (5) for [,T].Thsmeansha ε should obey an upper bound condon afer whch dvergence s obaned. 3. The Equvalen Deermnsc Equaons Apply he prevous WHEP algorhm o ge he followng sysems of equaons of he quadrac (n = )nonlnear oscllaory equaon and frs-order (m = ) Gaussan

14 4 Journal of Appled Mahemacs (a) 3rd correcon (mean of nd and 3rd orders s concden) s order nd order 3rd order s order nd order 3rd order (b) 4h correcon (mean of nd and 3rd orders s concden) Fgure 3: Comparson beween he response mean and varance for he frs, second, and hrd orders wh numbers of correcons 3 and 4, ε =.3. approxmaon and dfferen number of correcons (NC). The nal condons are assumed deermnsc and hence only he zero-order and zero-correcon kernel equaon (L(x () )=f())wllhave he nal condons (x and x ). Oher kernels equaons wll have zero nal condons. For he quadrac nonlnear oscllaory sochasc equaon, he applcaon of he WHEP echnque wll resul n he followng se of equaons (Tables,, and3). The equaons wll be wren for he frs, second, and hrd orders. For a ceran correcon level, he deermnsc sysem wll nclude also he equaons from prevous levels. In case of zero nal condons, f() = and g() = ; ha s, RHS = εω x + N(), and we wll have he followng reduced sysem of equaons: L(x () )=, L(x() )=δ( ), L(x () )= ω [x () ( )] d, L(x () )=, R L(x () )=, L(x() )= ω x () x() ( ), L(x () 3 )= ω [x () ] ω x () ( )x () ( )d, R L(x () 3 )=, L(x () 4 )=,

15 Journal of Appled Mahemacs 5 Table : The equvalen deermnsc sysem for frs-order approxmaon wh dfferen number of correcons (NC). L(x () ) = f() NC = NC = NC = 3 NC = 4 NC = 5 L(x () ) = g()δ( ) L(x () )= ω [x () ] ω R [x () ( )] d L(x () )= ω x () x() ( ) L(x () )= ω x () x() ω R x () ( )x () ( )d L(x () )= ω x () x() ( ) ω x () x() ( ) L(x () 3 )= ω x () x() ω [x () ] ω R x () ( )x () ( )d ω R [x () ( )] d L(x () 3 )= ω x () x() ( ) ω x () x() ( ) ω x () x() ( ) L(x () 4 )= ω x () x() 3 ω x () x() ω R x () ( )x () 3 ( )d ω R x () ( )x () ( )d L(x () 4 )= ω x () x() 3 ( ) ω x () x() ( ) ω x () x() ( ) ω x () 3 x() ( ) L(x () 5 )= ω x () x() 4 ω x () x() 3 ω [x () ] ω R x () ( )x () 4 ( )d ω R x () ( )x () 3 ( )d ω R [x () ( )] d L(x () 5 )= ω x () x() 4 ( ) ω x () x() 3 ( ) ω x () x() ( ) ω x () 3 x() ( ) ω x () 4 x() ( ) L(x () 4 )= ω x () x() ( ) ω x () 3 x() ( ), L(x () 5 )= ω x () x() 3 4. Oscllaory Equaon and he Numercal Solver For he lnear oscllaory equaon, ω x () ( )x () 4 ( )d R ω [x () ( )] d, R L(x () 5 )=. (6) x+ωζ x+ω x=f(), (7) he lnear operaor L wll be L= d d +ωζd d +ω. (8) Ths means ha he lnear oscllaory equaon can be wren as L(x) = f(). The parameer ω s he undamped angular frequency of he oscllaor and ξ s he dampng rao. The nal condons wll be consdered as x() = x and x () = x. In case of zero nal condons, he exac soluon ha canbeobanedusngdfferenmehodssuchasheheory of lnear dfferenal equaons or he Laplace ransform wll be he convoluon x () =h() f(), (9) where h() = (/ω d )e ωξ sn(ω d ) wh ω d = ω ξ, assumng underdampng (ξ <). For f() = e,hesoluonwllbex() = h( τ)f(τ)dτ,whchresulsn x () = ωξ+ω (e e ωξ cos (ω d ) + ωξ ω d e ωξ sn (ω d )). (3) The numercal soluon can be obaned for a model equaon andhenusedforallkernelsnhepropersequence.the modelequaonnhscasewllakeheform x+a x+a x=f(), (3) where a and a are assumed consan. We can use any dfference scheme, bu as we are workng wh a whe nose, he Drac dela funcon δ( a) sexpecedoappearnhe equaons of some kernels. So, an negral numercal scheme such as FEM or FVM wll be more suable as he negraon of he Drac dela funcon s easer o be handled. In our case,hefnevolumemehod(fvm)wllbeconsdered.the me axs where [,T] wll be dscrezed no n equal nervals of sze Δ. The nerval exendng from o s aken as he conrol volume. Ths echnque n one dmenson wll be equvalen o he rapezodal negraon rule. The second-order lnear equaon (3) wllbedecomposedno wo smulaneous frs-order equaons. Ths can be done by subsung x=y,andhenwewllhave x=y wh x () =x, y+a y+a x=f() wh y = x (). (3)

16 6 Journal of Appled Mahemacs Table : The equvalen deermnsc sysems for second-order approxmaon wh dfferen number of correcons (NC). NC = NC = NC = 3 NC = 4 L(x () ) = f() L (x () ) =g() δ ( ) L (x () ) = L(x () )= ω [x () ] ω R [x () ( )] d L (x () ) = ω x () x() ( ) L (x () ) = ω x () ( ) x () ( ) L(x () )= ω x () x() ω R x () ( )x () ( )d L(x () )= ω x () x() ( ) ω x () x() ( ) 4ω R x () ( )x () (, )d L (x () )= 4ω x () x() (, ) 4ω x () ( )x () ( ) L(x () 3 )= ω x () x() ω [x () ] ω R x () ( )x () ( )d ω R [x () ( )] d ω R [x () (, )] d d L(x () 3 )= ω x () x() ( ) ω x () x() ( ) ω x () x() ( ) 4ω R x () ( )x () (, )d 4ω x () ( )x () (, )d R L (x () 3 )= 4ω x () x() (, ) 4ω x () x() (, ) 4ω x () ( )x () ( ) ω x () ( )x () ( ) 8ω R x () (, 3 )x () (, 3 )d 3 L(x () 4 )= ω x () x() 3 ω x () x() ω R x () ( )x () 3 ( )d ω R x () ( )x () ( )d 4ω R x () (, )x () (, )d d L(x () 4 )= ω x () x() 3 ( ) ω x () x() ( ) ω x () x() ( ) ω x () 3 x() ( ) 4ω R x () ( )x () 3 (, )d 4ω R x () ( )x () (, )d 4ω R x () ( )x () (, )d L (x () 4 )= 4ω x () x() 3 (, ) 4ω x () x() (, ) 4ω x () x() (, ) 4ω x () ( )x () 3 ( ) 4ω x () ( )x () ( ) 8ω R x () (, 3 )x () (, 3 )d 3 8ω R x () (, 3 )x () (, 3 )d 3 Inegrae (3) along he conrol volume o ge xd = yd. (33) Approxmae he negral wh he rapezodal rule whch s of accuracy proporonal o (Δ) o ge x =x + Δ (y +y ). (34) Also, negrae (3) along he conrol volume and subsue wh x from (34)oge y = 4F +y (4 a Δ a (Δ) ) 4a Δx 4+a Δ + a (Δ), (35) where F = f()d =.5(f +f )Δ. If he Drac dela funcon appears n he rgh hand sde, a specal reamen for F s consdered. In hs case F = q()δ( )d = q( ) when = and F =when =.Equaon(35)wll be solved a each node o ge y and hen (34)compuedo ge x. The numercal soluon can be valdaed wh he exac soluon n case of f() = e (see (3)). Wh x = x =, ω=, ξ =.5, T=,andΔ =., hecomparson n Fgure shows ha he numercal soluon has suffcen accuracy wh a relave error of.3% a Δ =..Theoverall convergence rae of he developed scheme was esed, and has a convergence order of.5 as shown n Fgure. 5. Resuls The followng oupu s smulaed usng he prevous developed numercal solver. The soluon (34) of he model equaon (3) susedogeallkernelswhheproperrgh handsde.themeanresponseandheresponsevaranceare hen calculaed from he kernels usng (4). Fgures 3 and 4 show he frs-order response mean and varance for dfferen values of ε. The smulaons are done for he case of zero nal condons, zero deermnsc excaon, and un envelope funcon mulpled by he whe nose. The angular frequency s ω=and he dampng rao s ξ =.5. For he response mean, he 3rd and 4h correcons are concden. For he varance, he nd and 3rd correcons are concden and also he 4h and 5h correcons. As s shown n he fgures, he nonlneary srengh ε grealy affecs he ampludes of he mean and varance. I should no be ncreased afer a ceran value o oban a convergen soluon. Ths value depends on he dfferen parameers of he problem. As he nonlneary srengh ε ncreases, we need hgher number of correcons. For ε =. and a = seconds, he rao beween he response mean of each correcon and he proceedng one s around

17 Journal of Appled Mahemacs 7 Table 3: The equvalen deermnsc sysem for hrd-order approxmaon wh dfferen number of correcons (NC). NC = NC = L(x () ) = f() L(x () ) = g()δ( ) L (x () )= 6L (x (3) )= L(x () )= ω [x () ] ω R [x () ( )] d L(x () )= ω x () x() ( ) L (x () )= ω x () ( )x () ( ) 6L (x (3) )= L(x () )= ω x () x() ω R x () ( )x () ( )d L(x () )= ω x () x() ( ) ω x () x() ( ) 4ω R x () ( )x () (, )d L (x () )= 4ω x () x() (, ) 4ω x () ( )x () ( ) 6L (x (3) ) = ω x () ( )x () (, 3 ) L(x () 3 )= ω x () x() ω [x () ] ω R x () ( )x () ( )d ω R [x () ( )] d ω R [x () (, )] d d L(x () 3 )= ω x () x() ( ) ω x () x() ( ) ω x () x() ( ) NC = 3 NC = 4 4ω R x () ( )x () (, )d 4ω R x () ( )x () (, )d L (x () 3 )= 4ω x () x() (, ) 4ω x () x() (, ) 4ω x () ( )x () ( ) ω x () ( )x () ( ) ω R x () ( 3)x (3) (,, 3 )d 3 8ω R x () (, 3 )x () (, 3 )d 3 6L (x (3) 3 ) = ω x () x(3) (,, 3 ) ω x () ( )x () (, 3 ) ω x () ( )x () (, 3 ) L(x () 4 )= ω x () x() 3 ω x () x() ω R x () ( )x () 3 ( )d ω R x () ( )x () ( )d 4ω R x () (, )x () (, )d d L(x () 4 )= ω x () x() 3 ( ) ω x () x() ( ) ω x () x() ( ) ω x () 3 x() ( ) 4ω R x () ( )x () 3 (, )d 4ω R x () ( )x () (, )d 4ω R x () ( )x () (, )d ω R x () (, 3 )x (3) (,, 3 )d d 3 L (x () 4 )= 4ω x () x() 3 (, ) 4ω x () x() (, ) 4ω x () x() (, ) 4ω x () ( )x () 3 ( ) 4ω x () ( )x () ( ) ω R x () ( 3)x (3) 3 (,, 3 )d 3 ω R x () ( 3)x (3) (,, 3 )d 3 8ω R x () (, 3 )x () (, 3 )d 3 8ω R x () (, 3 )x () (, 3 )d 3 6L (x (3) 4 ) = ω x () x(3) 3 (,, 3 ) ω x () x(3) (,, 3 ) ω x () ( )x () 3 (, 3 ) ω x () ( )x () (, 3 ) ω x () ( )x () (, 3 ) 4ω R x () (, 4 )x (3) (, 3, 4 )d 4 4ω R x () (, 4 )x (3) (, 3, 4 )d 4 4ω R x () ( 3, 4 )x (3) (,, 4 )d 4.98, whch s greaer han ε. Ths means ha he condon of convergence s sasfed and hence he soluon converges n hs case. Also, he convergence condon s sasfed for ε =.3 and ε =.5. In case ε =.7, he rao beween he 4h and 5h correcons s.6 and beween he nd and 3rd correcons s.58. Boh raos are lesser han.7, whch means ha we wll have a dvergen soluon for ε=.7.also, ε =. wll produce a dvergen soluon. Fgures 5 and 6 show he frs-order response mean and varance usng he same parameers n Fgures 3 and 4, bu heenvelopefuncong() s aken as e.5.assclear from he fgures, he effec of he nonlneary srengh ε

18 8 Journal of Appled Mahemacs (a) s correcon (mean of s, nd, and 3rd orders s concden) (varance of nd and 3rd orders s concden) s order nd order 3rd order s order nd order 3rd order (b) nd correcon (mean of s, nd, and 3rd orders s concden) Fgure 4: Comparson beween he response mean and varance for he frs, second, and hrd orders wh numbers of correcons and, ε =.. on he response varance s neglgble. The mulplcaon of e.5 aenuaes he effec of he whe nose and becomes neglgble wh he me ncrease. The varance vanshes wh he me and he soluon becomes nearly deermnsc. Also, he response varance s nearly nvaran wh he number of correcons. Fgures 7 and 8 show he second-order response mean and varance usng he same parameers n he frs-order smulaons, Fgures 3 and 4. Fgure 9 shows he second-order response mean and varance for longer me nerval, T= seconds. Ths was done o ensure ha T=secondssa suffcen nerval for he response mean and varance o reach her seady sae values. For he second-order approxmaon, heresponsevarancescompuedpraccallyas Var [x ()] NC = (( R = ) ε x () NC ( + R = ) ε x () d )d. (36) Fgures and show he hrd-order response mean and varance wh he same parameers used n he frs- and second-orders smulaons. For he hrd-order approxmaon, he response varance s compued praccally, for me

19 Journal of Appled Mahemacs (a) 3rd correcon (mean of nd and 3rd orders s concden) s order nd order 3rd order s order nd order 3rd order (b) 4h correcon (mean of nd and 3rd orders s concden) Fgure 5: Comparson beween he response mean and varance for he frs, second, and hrd orders wh numbers of correcons 3 and 4, ε =.. and memory savng, as Var [x ()] = R NC (( = + R ) ε x () NC (( = ) ε x () +3 R NC ( = ) ε x (3) d 3 )d )d. (37) Table 4 shows a comparson beween he compung mes for dfferen levels of correcon for he hrd-order approxmaon. The compung mes for he response mean and varancearealsoshown.theabledsplaysalsoheorder of negrals n each level. The me sep was. seconds andhecompuaonsaredoneoninelcore5,.4ghz, machne wh W7, 3 bs. We can noe ha around 9% of he compung me s consumed n level 4 (4h correcon level).thssdueohehgherordersofnegralscompued n hs level. As he level of correcon ncreases, he order of negral ncreases and hence he compuaonal me wll also ncrease.

20 Journal of Appled Mahemacs Table 4: The compung me for each level and for he response mean and varance compared wh he oal me. Tme (sec.) Order of negrals Level.4 Level.5 Level.9 & Level3.8 &&3 Level &&3&4.9.6 &&3 Toal me.88 Fgures and 3 show a comparson, a ε =.3, beween he response mean and varance for dfferen orders and dfferen number of correcons. As was descrbed earler, he soluon s convergen a ε =.3. As s shownnhefgures,heresponsevarancesmoresensve o he approxmaon order han he response mean. The varance converges as he approxmaon order ncreases. Ths means ha, for a convergen soluon, hgher-order approxmaons are requred for accurae predcon of he sochasc response. Fgures 4 and 5 show a comparson, a ε =., beween he response mean and varance for dfferen orders and dfferen number of correcons. In hs case, he soluon s dvergen. The response varance dverges as he approxmaon order ncreases and also as he correcon level ncreases. I s worh o noe ha he WHEP echnque used n he curren work can be exended o solve sochasc PDEs wh whe nose n mulple dmensons and of dfferen colors as descrbed n [6]. 6. Summary and Conclusons The mean response of he quadrac nonlnear oscllaory sysem subjeced o nonsaonary random excaon was nvesgaed usng WHEP echnque. The equvalen deermnsc equaons are derved up o hrd order. The soluon s approxmaed up o ffh correcon for he frs order and up o fourh correcon for he second and hrd orders. Numercal negral soluon of he equvalen deermnsc seofequaonswasappledusnghefvm.thenumercal reamen s valdaed afer comparng he resuls wh he analycal soluon. The numercal solver s ulzed n smulang he mean and varance of he nonlnear sochasc oscllaory moon wh hgher order, hgher number of correcons, and dfferen srenghs of he nonlnear erm. The values of he nonlneary srengh requred for convergen soluon are esmaed. I was found ha he numercal soluon s effcen, and hgher-order approxmaons are requred for accurae predcon of he sochasc response. References [] A. Jahed and G. Ahmad, Applcaon of Wener-Herme expanson o nonsaonary random vbraon of a Duffng oscllaor, Transacons of he ASME. Journal of Appled Mechancs,vol.5,no.,pp ,983. [] J.C.Cores,J.V.Romero,M.D.Rosello,andR.J.Vllanueva, Applyng he Wener-Herme random echnque o sudy he evoluon of excess wegh populaon n he regon of Valenca (Span), AmercanJournalofCompuaonalMahemacs,vol., no. 4, pp. 74 8,. [3] W. Lue, Wener chaos expanson and numercal soluons of sochasc paral dfferenal equaons [Ph.D. hess], Calforna Insue of Technology, Pasadena, Calf, USA, 6. [4] M. A. El-Tawl, The applcaon of WHEP echnque on sochasc paral dfferenal equaons, Inernaonal Journal of Dfferenal Equaons and Applcaons,vol.7,no.3,pp , 3. [5] M. A. El-Tawl and A. S. Al-Johan, Approxmae soluon of a mxed nonlnear sochasc oscllaor, Compuers & Mahemacs wh Applcaons,vol.58,no.-,pp.36 59,9. [6] M. A. El-Tawl and A. S. El-Johan, On soluons of sochasc oscllaory quadrac nonlnear equaons usng dfferen echnques, a comparson sudy, Journal of Physcs, vol.96,no., 8. [7] M.A.El-TawlandA.F.Fareed, Soluonofsochasccubc and qunc nonlnear dffuson equaon usng WHEP, Pckard and HPM mehods, Open Journal of Dscree Mahemacs,vol., no., pp. 6,. [8] M. A. El-Tawl and A. El-Shkhpy, Approxmaons for some sascal momens of he soluon process of sochasc Naver- Sokes equaon usng WHEP echnque, Appled Mahemacs & Informaon Scences, vol. 6, no. 3, pp. 95,. [9]M.A.El-TawlandA.A.El-Shekhpy, Sascalanalyss of he sochasc soluon processes of -D sochasc naversokes equaon usng WHEP echnque, Appled Mahemacal Modellng,vol.37,no.8,pp ,3. [] A. S. El-Johan, Comparsons beween WHEP and homoopy perurbaon echnques n solvng sochasc cubc oscllaory problems, n Proceedngs of he AIP Conference, vol. 48, pp ,. [] N. Wener, NonlnearProblemsnRandomTheory, MIT Press, John Wley, Cambrdge, Mass, USA, 958. [] R. H. Cameron and W. T. Marn, The orhogonal developmen of non-lnear funconals n seres of Fourer-Herme funconals, Annals of Mahemacs,vol.48,pp ,947. [3] T.Imamura,W.C.Meecham,andA.Segel, Symbolccalculus of he Wener process and Wener-Herme funconals, Journal of Mahemacal Physcs,vol.6,pp ,965. [4] W. C. Meecham and D. T. Jeng, Use of he Wener-Herme expanson for nearly normal urbulence, Journal of Flud Mechancs,vol.3,pp.5 35,968. [5]E.F.Abdel-GawadandM.A.El-Tawl, Generalsochasc oscllaory sysems, Appled Mahemacal Modellng,vol.7,no. 6, pp , 993. [6] M. A. El-Belagy and M. A. El-Tawl, Toward a soluon of a class of non-lnear sochasc perurbed PDEs usng auomaed WHEP algorhm, Appled Mahemacal Modellng,vol.37,no. -3, pp , 3. [7] X. Yong, X. We, and G. Mahmoud, On a complex duffng sysem wh random excaon, Chaos, Solons and Fracals, vol.35,no.,pp.6 3,8. [8] P. Spanos, Sochasc lnearzaon n srucural dynamcs, Appled Mechancs Revews, vol. 34, pp. 8, 98. [9] W. Q. Zhu, Recen developmens and applcaons of he sochasc averagng mehod n random vbraon, Appled Mechancs Revews,vol.49,no.,pp.7 8,996. [] J. Aknson, Egenfuncon expansons for randomly exced nonlnear sysems, Journal of Sound and Vbraon, vol. 3, no.,pp.53 7,973.

21 Journal of Appled Mahemacs [] A. Bezen and F. Klebaner, Saonary soluons and sably of second order random dfferenal equaons, Physca A,vol.33, no. 3-4, pp , 996. [] A. H. Nayfeh, Problems n Perurbaon, JohnWley&Sons, New York, NY, USA, 993. [3] MahML, hp://

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