Riesz ( ) Vol. 47 No u( x, t) 5 x u ( x, t) + b. 5 x u ( x, t), 5 x = R D DASSL. , Riesz. , Riemann2Liouville ( R2L ) = a

Transcript

1 47 () Vo. 47 No. 008 Joura of Xiame Uiversiy (Na ura Sciece) Ja. 008 Riesz, 3 (., ;.,,400, ) : Riesz. Iic,Liu, Riesz. Riesz.,., Riesz.. : Riesz ; ; ; ; :O 4. 8 :A : (008) ,, [ - 3 ].,. [4-5 ] ; [6 ].,, [6 ], [ ], [3 ].,. Iic,Liu [ ] Riesz.. Riesz [4-5 ],., (0 << ) ( < ) Riesz. Riesz Riesz, RiemaLiouvie ( RL ), Riesz [ 6-7 ]. Riesz, Iic,Liu [ ], [ ],Riesz : : (07098), (L P ) 3 edu. c, Riesz. DASSL., Liu FokkerPack [8 ], [ ].,Riesz. Riesz : 5 u( x, ) 5 5 = a 5 x u ( x, ) + b 5 x u ( x, ), 0 < x <, > 0 () u(0, ) = u(, ) = 0, > 0 () u( x,0) = g ( x),0 < x < (3) 5 5 x = R D x = - ( - ) 5 5 x = R D x = - ( - ) Riesz,0 < <, <, Riesz [6-7 ]. a b. u( x, ), g ( x). [6,9 ] ( a x b) Chia Academic Joura Eecroic Pubishig House. A righs reserved. hp://

2 :Riesz Riesz : 5 5 x u ( x, ) = R D x u ( x, ) = - c( ad x + x D b) u( x, ) (4) c = cos,, ad x u ( x, ) = ( - ) 5 5 x x a u (, ) d ( x - ) +, - = + (5) bu ( x, ) = ( - ) ( - ) x D 5 5 x b x u (, ) d (- x) +, - = + (6) [ ] ( - ),( - ) = ;( ) = 0 5,( ) Diriche. F= { f = c, c,= max,0 }, f ( - ) =f, c F, ( - ) f = < c (7) [ ] T = ( - )., f = a, g = Tf, g= b, ab =f, Tg (8) 3 [ ] ( - ). ( - ) f = ( - ) m f,= m, m = 0,,, ( - ) - m ( - ) m f, m - < < m, m =,, f,,< 0 [ ] 3. (9) [0 ] A ( N - ) ( N - ), P( N - ) ( N - ), A = PP - (0) = diag,,, N - A ( N - ) ( N - )., i i =,, N u = - a x u,0 < x <, > 0 () u(0, ) = u(, ) = 0, > 0 () ( x,0) = g ( x),0 < x < (3). d ui = - a h - ui+ + ui - ui-, i =,,, N - (4) u0 = un = 0 (5) u( x i,0) = g ( x i), i =,,, N - (6) ui = u( x i, ), h, N = h. : A = h = - aa U (7) ω ω ω A ( N - ),., A A = PP Riesz () : 5 u b - = - a x A = m x x u x u (8) 5 5 x, Riesz () (3), m x - = A -, m x () : - = A -, = - aa - AU - ba - AU (9) Chia Academic Joura Eecroic Pubishig House. A righs reserved. hp://

3 () 008, A ( ), (MM T). (9) : = - ap P - U - bp P - U (0) = diag (,,, N - ), = diag (,,, N - )., Riesz. DASSL [8 ]. DASSL k ( k =,,,5).,, k. [,8 ]. 3 Riesz - f, g= L f ( x) g ( x) d x, x u(0) = u( ) = 0 = (,, ), u ( x, ) = si x Riesz. () (3),: sgu - g = - a - b x 5 5 x gu ( x, s) - gu ( x, s) () gu (0, s) = gu (, s) = 0 () u : su, gu ( x, s) -u, g= - au, - bu, x (3) u, -, u, x. 5 5 x gu - gu (3) gu=u, x 5 5 x gu, x gu= x - u, - u ( x, ) = 5 5 x gu= - u, - gu si x u, - gu = [ u ( ) gu ( ) - u (0) gu (0) ] - guu d x = - guu 0 0 (4), d x (5) - guu d x = -u, gu=( - ) u, gu= 0 u, gu= u, gu (6) (6) (5), u, - gu = u, gu (7) (4) u, - u, x 5 5 x gu= u, gu (8) gu= u, gu (9) (8) (9) (3), su, gu ( x, s) -u, g= - a u, gu- b u, gu (30) u, gu ( x, s) = C (s),u, g= G, (30) sc (s) - G = - a C (s) - b C ( s) (3) (3) G C (s) = s + a + b gu ( x, s) = C ( s) u ( x) = G u ( x) = s + a + b G s + a+ b si x (3) (33), (33), : u( x, ) = 4 Ge - ( a +b ) si x (34) : =, g ( x) = x (- x). G =u, g= 0 x (- x) si ( x ) d x = Chia Academic Joura Eecroic Pubishig House. A righs reserved. hp://

4 :Riesz 3 si ( x ) 0 x d x -,, si ( x ) 0 x d x = ( - ) + si ( x ) 0 x3 d x = si ( x ) 0 x3 d x (35) + 3 [ ( - ) - ] (36) 3 ( - ) ( - ) (37) G = 3 [ ( - ) - ] ( - ) -, (38) (34),: u( x, ) = ( 3 - ) + e - a 3 e - a ( 3 - ) + e - 3 e - a +b 6 3 ( - ) = 3 (38) +b a +b +b si ( x ) = si ( x ) - si ( x ) - si ( x ) (39) = 0.,0. 4,0. 6,0. 8,. 0,,a = b = 0. 5,= 0. 4,=. 8. ( ) ( ) = 0., 0. 4,0. 6,0. 8,. 0 Fig. Compariso of umerica souio (symbos) ad aaysis souio (curves) a = 0.,0. 4,0. 6, 0. 8,. 0,respecivey., Riesz. 5 Riesz,... : [ ] Iic M,Liu F, Turer I,e a. Numerica approximaio of a fracioaispace diffusio equaio ( ) wih oho mogeeous boudary codiios[j ]. Fracioa Cacuus & Appied Aaysis,006,9 (4) : [ ] Samko S G, Kibas A A, Marichev O I. Fracioa ie gras ad derivaives : heory ad appicaios [ M ]. Am serdam : Gordo ad Breach,993. [3 ] Poduby I. Fracioa differeia equaios [ M ]. New York :Academic Press,999. [4 ] Gorefo R,Maiardi F,Morei D. Time fracioa diffu sio :a discree radom wak approach[j ]. Joura of No iear Dyamics,000,9 :9-43. [5 ] Huag F,Liu F. The fudamea souio of he space ime f racioa adveciodispersio equaio [ J ]. J App Mah & Compuig,005,8 (/ ) : [6 ]. [D ]. :,006. [7 ] Meerschaer M M,Scheffer H, Tadjera C. Fiie differ ece mehods for wodimesioa f racioa dispersio e quaio[j ]. J Comp Phys,006, :49-6. [8 ] Liu F,Ah V, Turer I. Numerica souio of he space f racioa FokkerPack equaio [ J ]. J Comp App Mahemaics,004,66 :09-9. [9 ] She S,Liu F,Ah V,e a. Deaied aaysis of a expic i coservaive differece approximaio for he ime f rac ioa diffusio equaio [ J ]. J App Mah Compuig, 006, (3) : - 9. [0 ] Liu F,Zhuag P,Ah V,e a. Sabiiy ad covergece of he differece mehods for he spaceime f racioa adveciodiff usio equaio[j ]. App Mah Comp,007, 9 : - 0. [ ] Liu Q,Liu F, Turer I,e a. Approximaio of he L vy Feer AdvecioDispersio process by radom wak ad fiie differece mehod[j ]. Phys Comp,007, : [ ] Roop J P. Compuaioa aspecs of FEM approximaio Chia Academic Joura Eecroic Pubishig House. A righs reserved. hp://

5 4 () 008 of f racioa advecio dispersio equaio o bouded domais i R [J ]. J Comp App Mah,006,93 () : [3 ] Yu Q,Liu F,Ah V,e a. Sovig iear ad oiear spaceime fracioa reaciodiffusio equaios by Adomia decomposiio mehod [ J ]. Ieraioa J for Numer Meh I Eg,007,DOI:0. 00/ NME. 65. [4 ] Zasavsky G M. Topoogica aspecs of he dyamics of fuids ad pasmas[ M ]. Kuwer :Dordrech,99. [5 ] Mezer R, Kafer J. The radom wak s guide o aom aous diff usio :a fracioa dyamics approach[j ]. Phys ics Repors,000,339 : [6 ] Gorefo R,Maiardi F. Radom wak modes for space f racioa diff usio processes [J ]. Fracioa Cacuus & Appied Aaysis,998, :67-9. [7 ] Gorefo R, Maiardi F, Morei D, e a. Discree ra dom wak modes for spaceime fracioa diff usio[j ]. Chemica Physics,00,84 :5-54. [8 ] Liu F,Ah V, Turer I. Numerica souio of he space f racioa FokkerPack equaio[j ]. J Comp ad App Mah,004,66 :09-9. [9 ] Gorefo R, Maiardi F. Approximaio of L vyfeer diffusio by radom wak [ J ]. J Aa App ( ZAA ), 999,8 :3-46. [0 ]. [ M ]. :, 000. A Compuaioay Eff icie Souio Mehod f or a Riesz Space Fracioa AdvecioDispersio Equaio SH EN Shuju,L IU Fawag 3 (. Schoo of Mahemaica Scieces,Xiame Uiversiy,Xiame 36005,Chia ;. Schoo of Mahemaica Scieces,Queesad Uiversiy of Techoogy,Qd. 400,Ausraia) Absrac : I his paper,a Riesz space f racioa adveciodispersio equaio ( RSFADE) is cosidered,which is derived f rom he kieics of chaoic dyamics. Foowig work by Iic ad Liu e a,a ew compuaioay efficie mehod for sovig he RSFADE o a bouded domai is proposed. The mehod is based o he marix represeaio of boh he Riesz space f racioa operaors. The ovey of his mehod is ha a sadard discreisaio of he operaor eads o a sysem of ordiary differeia equaios (ODEs) wih he marix raised he same f racioa power. The he ODEs is soved by a compuaioay efficie f racioa mehod of ies. Usig a specra represeaio of he f racioa derivaives ad he Lapace rasform,he aaysis souio of his equaio is aso de rived. Fiay,a umerica exampe is give o demosrae ha his umerica mehod is compuaioay efficie. Key words : Riesz space f racioa derivaive ;marix rasfer echique ;Lapace rasform ;adveciodispersio equaio ; mehod of ies Chia Academic Joura Eecroic Pubishig House. A righs reserved. hp://