[Hussain* et al., 5(8): August, 2016] ISSN: IC Value: 3.00 Impact Factor: 4.116

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1 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY NUMERICAL SOLUTIONS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA ACCELERATED GENETIC ALGORITHM Dr. Ema A. Hussai*, Dr. Yasee M. Alrajhi * Al-Musasiriyah Uiversiy, College of Sciece, Deparme of Mahemaics, Iraq Al-muhaa Uiversiy, College of Sciece, Deparme of Mahemaics, Iraq DOI:.58/zeoo ABSTRACT This paper irouce a ew accelerae Geeic Algorihms (GAs) meho o fi a umerical soluios of sochasic Parial iffereial equaios rive by space-ime whie ose wieer process. The umerical scheme is base o a represeaio of he soluio of he equaio ivolvig a sochasic par arisig from he oise a a eermiisic parial iffereial equaio. By usig Doss-Sussma rasformaio ha eables us o work wih a parial iffereial equaio isea of he sochasic parial iffereial equaio. The compare hese soluios obaie by our meho wih saul'yev meho a eermiisic soluio. KEYWORDS: SPDS, Accelerae Geeic Algorihm meho, Numerical soluio of sochasic parial iffereial equaios. INTRODUCTION I his paper we wa o ake a quicker look a he umerical soluios for sochasic parial iffereial equaios (SPDEs). Workig o he umerical soluios for SPDEs we face may ifficulies. O he oe ha we have o cosier problems kow from umerically solvig eermiisic parial iffereial equaios. O he oher ha we are face wih problems riggere by umerically solvig sochasic oriary iffereial equaios (SODEs). A aiioally ew issues arise resulig from he ifiie imesioal aure of he uerlyig oise processes,[]. Sochasic parial iffereial equaios (SPDEs) are use as a moel i may applicaios. This area of mahemaics is especially moivae by he ee o escribe raom pheomea suie i aural scieces like physics, chemisry, biology, a i corol heory, []. So, we ca efie SPDEs, by combie eermiisic parial iffereial equaios wih some ki of oise. Cosier he SPDE wih space-ime whie oise,[4]. u(, x) = u xx (, x) + g(u(, x))w(, x) () wih < x <, a u(, x) = u, u(, ) = u(, ) =, a u (, ) = u (,) = The, wo iffere ways of wriig his equaio () are : u = u x + g(u) W () x or u = u xx + g(u)h k (x)w k () k= Such ha, hree kis of space-ime whie oise as i [4] are : (3) Browia Shee W(, x) = μ([, T] [, x]) Cylirical Browia moio family of Gaussia raom variables hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [68]

2 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 B = B (h), h H a Hilber space, s.. E[B (h)] =, E[B (h)b s (g)] = h, g H ( s) Space-ime whie oise W(, x) = W = x k= hk (x)w k (), where {h k } is assume a Basis of he Hilber space we re i, if {h k, k > } is a complee orhoormal sysem, he {B (h k ), k > } iepee saar Browia moio. The coecio bewee he hree kis: If H = L (R) or H = L (, ), he B (h) = W x h(x)x a x B (x) = B (X [,X] ) = (h k (y)yw k ()) = W(, x) k= where we assume ha h k (x) = si(kπx) (4) The we ge equaios of he form : Or i iegral form U(, x) = [AU(, x) + c(x)u(, x)] + h l ()U(, x)b l U(, x) = U (x) + AU(s, x)s + c(x)u(s, x)s + h l l (s)u(s, x)b s For T. The process B = (B l,, B ) T is a -imesioal Browia moio. The operaor A is efie as : Au = a u u ij + b x i x i j x i i,j= i= l= l= m, wih a ij (x) = σ ik σ jk Where he iffusio marix σ: R R m a he rif coefficie : R R. he iiial coiio U, he fucios c, σ a b are suppose o be smooh fucios of he space variable, (h l ()) l are boue a holer coiuous of orer /. Thus he equaio (5) has a uique regular srog soluio. I his paper, we focus o he sochasic hea equaio. Thus, we simplify he above equaio o : U(, x) = AU(, x) + S(U(, x))w(, x) (8) where S is a muliplicaio operaor of he form (S(v)u)(x) = b(x, v(x)). u(x) Takig a closer look a he oise i his equaio we see ha we ca spli i io wo ypes, aiive a muliplicaive oise. We speak of aiive oise if he operaor S is a cosa operaor a of muliplicaive oise if S is o cosa. The objecive of our work is o evelop a umerical scheme for he raom fiel U. The problem of umerical soluios of (5) has bee suie by may auhors wih iffere approaches. The ieas ha lea us o propose a ew scheme are wofol. O he oe ha we wish o propose a umerical scheme ha separaes he oise S from he seco orer operaor A. This iea has bee use i [],[5] i a filerig coex i which he auhors scheme firs performs off-lie a wie umber of soluios of parial iffereial equaios by he fiie eleme meho. The sochasic par of he simulaio is oe afer his firs sep. O he oher ha we wa o use he accelerae geeic algorihm meho o fi umerical soluio for he parial iffereial equaios ha may appear i our scheme. I orer o impleme he above ieas, we ee o irouce he -imesioal Markov process X = (X ) T whose ifiiesimal geeraor is give by he seco orer operaor i equaio (7). The Markov process X is govere by ifiiesimal geeraor A of he sochasic iffereial equaio is : k= (5) (7) (6) hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [69]

3 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 where he iiial coiio x i R. X = x + b(x s )s + σ(x s ) W s, T () TYPES OF SOLUTIONS OF SPDES Sochasic parial iffereial equaios of he form X (x) = [AX + F(X )] + S(X )W (x), X() = ξ () have iffere oios of soluios. As i [] we fi : Defiiio : D(A)-value preicable process X(), [, T] is calle a aalyical srog soluio of he problem (6) if X() = [AX s + F(X s )]s + S(X s )W s I paricular,he iegral i he righ-ha sie have o be well-efie,[],[4]., P a. s. () Defiiio : H-value preicable process X(), [, T] is calle a aalyical weak soluio of he problem (6) if < X(), ζ >= [< X(s), A ζ > +< F(X s ), ζ >]s + < ζ, S(X s )W s >, P a. s. (3) For each εd(a ), i paricular,he iegral i he righ-ha sie have o be well-efie. Defiiio 3: H-value preicable process X(), [, T] is calle a mil soluio of he problem (6) if X() = [e A( s) F(X s )]s + e A( s) S(X s )W s I paricular,he iegral i he righ-ha sie have o be well-efie,[],[4]., P a. s. (4) STOCHASTIC INTEGRAL WITH RESPECT TO CYLINDRICAL WIENER PROCESS We eoe by L = L (U, Y) he space of Hilber-Schmi operaors acig from U io Y, a by L = L(U, Y), we eoe he space of liear boue operaors from U io Y,[],[4]. Le us cosier he orm of he operaor ψ L : ψ L = < ψg h, f k > Y h,k= = λ h < ψe h, f k > Y = ψq = r(ψqψ ) (5) HS h,k= Where g i = λ i e i a {λ i }, {e i } are eigevalues a eigefucios of he operaor, {g i }{e i } a {f i } are orhoormal bases of spaces U, U a Y, respecively. The space L is a separable Hilber space wih he orm ψ L = r(ψqψ ) (6) I paricular.whe Q = I he U = U a he space L becomes L (U, Y).. Whe Q is a uclear operaor, ha is rq < +, he L(U, Y ) L (U, Y). For, assume ha K L(U, Y), ha is K is liear boue operaor from he space U io Y. Le us cosier he operaor ψ = K U, ha is he resricio of operaor K o he space U, where U = Q (U). Because Q is uclear operaor, he Q is Hilber- Schmi operaor. hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [7]

4 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 Proposiio. he formula W c () = g j β j () j=, (7) efies Wieer process i U wih covariace operaor Q such ha rq < +. Proposiio. For ay a U he process < a, W c () > U = < a, g j > U β j () j= (8) is real-value Wieer process a E < a, W c () > U < b, W c () > U = ( s) < Qa, b > U, a, b U Aiioally, ImQ = U a u U = Q. u U I he case whe Q is uclear operaor, Q is Hilber-Schmi operaor. Takig U = U, he process W c (),, efie by (7) is he classical Wieer process irouce. Defiiio.4 The process W c (),, efie i (7), is calle cylirical Wieer process i U whe rq < +. As show i Fig. below. Fig. Cylirical Whie oise wih is isribuio a specral esiy The sochasic iegral wih respec o cylirical Wieer process is efie as follows. As we have alreay wrie above, he process W c (), efie by () is a Wieer process i he space U wih he covariace operaor Q such ha rq < +. The he sochasic iegral, g(s)w c (s) Y (9) where g(s) L(U, Y), wih respec o he Wieer process W c () is well efie o U. We eoe by N(Y ) he space of all sochasic processes : [, T] Ω L [U, Y] Such ha T E ( () L [U,Y] ) < + () a for all u U, ()u is a Y-value sochasic process measurable wih respec o he filraio F. The sochasic iegral (s)w c (s) Y wih respec o cylirical Wieer process, give by () for ay process N(Y) ca be efie as he limi hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [7]

5 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 m (s)w c (s) = lim (s)g j β j (s) m i Y () I L (Ω) sese. j= MATHEMATICAL SETTING AND ASSUMPTIONS,[] Le T > a le (Ω, F, Ρ) be a probabiliy space wih a ormal filraio F, [, T]. i aiio le (H, <.,. >) be a separable Hilber space wih orm eoe by..we will ierpre he SPDE () i such a space H. The objecs, x, F, W, here are specifie hrough he followig assumpios. Assumpio : liear operaor A. There exis sequece of real eigevalues λ λ a eigefucios {e } of A such ha he liear operaor A: D(A) H H is give by : For all Av = λ < e, v)e, = () v D(A)wih D(A) = {v H λ < e, v > < } Le D(( A) r ) wih r R eoe he ierpolaio space of he operaor ( A),[8]. Assumpio : Cylirical Browia moio W. here exis a sequece of q,, of posiive real umbers γ (,) such ha = (λ ) γ q < (3) = A iepee real value F Browia moio β,,, i.e. each β is F -aape a he icremes β +h β, h >, are iepee of F. The he cylirical Browia moio W is give by : W (x) = q e (x)β = Remark. The above series may o coverge i H, bu i some space U io which H ca be embee, ([7] a [8]). I our example wih he Laplace operaor i oe imesio, we will have λ = π a q, for. This is he impora case of space ime whie oise. Assumpio 3: olieariy f. The olieariy f: H H is wo imes coiuously iffereiable, i a is erivaives saisfy f (x) f (y) L x y, ( A) ( r) f (x)( A) r v L v For all x, y H, v D( A) r a r =,, a hey saisfy (4) A f (x)(v, w) L ( A) ( ) v ( A) ( ) w, for all v, w, x H, L > Remark. The fucio f is usually give as a real-value fucio of a real variable, bu i he SPDE () i is cosiere as a fucio efie o H a akig values i some fucio space such as a subspace of H. Assumpio 4: iiial value X. The iiial value X is a D(( A) r ) value raom variable, which saisfies E ( A) ( ) X 4 < (5) where γ > is give i assumpio.. Wih he above assumpios we ge by [JK] ha Wih X = [k X + f(x, x )] + b(x, X )W (x) (6) hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [7]

6 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 X (x) = a X () = X () = (7 ) for x (,), [, T) has uique mil soluio X: [, T] Ω H β+ (8). A RELATED PARTIAL DIFFERENTIAL EQUATION We prese i his secio a rasformaio ha eables us o work wih a parial iffereial equaio isea of he sochasic parial iffereial equaio (5). This meho is classical a i is kow as he Doss-Sussma rasform whe oe applies i o sochasic iffereial equaio ([7] a [8]). I is a useful rick ha permis o rewrie a large class of oe imesioal sochasic yamic as a oe imesioal raom oriary yamic (by sochasic yamic we mea sochasic iffereial equaio or sochasic parial iffereial equaio). I has bee successfully use i [8] i which he auhors have esimae he probabiliy of fiie-ime blowup of posiive soluios of sochasic parial iffereial equaios wih Dirichle bouary coiio. Doss-Susma rasform The paricular form of (5) will allow us o use a Doss-Susma rasform. We may wrie ha,[5]. U(, x) = exp ( h l l (s)b s ) v(, x) (9) l= wih v ha solves he parial iffereial equaio v(, x) = Av(, x) + (c(x) (hl ()) ) v(, x) (3) As regar o he expressio of he fucio v, i is clear ha v ca be simulae off-lie. Iee he coefficies i he above parial iffereial equaio are c, (h l ) <l< a he coefficies of he Markov process X. They are all suppose o be kow. Cosequely, we ca perform a wie umber of compuaios relae o he parial iffereial equaio saisfie by v. The we shall come back o he simulaio of U iself a we use as much as we wa he previous compuaios. Thus we have spli our scheme io a eermiisic par (he approximaio of v) a a sochasic par (he immeiae compuaio of U whe oe simulaes he Browia moio B). The approximaio of v a he Markov process X will be achieve by a accelerae geeic algorihm. We have he followig proposiio. Proposiio 3. Le u be he soluio of (5). The he fucio v efie almos-surely by v(, x) = u(, x) exp ( h l l (s)b s ) (3) l= is he uique srog soluio of he followig parabolic parial iffereial equaio v(, x) = Av(, x) + (c(x) (hl ()) ) v(, x) (3) T, The above equaio is uersoo rajecory wise sice i is vali for almos-all ω. Proof. We eoe E = (E ) T he process efie by E = exp ( h l l (s)b s ) (33) l= I is a semi-marigale wih he ecomposiio l= l= E = h l l (s)e s B s + (hl (s)) E s s l= I view of (), for all x R, (u(, x)) T is a semi-marigale a we have l= E, u(., x) = (h l (s)) E s u(s, x)s = (h l (s)) v(s, x)s l= l= (34) (35) hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [73]

7 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 Sice E oes o epe o he space variable x, i hols ha E Au(, x) = A(E u(, x)) = Av(, x) a he iegraio by pars formula yiels he resul. ACCELERATED GENETIC ALGORITHM The priciples of geeic algorihm are iscusse i previous paper [9]. Where he compoes of he geeic algorihm,,[],[],[]are :. Iiializaio The value of muaio rae a selecio rae are sae,[9]. The iiializaio of every chromosome is performe by raomly selecig a ieger for every eleme of he correspoig vecor.. Fiess-evaluaio Expressig he Parial iffereial equaio i he followig form: f (x, y, u u (x, y), x y (x, y), u x (x, y), u (x, y)) = (36 ) y x [x, x ], y [y, y ] The associae bouary coiios are expresse as: u(x, y) = f (y), u(x, y) = f (y), u(x, y ) = g (y), u(x, y ) = g (y) (37 ) The seps for he fiess evaluaio of he populaio are he followig:. Choose N equiisa pois i he box [x, x ] [y, y ], N x equiisa pois o he bouary a x = x a a x = x, N y equiisa pois o he bouary a y = y a a y = y. For every chromosome i : (i) Cosruc he correspoig moel M i (x, y), expresse i he grammar escribe earlier. (ii) Calculae he quaiy E(M i ) = N (f(x j, y j, M x i(x j, y j ), M y i(x j, y j ), M x i(x j, y j ), j= M y i(x j, y j )) (38) (iii) Calculae a associae pealy P i (M i ). The pealy fucio P epes o he bouary coiios a i has he form: N P (M i ) = x (M i (x, y j ) f (y j )) j= N x P (M i ) = j= (M i (x, y j ) f (y j )) (39) N y j= N y j= P 3 (M i ) = (M i (x j, y ) g (x j )) P 4 (M i ) = (M i (x j, y ) g (x j )) (iiii) Calculae he fiess value of he chromosome as: v i = E(M i ) + P (M i ) + P (M i ) + P 3 (M i ) + P 4 (M i ) (4) 3. Geeic operaors The geeic operaors ha are applie o he geeic populaio are he iiializaio, he crossover a he muaio. A raom ieger of each chromosome was selece o be i he rage [..55]. The pares are selece via ourame selecio, i.e. : - Firs, creae a groups of K raomly selece iiviuals from he curre populaio. - The iiviuals wih he bes fiess i he group is selece, he ohers are iscare. The fial geeic operaor use is he muaio, where for every eleme i a chromosome a raom umber i he rage [, ] is chose,[9]. hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [74]

8 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: Termiaio corol Creaig ew geeraio require for applicaio geeic operaors o he populaio i orer o fi he bes chromosome havig beer fiess or wheever he maximum umber of geeraios was obaie. 5. Techical of he Accelerae Meho To make he meho is faser o arrive he exac soluio of he parial iffereial equaios by he followig : - Iser he bouary coiios of he parial iffereial equaio as a par of chromosomes i he our populaio of he problem, he algorihm gives he exac soluio or umerical approximaio soluio i a few geeraios. - Iser a par of exac soluio ( or paricular soluio ) as a par of a chromosome i he populaio, fi he algorihm ha gives a exac soluio i a few geeraios. 3- Iser he vecor of exac soluio ( if exis ) as a chromosome i he our populaio of he problem, he algorihm gives he exac soluio i he firs geeraio. APPLICATION OF THE ACCELERATED GENETIC ALGORITHM I his secio we applie our algorihm o some SPDEs rive by cylirical Browia moio wih aiive a muliplicaive cases.. Sochasic Parial iffereial equaios wih aiive oise. We firs look a SPDEs wih aiive oise o ge a referece abou how well he earlier presee meho work. We cosier he sochasic hea equaio wih aiive space ime whie oise o he oe-imesioal omai [,] over he ime ierval [, T] wih T =. Cosier he followig SPDE X (x) = [κ X (x) + f (x, X (x))] + b(x, X (x))w (x) (4) wih X (x) = a X () = X () = for x (,), [, T). a f(x, y) =, b(x, y) =, where he oise W (x) here is he space-ime whie oise wieer process W (x) = e (x)β = wih q for all i view of assumpio.. (The summaio here is jus formal, i oes o coverge i H.) Therefore, we have γ = ( ) ε,wih a arbirary small ε > i our siuaio. 4 The he SPDE X (x) = [κ X (x) ] + W (x) (4) has uique mil soluio X: [, T] Ω H β+. where escribe i [Kru] ca be wrie as X = e A( s) W s = e e λ( s) β s (43) where we use he eigevalues λ = π (44) a eigevecors e (x) = si(πx) (45) for all of he operaor A. = Example Le we ry o fi he umerical soluio of he SPDE wih aiive oise. U (x) = [ U(, x) x ] + W (x) (46) Wih U (x) = a U () = U () = (47 ) for x (,), [, T] where W (x) is space-ime whie oise wieer process. By usig Doss-Susma rasform (3). We fi hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [75]

9 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 v(, x) = v x (, x) + ( (hl ()) ) v(, x) (48) wih AU = a U U ij + b x i x i = U + U j x i x i x j x i i,j= i= l= i,j= i= = U x (49) The b(x ) = a σ ij =,a he Markov process X was govere by he operaor A of his sochasic parial iffereial equaio is: X = x + B s, T (5) where he iiial coiio x i R. if h() = a =, he (48 ) became : v(, x) = v x (, x) v(, x) (5) Wih v (x) = a v () = v () = for x (,), [, T) Now fi he umerical soluio of he parial iffereial equaio (PDE)(5) by usig a accelerae geeic algorihm. We fou ha Gp(, x) = exp ( 3 ) six (5) A he soluio of he sochasic oriary iffereial equaio (5) (Markov process) geerae by he ifiiesimal geeraor A by accelerae geeic algorihm is : X = x + B() (53) The, he soluio of he origial equaio (46) is obaie by subsiuig (5),(53) i equaio (9): U(, x) = exp ( h l l (s)b s ) GP(, X ) = exp ( h l l (s)b s ) exp ( 3 ) si (x + B()) (54) l= Fig. show his soluio l= Fig. soluio of SPDE (46) a he compare his soluio by our meho wih he soluio obaie by Saul'yev meho,[3]. A wih is correspoig eermiisic soluio. (I his problem a oher es examples, by a eermiisic soluio we mea he umerical soluio of he uperurbe problems). This compariso show i Fig. below : hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [76]

10 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 Fig. compariso of soluios of SPDE (46). Sochasic Parial iffereial equaios wih muliplicaive oise. Look a SPDEs wih muliplicaive oise o ge a referece abou how well he earlier presee meho work. We cosier he sochasic hea equaio wih muliplicaive space ime whie oise o he oe-imesioal omai [,] over he ime ierval [, T] wih T =. Example Le us ry o fi he umerical soluio of he SPDE wih muliplicaive oise. Cosier he SPDE U = κδu (x) + U (x)w (x) ( 55) Wih U (x) = x ( si ( π x) ) a U () = U () = (56) for x (,), [, T) where W (x) is space-ime whie oise wieer process. where κ is a small parameer, we will have κ =. By usig Doss-Susma rasform (3). We fi v(, x) = v x (, x) + ( (hl ()) ) v(, x) (57) Wih AU = a U U ij + b x i x i = U + U = U j x i x i x j x i x (58) i,j= i= i,j= The b(x ) = a σ ij =.,a he Markov process X was govere by he ifiiesimal geeraor of his sochasic iffereial equaio A is: X = x +. B s, T (59) where he iiial coiio x i R. if h() = a =, he (57) became : Wih v(, x) l= i= =. v x (, x) v(, x) (6) v (x) = x ( si ( π x) ) a v () = v () = (6) for x (,), [, T). Now fi he umerical soluio of he parial iffereial equaio (PDE)(6) by usig accelerae geeic algorihm. We fou ha a geeraio 6, he umerical soluio is: Gp6(, x) = e si (3x ) (6) hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [77]

11 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 a he soluio of he sochasic oriary iffereial equaio (59) (Markov process) geerae by he ifiiesimal geeraor A by accelerae geeic algorihm is : X = x +.B() (63) The, he soluio of he origial equaio (55) is obaie by subsiuig (6),(63) i equaio (9), we fi : U(, x) = e W (x) GP6(, X ) = e W(x) e si(3(x +.B()) ) (64) Fig.3 show his soluio Fig.3 The soluio of SPDE (55 ) a he compare his soluio by our meho wih he soluio obaie by Saul'yev meho a wih is correspoig eermiisic soluio. This compariso show i Fig. 4 below : Fig.4 compariso of soluios of SPDE (55) The comparisos of errors of hese soluios was show i able (4.). Table (4.) Comparisos of he errors. x saul'yev-gp6 iermeisic-gp hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [78]

12 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: Example 3 Le we ry o fi he approximaio soluio of he SPDE wih muliplicaive oise. U (x) = κδu (x) U (x)w (x) ( 65) Wih U (x) = x ( x ) a U () = U () = (66) for x (,), [, T),where W (x) is space-ime whie oise wieer process, a where κ is a small parameer, we will have κ =. By usig Doss-Susma rasform (3). We fi Wih v(, x) =. v x (, x) + ( (hl ()) ) v(, x) (67) AU = a U U ij + b x i x i = U + U = U j x i x i x j x i x (68) i,j= i= i,j= The b(x ) = a σ ij =.,a he Markov process X was govere by he ifiiesimal geeraor A of his sochasic iffereial equaio is: X = x +. B s, T (69) where he iiial coiio x i R. if h() = a = he (67) becomes : v(, x) =. v x (, x) + v(, x) (7) Wih v (x) = x ( x ) a v () = v () = (7) for x (,), [, T) Now fi he umerical soluio of he parial iffereial equaio (PDE)(65) by usig accelerae geeic algorihm. We fou a geeraio ha : Gp(, x) = exp( exp()) six (7) A he soluio of he sochasic oriary iffereial equaio (69) (Markov process) geerae by he ifiiesimal geeraor A by accelerae geeic algorihm is : X = x + B() (73) The, he soluio of he origial equaio (65) is obaie by subsiuig (7),(73) i equaio (9): l= i= U(, x) = e W (x) Gp(, X ) = e W (x) exp( exp()) si (x +.B()) (74) Fig.5 show his soluio hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [79]

13 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 Fig.5 soluio of SPDE (65 ) A he compare his soluio by our meho wih he soluio obaie by Saul'yev meho a wih is correspoig eermiisic soluio. This compariso show i Fig. 6 below : Fig.6 compariso of soluios of SPDE (65) The comparisos of errors of hese soluios was show i able (4.). Table (4.) Comparisos of he errors. x saul'yev-gp iermeisic-gp CONCLUSIONS Applicaio of a ew echique for solvig sochasic parial iffereial equaios. Such as applie of accelerae geeic algorihm (AGA) o fi he umerical soluios of sochasic parial iffereial equaios wih aiive a muliplicaive cylirical Browia moio ( or space-ime whie oise ), usig Doss-Susma rasformaio, o rasform hese equaio io parial iffereial equaios a sochasic oriary iffereial equaio, he applie he AGA o fi he umerical soluios of rasforme equaios a he he soluio of origial equaios. We hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [8]

14 [Hussai* e al., 5(8): Augus, 6] ISSN: IC Value: 3. Impac Facor: 4.6 oe ha his meho has geeral uiliy for applicaios, a we fou ha iserio of bouary coiio as a chromosomes i he populaio quick he algorihm o approximae he umerical soluios. I orer o compare he resuls ha have bee obaie by usig accelerae geeic algorihm, valiaig, i has compariso wih some umerical mehos (such as fiie ifferece meho a he saul'yev meho), where hese mehos are use o solve his ki of sochasic parial iffereial equaios a i's always covergece. I urs ou ha he resuls ha have bee obaie by usig accelerae geeic algorihm are goo resuls a covergece wih hese mehos. The mai problem ha we face urig he applicaio of he (AGA) o fi umerical soluios of sochasic iffereial equaios, are oise-geeraig process, such as (Browia moio or cylirical Browia moio ). Where he values of he oise mus be ormally isribue wih zero mea a variace equal o i.e. N(, ). To achieve his value of mus be very small chage so ha we ge he larges umber of values wihi he specifie ierval, hese issues ha affec o he shape a isribuio of he oise a shows is ifluece is clear i he fial soluios. REFERENCES [] Jeze, P. E. Kloee, "Taylor Approximaios for Sochasic Parial Differeial Equaios", by he Sociey for Iusrial a Applie Mahemaics, (). [] S. Loosky, R. Mikulevicius & B. L. Rozovskii, "Noliear filerig revisie: a specral approach", SIAM J. Corol Opim. 35 (997), o., p [3] S. V. Loosky,"Wieer chaos a oliear filerig", Appl. Mah. Opim. 54 (6) [4] J.B. Walsh,"A iroucio o sochasic parial iffereial equaios", Lecure Noes i mahemaics, Volume 8, (986), pp [5] B. Saussereau, "A ew umerical scheme for sochasic parial iffereial equaios wih muliplicaive oise"', bruo.saussereau@uiv-fcome.fr December 9, (). [6] M. Dozzi & J. A. Lopez-Mimbela,"Fiie-ime blowup a exisece of global posiive soluios of a semiliear SPDE", Sochasic Process. Appl. (), o. 6, p [7] E. Paroux & S. Peg,"Backwar sochasic iffereial equaios a quasiliear parabolic parial iffereial equaios, i Sochasic parial iffereial equaios a heir applicaios",(charloe, NC, 99), Lecure Noes i Corol a Iform. Sci., vol. 76, Spriger, Berli, 99, p. 7. [8] H. J. Sussma," O he gap bewee eermiisic a sochasic oriary iffereial equaios", A. Probabiliy 6 (978), o., p [9] E.A. Hussai a Y.M. Alrajhi, "Soluio of parial iffereial equaios usig accelerae geeic Algorihm", I.J. of Mahemaics a Saisics Suies, Vol., No., pp , March 4. [] D.E. Golberg, "Geeic algorihms i search, Opimizaio a Machie Learig", Aiso Wesley, 989. [] G. Tsoulos. I. E, "Solvig iffereial equaios wih geeic programmig", P.O. Box 86, Ioaia 45, 3 [] P. Naur, Revise repor o he algorihmic laguage ALGOL, 963. [3] A. R. SOHEILI, M. B. Niasar a M. Arezoomaa, "Approximaio of sochasic parabolic iffereial equaios wih wo iffere fiie ifferece schemes", Special Issue of he Bullei of he Iraia Mahemaical Sociey Vol. 37 No. Par (), pp 6-83 hp: // Ieraioal Joural of Egieerig Scieces & Research Techology [8]