Construction of mixed estimation for solving the first edge problem for the equation of non-isotropic diffusion

Transcription

1 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 Costuctio of mixed estimatio fo sovig the fist edge pobem fo the equatio of o-isotopic diffusio okhi ozhiev, Kuvvatai Rahimov, Jasubek Soioov PhD, Depatmet of Ifomatio echoogy, Fegaa State Uivesity, Fegaa, Uzbekista PhD studet, Depatmet of Ifomatio echoogy, ashket Uivesity of Ifomatio echoogies, ashket, Uzbekista Studet, Fegaa State Uivesity, Fegaa, Uzbekista ABSRAC: Usig method of Mote Cao is costaty expadig i vaious fieds as we as the deveopmet of computig techoogy. Iceasig the speed ad stoage capacity of mode computig systems wi aow us to sove vaious pobems usig the statistica modeig method. I this atice, usig the theoy of matigaes ad Makov momets, a agoithm is costucted fo sovig iitia-bouday vaue pobems fo the geeaized o-isotopic diffusio equatio. KEY WORDS: Mote Cao method, estimatio, diffusio, matigae, ubiasedess, dispesio. I. INRODUCION o sove may cassica pobems of mathematica physics, a umbe of pobabiistic epesetatios ae kow. Howeve, they do ot aways diecty ead to a simpe Mote Cao agoithm fo sovig the pobem. I a cetai sese, it ca be cosideed that iea taspot pobems ae a exceptio to the geea ue that may pobems of mathematica physics obey, ad i paticua, bouday vaue pobems fo eiptic paaboic equatios. he pesece of a deep coectio betwee diffeetia equatios ad adom pocesses equies a compehesive study of it ad opes up the pospect of ceatig ew efficiet umeica methods fo sovig pactica pobems. Such a coectio has bee kow fo a og time ad at fist a we-deveoped theoy of diffeetia equatios was widey used i pobabiity theoy. It shoud be oted that the Mote Cao method was used maiy fo sovig statioay pobems of mathematica physics. hee is a sma umbe of woks devoted to the deveopmet of the Mote Cao method fo sovig ostatioay pobems. Amog them ae the woks [], [6]. I ou atice, a agoithm fo sovig bouday vaue pobems fo the geeaized oisotopic diffusio equatio is costucted usig a shaoid agoithm based o the mea vaue fomuas obtaied i [6]. Coside i dimesioa space u y, z, t Lu y, z, t f y, z, t t i m,, i k, m, II FORMULAION OF HE PROBLEM y, z, t, yr, z R, tr, k equatio R poits k with diffeetia opeato with costat coefficiets,, u y, z, t u y z t Lu y, z, t y. k k i m i m i y y m z Copyight to IJARSE

2 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 i m Let k k be matix, k - matix, D R Combie spatia vaiabes yz, i oe spatia vaiabe - coum,. is imited domai with bode D x y z R. Coside i space vaiabes x, t x, x,..., x, t cyide D* 0,. We wi coside the foowig iitia-bouday pobem. Fo fuctios x, t C D* 0, x CD, fid fuctio ux, t CD* 0,, C D* 0, R ad satisfyig i the cyide to the equatio u( y, z, t) Lu y, z, t f y, z, t, x, t () t magia coditios u x, t x, t, xd, t [0, ) () ad iitia coditios u 0, t x, t, x D (3) Futhe, fo coveiece, we itoduce matices of size * (i a bock ecod), which have the foowig fom m m Ik Ik 0 a, m m C exp, d 3 3 I 0 0 I bock m size to k k, bock 3 m - size to k, m - size to k, m3 3 - size to, I, o,, I - simia sizes. k he matix a is symmetic ad positive defied ad theefoe it ca be epeseted as a poduct of some matix o its tasposed b, videicet a b b, d p - diagoa matix,. III. REPRESENAION OF SOLUIONS he Z x, t; y, - fudameta soutio of equatio () with poit siguaity o the poit, [] Z x, t; y, a t exp y Cx d ad y cx t t k. B,, :, ;, y Z y a, t,, B x, t whee a detemiat of the matix a, 3 We itoduce a domai depedig o the paamete 0 y has the fom which wi be caed the shaoid of adius with the cete at the poit xt ad its bode spheoid. Fo 0 B x, t ad B x, t mootoousy tighteed to xt,. heefoe thee exists which B x, t fo xt,. We peset oe of the ways to seect paamete 0., 0, Copyight to IJARSE

3 Let ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Rx be the distace fom o the boud of D, 3 ma, t, Vo. 5, Issue, Decembe 08 x max x, - the agest eigevaue of the matix a, i k i max, V, i i, 8 e i V V 4R x V V x, x. V Lemma. If x, t mi{ ( x), t } at x, t the B, Evidece: Fom y x R () x it foows that B ( x, t).. y x y Cx Cx x R() x (4) Fom the defiitio of B ( x, t ) ad usig some mathematica tasfomatios fo the fist tem of (4) we get Sice Let dt ( ). 4 ( ) ( t ) g t Fom the iequaity d( t )( y Cx) a d( t )( y Cx) d( t )( y Cx) Fo the secod tem (4) we have Fiay fo the iequaity (4), we get Sovig the iequaity d( t ) y Cx y Cx. 8 t, the e 4 max gt ( ) fo t e. 4 8 e y Cx we get 8 8 e y Cx o k k x Cx x x o v v R x, we obtai 4 ( ) y x v v R x 4 ( ). v v 4 R( x) v v t ad t it foows that t. So fo Fom 0 B ( x, t). he emma has bee poved. x, t 0 B x, t. Let be such that 4 e y Cx. 8 x Cx x k ( x). mi{ ( x), t}. (5) he, usig the fomuas of the paaboic aveage [], fo sovig the pobem () - (3) we get the foowig epesetatio. Copyight to IJARSE

4 whee ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 0 S (0) u( x, t) P( ) P H u( y( e ), ( e )) dsd f ( x, t), (6) f ( x, t) (, ;, ) Z y a f ( y, ) dyd B x, t - dimesioa uit sphee, H S 0 - uit - dimesioa vecto, P S 0 - is the gamma desity of a distibuted adom vaiabe with the paamete, ( ) H b bh P H - is the desity of a t, y, H e x ( ) d b H adom vecto, Let be whee (0)( ) S H IV. CONSRUCING MARKOV CHAINS P ( H ) P ( H, H,..., H ) ( q H H ) ( H ) i i S (0) i, suface of the uit sphee. is the idicato of the set S (0), is the suface of the uit sphee, q 4 b b. Sice H, H,..., H ae the coodiates of the uit vecto ad H H... H we get H qh, whee is the agest eigevaue of the matix q. he oe ca simuate a adom vecto with distibutio desity (6) by method of Neuma. We wi peset a agoithm of modeig. Agoithm: ) Simuated (,,..., ) isotopic vecto ad is a uifomy distibuted adom vaiabe; ) E f ( x) dx ; 3) If i x ( qii ) / E, the is accepted, othewise it is epeated paagaph ). Let, be a sequece of idepedet gamma distibuted adom vaiabes with the paamete is a sequece of idepedet adom vectos with distibutio desity P( H ). x, t I we defie a Makov chai 0 usig the foowig ecuece eatios: 0 0 x x, t t, t t exp, i i imi m x x exp b, Copyight to IJARSE

5 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 k 3 3 k p k p pc c k p c x x exp( ) x exp( ) b, We defie a sequece of adom vaiabes 0 by the foowig equaity h( x, t ) f ( y, ) u( x, ) (7) whee ( y, ) is a adom poit of the B ( x, t ) which has distibutio desity fo the fixed ( x, t ). Let 0 be a sequece of -agebas geeated by adom vaiabes,,...,, sequece of vectos 0 0 0,,..., ad adom poits y y y coespodig to the give f,,, heoem. a). Sequece 0 b) If,0,0 (, ) f,,,,...,,, u [ Z( x, t ; y, ) a ], h( x, t ) B ( x, t) f,,, h( x, t) [ Z( x, t; y, ) a ] dyd u ad u ( x, t ), the,0,0 Poof. Fisty, we pove that 0 f i - soutio of pobem (3)-(5), foms a matigae with espect to sequeces of -ageba 0 wi be quadatic itegabe. foms a matigae. Fom the defiitio of. it is obvious that measuabe. E( x, t) ( ) E( x, t) ( h( x, t ) f ( t, ) u( x, t )) 0 E( xt, ) ( h( x, t ) f ( t, ) h( x, t ) f ( t, ) u( x, t )) 0 E( xt, ) h( x, t ) f ( t, ) E( x, t) h( x, t ) f ( t, ) E( x, t) ( u( x, t ) ). 0 Sice h( x, t ) f ( t, ) is 0 is is measuabe, the fom the popeties of the coditioa mathematica expectatio it foows that E( xt, ) ( h( x, t ) f ( y, )) h( x, t ) f ( y, ) 0 0 ( h x, t ) - is measued ad f ( x, t ) does ot deped o ad E( x, t) h( x, t ) f ( y, ) h( x, t ) E( x, t) f ( y, ), Copyight to IJARSE

6 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 u x (, t ) does ot deped o, it meas E ( u( x, t ) ) E u( x, t ) ( x, t) ( x, t) E h f y h E f y E u ad fiay we get ( x, t) ( ) (, ) (, ) (, ) ( x, t) (, ) ( x, t) (, ). 0 Usig the fomuas (6) ad (7) we get E h f y u ( x, t) ( ) (, ) (, ) (, ). 0 It foows that 0 foms a matigae with espect to. We wi pove that E ( x, t). o do this it is eough to show that I E( xt, ) h( x, t ) f ( y, ). 0 Sepaatig I ito two tems I, I, we wi show the fiiteess of I : k k k k I E( xt, ) h ( x, t ) f ( y, ) E( xt, ) h( x, t ) h( x, t ) f ( y, ) f ( y, ) 0 0 k Fom (7) ad the coditio (x, t) t we obtai that (, ) ( ) h t ( x, t) ( x, t) 0 0 I E h ( x, t ) f ( y, ) ( ) te h( x, t ) f ( y, ) tu f,0,0 ( ) ( x, t). k k k k ( xt, ) (, ) (, ) (, ) (, ) f,0,0 f,0,0 xd, t 0 k I E h h f y f y he theoem has bee poved. ( max u ( x, t)) u ( x, t). Lemma. Fo the fuctio f x, t the foowig eatio is vaid whee f x, t E f y,,,,, y,, e exp x d exp b,, t exp. (8) Copyight to IJARSE

7 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 Hee is gamma distibuted adom vaiabe with the paamete, is beta distibuted adom vaiabe with the paamete (,) ad is adom uit vecto. Poof. We itoduce a domai B {( y, t) : y d( ) ad( ) y, 0} the esutig mio image of the B (0,0) shaoid with espect to the pae 0. hese egios wi aso be caed shaoids(with adius ). he we have a f ( x, t) [ exp( y d( ) ad( ) y) ) f ( e x y, t ) dyd. B I this itega we wi chage itegatio vaiabes ( y, ) by (,, ) usig the fomua ( ( )) ( ) ( ),. y d d H By simpe computatios, it ca be show that dyd a (( )) ddds exp( y d( ) ad( ) y) he domai of itegatio is tasfomed ito the cyide (0 ) (0 ) S(0). Fom hee we get 0 0 f ( x, t) ( ) d ( ) d f ( e x ( ) d( ) b H( ), t ) ds S (0) usig chagig vaiabes we get: z 0 0 S (0) f ( x, t) ( ) v ( v) dv e z dz f ( y( z, v, H( )), ( z, v)) ds S (0) ( ) P( v) dv P ( z) dz P ( H) f ( y( z, v, H ( )), ( z, v)) ds ( ) Ef ( y(,, ), (, )). Copyight to IJARSE 766

8 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 Whee is a adom vaiabe with distibutio desity is a adom vaiabe with a distibutio desity P( H( )) 3 ( ( ) Let N if : ( x, t ) ( ) v Pv () B ( v) (,) - is the uit suface sphees). he emma is poved. - the momet of the fist hit of the pocess (, ) momet of stoppig the pocess (Makov momet). t Lemma 3. he foowig iequaity hods: E, N xt. Poof. akig u( x, t) t ad appyig the fomuas of eatios (6) ad (7) we get Fom the defiitio of ( x, t ) it foows that N N,0,0 ( x, t) ( x, t) z e z P () z (gamma - distibutios), ( ) t u ( x, t) E h( x, t ) ( ) E ( x, t ). ( 4 ( )), - adom vecto with distibutio desity. At,i.e. N the v v v R x ( x, t ) mi{ ( x ), t } mi{ v ( v 4 v ) } mi{, } ( ). v v heefoe E( xt, ) N t. ( ) he emma has bee competed. heoem. Let the coditios of heoem ae fufied. he wi be a ubiased estimate fo, N u. he vaiace is fiite. Poof. Fom the heoem it foows that is squae itegabe ad hece is uifomy itegabe ad N, ad the momet stoppig the pocess is Makov momets. heefoe, accodig to Doob's theoem "O fee choice tasfomatios" [5] ad the fomua eatios u ( x, t ) E u ( x, t ) f ( x, t ) ad D E u N (, ) i.e N ad you ca see that D N x, t is a ubiased estimate fo u( x, t ). Fom the defiitio of adom vaiabes D. ( ( i, i ) ( i, i ) (, )) ( ( i, i ) ( i, i h f y u h f y )) i0 i0 (, ) ( i, i ) ( i, i u h f y ) u ( x, t ) h ( x i, t i ) f ( y i, i ) i0 i i0 i0 ( i, i ) ( i, i ) (, ) (, h f y h f y ) (, ) ( i, i ) ( i, i u h f y ) u ( x, t ). i0 N Copyight to IJARSE 766

9 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 ( xt, ) f,0,0 x D f,0,0 f,0,0, t E ( ) tu ( x, t) ( max u ( x, ) u ( x, t) u ( x, t) u ( x, t) u ( x, t). 0,, f,0,0 0,, ( ) (, ) i.e. D N. D E E E u Fom * N, the biased but actuay eaizabe estimate is costucted i the stadad way N. Let x patia D 0,, x x,, bouday poits of. I the assessmet N N 0,0 N N * * epace u( x, t ) with (, ) ad we get N N, x D * * ad, N N h( x, t ) f ( x, ) u( x, t ), N N N * * * N h f y 0 (, ) (, ) (, ). heoem 3. Let u( x, y ) satisfies the coditio Lipschitz ad adom vaiabe is a biased estimate fo, * N Poof. Sice E( x, t) u( x, t), N e u. * * ( x, t) N ( x, t) Ne ( x, t) N * N is the cosest to xt, A is moduus of cotiuity of, D bouded fuctio of paamete. * * (, ) (, ) (, ) ( N N N N, ) u( x, t) E E E E u E * * (, ) (, ) ( N N N N xt, ) ( ), E u u A that is, is mixed estimate (mixig of which does ot exceed A( ) ). * N D E ( E ) E ( u( x, t) u( x, t) E ) he theoem is poved. * * * * * N ( x, t) N ( x, t) N ( x, t) N Ne Ne ( x, t) N * * 4 E( x, t) ( N u( x, t)) 4 E(, )( N ) 4 (, )( (, ) (, ) N ) e N E e u E * * 4 (, )( ( N N N N E x, t ) u( x, t )) 4 A ( ) 8 A ( ) 4 D N. e V. CONCLUSION u. he the I the peset wok, a ubiased ad biased estimate fo iitia ad bouday-vaue pobems fo the geeaized o-isotopic diffusio equatio with usig the theoy of matigaes ad Makov momets. REFERENCES [] Komogooff A.N., U be die aaytische Methode i de Wahscheiichkeitsechug, Mathematische Aae, 93, Decembe, Voume 04, Issue, pp [] Yemakov S.M., Netutki V.V., Sipi A.A., Radom pocesses fo sovig cassica equatios of mathematica physics, 984, Moscow: Sciece, 06 p. [3] Kuptsov L. P., O the popety of the mea fo a geeaized equatio A. N. Komogoova I, Diffeetia equatios, 983, Vo. XIX, N'. with [4] Sobo I.M., Mote-Cao Numeica Methods, 973, Moscow: Mai editos of physica ad mathematica iteatue pubishig house, "Sciece". [5] Doob J.L., Cassica Potetia heoy ad its, 984, Pobabiistic Coutepat. Spige - Vaag. 846 p. Copyight to IJARSE

10 ISSN: Iteatioa Joua of AdvacedReseach i Sciece, Egieeig ad echoogy Vo. 5, Issue, Decembe 08 [6] Khadzhi-Sheikh A., Spaow EM., Pobem Sovig thema coductivity pobabiistic methods, 96, Ameica woks societies of mechaica egiees, v., p. -8. Copyight to IJARSE