Finite Difference Approximations for Fractional Reaction-Diffusion Equations and the Application In PM2.5

Transcription

1 Iteratoal Symposum o Eergy Scece ad Chemcal Egeerg (ISESCE 5) Fte Dfferece Appromatos for Fractoal Reacto-Dffuso Equatos ad the Applcato I PM5 Chagpg Xe, a, Lag L,b, Zhogzha Huag,c, Jya L,d, PegLag L,e Shaome Fag,f, * College of Mathematcs ad Iformatcs, South Cha Agrcultural Uversty, Guagzhou 564, Cha a @qqcom, bllag54@63com, c @qqcom, dduduwog@6com, e @qqcom, fdz9@scaueduc *Correspodg author Keywords: PM5; fractoal reacto-dffuso equatos; Crak-Ncolso method; umercal smulato Abstract I ths paper, fractoal reacto-dffuso equatos are used to model the dffuso of PM5 the ar Frst, based o the shfted Grüwald formula, we propose the fractoal Crak-Ncolso method to solve the fractoal reacto-dffuso equatos The we prove the estece ad uqueess of umercal solutos, ad establsh the stablty ad covergece of the method Furthermore, umercal eamples are also provded to show the effcecy of the method Fally, the dffuso of PM5 Guagzhou s smulated by usg ths method uder approprate parameters Itroducto Wth the rapd developmet of ecoomy ad costat mprovemet of huma's requremets of lvg evromet, the ar polluto, especally PM5, has draw much atteto Cha The so called PM5, partcles of aerodyamc dameter less tha 5 mcrometers, ca lead to hazy weather ad cause health problems [] Therefore, t s mportat ad ecessary to predct precsely the PM5 cocetratos for the cotrol of ar polluto ad the mprovemet of the huma lvg codtos However, t s ot adequate to model ad predct the PM5 cocetratos usg tradtoal methods whe the fre or eploso takes place somewhere For ths case, t s crucal to fd a sutable method or model to predct the PM5 cocetratos Fractoal reacto-dffuso equatos ca be regarded as the geeralzato of classcal reacto-dffuso equatos [] ad have bee successfully appled to model problems physcs [3], bology [4], face [5] I ths paper, we wll use the followg fractoal reacto-dffuso equato to descrbe the dffuso of PM5: u (, t ) u (, t ) u(, t ) b( ) a( ) c( )u(, t ) f (, t ), t subect to the tal ad Drchlet boudary codtos gve by u (, ) q ( ), u (t, L) r (t ), u (t, R ) s (t ), L R, t, L R, t () () (3) where the dffuso coeffcet a( ), wd speed b( ), ad decay coeffcet c( ) are cotuous, f (, t ) s a cotuous fucto o L, R, T whch represets sources ad sks u(, t ) s the pollutat cocetrato u (, t ) Here ( ) s a Rema fractoal dervatve of order, whch s defed by[6] u (, t ) u (, t ) d (4) ( ) L ( ) 5 The authors - Publshed by Atlats Press 393

2 For ths equato, may researchers have studed the estece ad uqueess of the soluto ad acqured good results [6-7] Whle to establsh the umercal soluto for ths equato s also attractve There are two ways to dscretze the fractoal dervatves Oe s shfted Grüwald formula [8], the other s fte dfferece scheme [9] Ths paper takes the frst way to appromate the fractoal space dervatves, ad costruct the Crak-Ncolso scheme Below we prove the estece of the umercal soluto ad the we aalyze the stablty ad covergece of the scheme A eample wth kow eact soluto s also preseted to test the effcecy of the scheme Fally, we smulate the dffuso of PM5 Guagzhou by choosg approprate parameters The Crak-Ncolso scheme for the fractoal reacto-dffuso equato I ths secto, we propose the Crak-Ncolso umercal appromato scheme based o the shfted Grüwald formula [-], whch s defed as M ut (, ) lm gu k ( ( k ) lt, ), l M k L ( k ) where M s a postve teger, l ad gk, k,,3, M ( ) ( k ) We defe h to be the grd sze spatal drecto wth R L K, L h,,,, K ; h T ad to be the grd sze tme drecto, wth N, t,,,, N / Deote a a( ), b b( ), c c( ), f f(, t /), Du ( u u )/ h, u g u, s the umercal soluto of Eqs ()-(3), we have the followg scheme: u k k h k ( b ) ( a ) ( c ) ( ) / u u Du Du u u u u f, K, N (5) u q( ), K (6) (7) u r( t), uk s( t), N Therefore, Eqs (5)-(7) ca be wrtte as the followg matr form: / ( ) I AU ( I AU ) F, (8) where / f ( EBg )( rt ( ) rt ( )) c / 3 c f B g (( r t ) r( t)) C, F, / fk BKgK (( r t ) r( t)) c K / fk BK gk( r( t ) r( t)) BK g( s( t ) s( t)) ad matr A ( A, ) ( K) ( K) s defed as follows: gb A, E gb C (9) E gb g B 394

3 wth B a h, E b C h, c Stablty ad covergece Theorem The Crak-Ncolso scheme defed by Eqs (5)-(7) has a uque soluto ad s ucodtoal stable for all Proof Let be a egevalue of the matr A, ad AX X for some ozero vector X Choose such that K ma :,,, The Substtutg the values of A, to () we have K A, K,,,, ad therefore A A () E gbcgb ( EgB) g B E( ) B( g g ) C, The by [], we ca see that gk g(,,, ) Therefore k, k g g g g,,, Sce E, B, C are o-egatve, t follows that E( ) B( g g ) C, The we have,, whch meas that every egevalue of I A s o less tha Therefore, the spectral radus of ( I A) satsfes (( I A) ) Thus we prove that the Eqs (5)-(7) has a uque soluto To prove the ucodtoal stablty of the scheme, we assume to be the error U, the t ( I A) ( I A) wll result a error U gve by We deote by a egevalue of ( I A) ( I A), the we have Thus So the method s ucodtoal stable Theorem The scheme defed by Eqs (5)-(7) wth s ucodtoal coverget Proof I lght of [], we ca see the local trucato error of Eqs (5)-(7) s Oh ( ), therefore the scheme s cosstet The accordg to Theorem ad La equvalece theorem [], we ca show the method s coverget ad the order of covergece s Oh ( ) A umercal eample The followg fractoal reacto-dffuso equato 5 ut (, ) ut (, ) ut (, ) b ( ) a ( ) cut ( ) (, ) f( t, ),, t, 5 t was cosdered wth the tal ad boudary codto t u (,), ; u(, t), u(, t) e, t Here we take 5 3/ b ( ), a ( ) ( ), c ( ), f ( t, ) (3 ) e t

4 The the eact soluto to ths fractoal equato s gve by ut (, ) e t Fg shows the umercal soluto at tme t obtaed from the Crak-Ncolso method Eq (8) dscussed above wth h /4, / As ca be see from Fg, ths umercal soluto compares well wth the eact aalytc soluto Table shows that as the umber of spatal steps s quadrupled ad tme steps s doubled, the mamum error s quadrupled, as epected from the order Oh ( ) of the covergece of the scheme Fg Comparg the umercal soluto wth the eact soluto ( h /4, /) Table Mamum error behavor for the eample problem t (, h) (/,/8) (, h) (/ 4,/ 3) Error rate 5 657e 7356e e 647e e 537e e 39448e Numercal smulato for the dffuso of PM5 Below we wll gve a umercal smulato for the dffuso of PM5 Guagzhou by choosg proper parameters Here we choose the dffuso parameter of SO to replace the dffuso parameter of PM5 e 5 a ( ) m / s, as the dffuso parameter of PM5 s hard to determe ad SO s oe of the most mportat chemcal compostos of PM5 Guagzhou[3-4] For the wd velocty, from Cha meteorologcal data sharg servce system, we kow the average speed of wd Guagzhou, wth scale -3, s less tha 54 m/ s Wthout loss of geeralty, we take the drecto of wd as the -as, ad the speed of wd s 5 m/ s, e b ( ) 5 m/ s Fally, we determe decay rate of PM5 It ca be appromate to due to the fact that PM5 s dffcult to dsappear automatcally wthout cotrol e c ( ) I ths paper, we take the fractoal order equal to 5 e 5 The we have 5 u u 5 u 5 f(, t),, t, 5 t wth tal boudary codtos t t u (,) e,, ; u(, t) e, u(, t) e, t, ad the source/sk fucto 9 t f(, t) e Here we choose h, Fg ad Fg 3 preset the soluto surface ad the cotour les of PM5 cocetrato respectvely As ca be see clearly from Fg, PM5 cocetratos decrease wth the crease ad crease wth the crease t Ths dcates that the speed of pollutats released s more tha the speed of dffuso O the other had, from Fg 3, we ca see the 396

5 cotour s gettg more ad more sparse wth the crease of for fed t, whch dcates the hgher the cocetratos of PM5, the faster of the cocetratos decreased ad the smaller rage wth hgh PM5 cocetratos Moreover, Fg 3 also shows the cotour s gettg more ad more tesve wth the crease t for fed, whch meas PM5 cocetratos crease more rapdly wth the crease t Fg The dstrbuto of PM5 Fg 3 The cotour les of PM5 cocetratos Ackowledgemets Ths work was facally supported by the Guagdog specal fuds for cultvato of techologcal ovato by uversty studets (pdh5b94) ad the NSFC (74) Refereces [] N Martell, O Olver ad D Grell: Eur J Iter Med, 95 3 (3), p 4 [] R Metzler ad J Klafter: Phys Rep 77 (), p 339 () [3] Y Rosslh ad M Shtkova: Appled Mechacs Revews 5-67 (997) p 5 [4] SB Yuste ad K Ldeberg: Phys Rev Lett8-3 (), p 87 () [5] E Scalas, R Goreflo ad F Maard: Statstcal Mechacs ad ts Applcatos (), p 84() [6] I Podluby: Fractoal Dfferetal Equatos (Academc Press, New York 999) [7] DF Maar ad Y Luchko: Fractoal Calculus ad Appled Aalyss53-9 (), p 4() [8] Jghua Che ad Fawag Lu: Joural of Xame Uversty (Natural Scece) (I Chese) (6), p 45 (4) [9] MM Meerschaert, HP Scheffler ad C Tadera: J Comput Phys49 6(6), p () [] M M Meerschaert ad C Tadera: Joural of Computatoal ad Appled Mathematcs65-77(3), p 7() [] C Tadera, MM Meerschaert ad HP Scheffler: J Comput Phys 5-3(6), p 3() [] RD Rchtmyer, KM Morto: Dfferece Methods for Ital Value Problemsd ed (Joh Wley & Sos, New York 98) [3] CY Che, et al: Evrometal Motorg Cha 6-64(6), p 5 [4] ZL Cheg, KS Lam, LY Cha, T Waga d KK Cheg: Atmospherc Evromet (), p