A Quintic Spline Collocation Method for the Fractional Sub- Diffusion Equation with Variable Coefficients

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1 AMSE JOURALS-AMSE IIETA pbliation-07-series: Advanes A; Vol. ; ; pp 0-9 Sbmitted Jan. 07; Revised Mar, 07, Aepted April, 06 A Qinti Spline Colloation Metod for te Frational Sb- Diffsion Eqation wit Variable Coeffiients * Gao Li *Experimental enter of Liberal Arts, eijiang ormal University, eijing, Cina (939@qq.om) Abstrat For te time frational sb-diffsion eqation wit variable oeffiients, a qinti spline metod is presented, along te time diretion, te rersion formla obtained from te Lagrange interpolation fntions is sed, along te spae diretion, te qinti spline interpolation fntions, wi ave ig order aray wen being sed to approximate smoot fntions and teir,,3 order derivatives, are sed as te basis fntions. Teoretial analyses and nmerial examples sow tat order aray in spae an be aieved for tis seme. Key words Sb-diffsion eqation, qinti spline olloation metod, frational. Introdtion Wen stdying diffsion penomena in igly inomogeneos media, ones often find tat traditional integer-order diffsion models always lead to eavy tail penomena. By omparison, orresponding frational models beave better wen desribing probability density. Terefore, in reent years, te frational diffsion eqation wit varios bondary onditions and nonlinear terms as beome more and more important in many fields, s as biology, mediine, flid meanis, termodynamis and eletroemial reation. More and More researers are fosing on te nmerial soltion and simlation of frational models sbjet to varios 0

2 onditions. And many exellent resear aievements ave been presented, s as te finite differene metod, te finite element metod, te spetral metod, te finite volme metod and so on. However, in tese existing literatres we an find most of te nmerial metods are effiient only for frational eqations wit onstant oeffiients. Terefore, te nmerial soltion of varios frational diffsion eqations wit variable oeffiients remains an important area to be stdied. In tis paper, sing te qinti spline interpolation fntions, wi ave ig order aray wen being sed to approximate smoot fntions and teir derivatives, we stdy te nmerial soltion of te following frational diffsion eqations wit variable oeffiients D0, t k( x, t) f ( x, t),( x, t) I (0, T]; x ( x,0) ( x), x I [ a, b]; ( x, t) ( a, t) xa ( t), t (0, T]; x ( x, t) ( b, t) xb ( t), t (0, T], x () were 0, k( x, t) 0 are diffsion oeffiient fntion, ( x), ( t), ( t), f ( x, t) are given smoot fntions. D0, t is te Capto derivative of te form t D0, t ( t s) ds ( ) () 0 s Wit () te Gamma fntion. For Eq.(), we se te qinti spline interpolation fntions as basis fntion in spae, present a olloation metod ombining wit te L rersion formla in time, analyze te teoretial aray, and illstrate te effiieny by some nmerial examples.. Preparation Define respetively { } xi i 0 and {} t ti i t 0 as niform partitions of te interval [ ab, ] and [0, T ] wit b a T xi a i, i 0,,,,, t j j, j 0,,, t,. t Let S [ a, b] { v : vc [0,], v P,0 i } be te spae of qinti spline fntions, i were P is te set of polynomials of degree, andi [ xi, xi ].

3 For onveniene of sing qinti spline fntion as basis fntion in spae, several axiliary nodes xi a i( i,,,,, ) are added to te meses, wi leads to a new interval I [ a, b ]. By te reslts in, we an immediately obtain te expression of qinti spline fntion and some arateristis as follows: For i,,,,,te qinti spline fntions are defined as ( x xi 3 ), x [ xi3, xi]; ( x xi 3 ) 6( x xi ), x [ xi, xi ]; ( x xi 3 ) 6( x xi ) ( x xi ), x [ xi, xi ]; Bi ( x) 0 ( x xi 3 ) 6( x xi ) ( x xi ), x [ xi, xi ]; ( x xi 3 ) 6( x xi ), x [ xi, x ]; i ( x xi 3 ), x [ xi, xi3]. (3) And for enog smoot fntion x ( ) ( x) B ( x) satisfying te following interpolation reslt. s i i i, tere is niqe qinti spline fntion Lemma Sppose( x) C [ a, b],and s( x) S is te qinti spline fntion defined above satisfying ( x ) ( x ), i 0,,,, '( x ) '( x ), ''( x ) ''( x ), i 0, s i i s i i s i i () ten, for all nodes x, i 0,,,, it olds i ( xi ) s ( xi ) ( i 6i 66 j 6 j j), '( xi ) s ( xi ) O( ) ( i 0i 0 i i) O( ), ''( xi ) s ''( xi ) O( ) ( i i 6i i i) O( ), 6 (3) (3) ( xi ) ( xi ) O( ) ( 3 i i i i) O( ). () In order to deal wit te Capto derivative operator (), denote by a l ( l ), l,,,, and let l an ( ank ank ) a, k ( k,, n), n, ( ) ( ) ( ) (6)

4 Ten, te se of piee-wise linear Lagrange interpolation fntion will reslt in te following rersion formla : Lemma Wen formla of te form n 0, t ttn k k n k C [0, T], for te Capto derivative operator (),tere is a rersion D ( t) ( t ) R( ( t )), n,,, (7) ere te remainder term R( ( tn )) satisfies R( ( tn )) O( ). (8) 3. Qinti spline olloation metod k k For onveniene, let D x. For te n t k x first eqation of (), tere is level (0 n t ),sbstitting (7) into te f ( x, t ) D ( x, t ) ( x, t ) ( t, x) R( ( t )). n n n x n n j j n k( x, tn ) k( x, tn ) j k( x, tn ) obtain n f ( x, tn ) Denote by r( x) j( t j, x), dropping te error term R( ( tn )), we k( x, t ) k( x, t ) n j n D ( x, t ) ( x, t ) r( x). (9) n x n n k( x, tn ) In (9), te first level ( x, t 0) an be obtained by allating te initial ondition ( x,0) ( x) in (). Let s( x, tn ) be te obtained approximation soltion. First, sbstitting all nodes x, x,, x into (9), and onsidering () we an get: r x (0) i i i i i n i i i i i 6 k( xi, tn ) 0 ( i) O( ) Seond, by te two bondary onditions in (), te se of first two eqations of () leads to 3

5 ( 0 0 ) ( ) ( tn ), ( 0 0 ) ( ) ( ). t n () For obtaining enog linear eqations, we take derivative wit respet to x in (9) and ave D ( x, t ) ( )' ( x, t ) D ( x, t ) r '( x). () 3 n n x n n x n k( x, tn ) k( x, tn ) Taking x a, x b, and dropping te error term O ( ),() reads: '( ), 3 a a r a a a r'( b), (3) n n were a n ( )' xa, a, b n ( )' xb, b. k( x, t ) k( a, t ) k( x, t ) k( b, t ) n n n n By (0)-(3), we an obtain te olloation eqations A QB U R () 0 ( ) Here A, Q, B are all ( ) ( ) -dimensional matries: A, Q n k( x, t ) 0 n n k( x, t ) n, 0

6 B, wit , 0, 3, 0,, , 0, 3, 0,, a a 6 a 0a 6 a 0a a a,, 3 a,,, b b 6 b 0b 6 b 0b b b,, 3 b,, In addition,te rigt term R and nknown U are respetively T T R [ ( t ), r '( a),6 r( x ),,6 r( x ), r '( b), ( t )], U [,,,, ]. n 0 n For te olloation system (), we ave Teorem Sppose model () as niqe soltion, wen approximation soltion in () satisfies te errors: ( x, t) C ([ a, b] [0, T]),te, xt, ( x, t) s( x, t) O( ). () PROOF Wen system () as niqe soltion, tere is Green fntion G( x, s) at every time level. In Level n ( n0,, t ), for te exat soltion ( x, tn ) and approximation soltion s( x, tn ),denote by Dx( x, tn ) ( x), D xs( x, tn ) ( x), ten tere is m m b (, ) b m G x s m G( x, s) Dx ( x, tn ) ( s) ds, Dx s( x, tn ) ( s) ds, m 0, a m a m x x. n For onveniene, let F( x, ) ( x, tn ) r( x), defining te following operator : k( x, t ) n D : S [ a, b] R, D ( g( x)) ( g( x ), g( x ), g( x )) 0 M R S a b, K : C[ a, b] C[ a, b], K( g( x)) F( x, G( x, s) g( s) ds). : [, ] T b a

7 Based on te above denotations, (9) an be written as D F( x, ) ( x) K( ( x)) ( I K)( ( x)) 0, (6) x () as niqe soltion, so operator I K is reversible. And te reversible is a bonded operator. Based on Lemma, it as trnation error O ( ) in te proess of (0)-(), ene, tere is ( I K)( ( x)) O( ). (7) Based on(6) and (7), we obtain ( I K)( ( x) ( x)) O( ), Again based on te bonded reversibility of Operator I K, tere is ( x) ( x) O( ), Terefore, we get te eqality b ( x, tn ) s( x, tn ) G( x, s)( ( x) ( x)) ds O( ). a By ombining wit te Lemma, we omplete te proof of te teorem.. merial examples In order to investigate te teoretial analysis reslts abot te effiieny of te presented qinti spline olloation metod, in tis setion, by sing te Matlab R00, some nmerial examples are provided. In all given reslts, Table lists te orresponding preision wit every spae step and fixed time step. We take te frational order as 0.0 to minimize te impats from te time diretion. Table and Table 3 list te orresponding preision wit every time step and fixed spae step. Here we take te frational order 0.0,0.,0.99 to ek te impat of te frational order on te preision. In all tables, te errors are allated in terms of infinite norm, we se of te onvergene rate is: E to represents te errors at all olloation nodes. Here te omptational formla Rate log( E ( / ) / E ( )) / log(). 3 x xt Example. ( x, t) t (os x e ), k( x, t) e, T, a 0, b,and 6

8 () f x t t x e t e x e ( ) 3 x 3 x xt (, ) (os ) ( os ). Example... ( x, t) t x, k( x, t) os( x t), T, a, b 0, and f x t (3.) (3. ) t x t x x t.... (, ).7 os( ). Table. Errors ( /0000, 0.0) along te spae diretion M Example Example E Rate CPUtime (seond) E Rate CPUtime (seond) 8.0e-7.3.8e e e e e e e Table. Errors ( / 6 ) along te time diretion in EXAMPLE E Rate E Rate E 00.33e-.3e-.73e-3 Rate 00.7e e e e e e e e e e e e Table 3. Errors ( / 6 ) along te time diretion in EXAMPLE E Rate E Rate E e-.873e- 6.83e-3 Rate e e e e e e e e e e e e Te experimental reslts from Table, Table and Table 3 sow tat, wen soltion fntion ( x, t) C ([ a, b] [0, T]), qinti spline olloation seme () aieves aray of, xt, O ( ) and O( ) along te time and spae diretion respetively. Tese nmerial reslts are in agreement wit te teoretial reslts of Teorem.. Conlsion 7

9 By ombing wit te L rersion formla in time and qinti spline fntions in spae, we ave introded a olloation metod for te te frational sb-diffsion eqation wit variable oeffiients. Teoretial analysis and nmerial experiments sow tis metod an aieve te preision O Referenes ( ) nder ertain smoot onditions wit te teoretial analysis.. R. Metzler, J. Klafter, Te random walk's gide to anomalos diffsion: A frational dynamis approa, Pys. Rep., 339 (000), C. Ingo, R. Magin, T. Parris, ew insigts into te frational order diffsion eqation sing entropy and krtosis, Entropy 6 (0) I. Podlbny, Frational Differential Eqations, Aademi Press, California, USA, S. Das, R. Kmar, Frational diffsion eqations in te presene of reation terms, J. Compt. Complex. Appl. () (0) -.. S.B. Yste, L. Aedo, K. Lindenberg, Reation front in an AB C reation-sbdiffsion proess, Pys. Rev. E, 69 (00) 0366: G.H. Gao, Z.Z. Sn, H.W. Zang. A new frational nmerial differentiation formla to approximate te Capto frational derivative and its appliations. J. Compt. Pys., 9()(0), G.-H. Gao, Z.-Z. Sn, A ompat finite differene seme for te frational sb-diffsion eqations, J. Compt. Pys., 30 (0), C.-C. Ji, Z.-Z. Sn, A ig-order ompat finite differene seme for te frational sbdiffsion eqation, J. Si. Compt., 6 (3) (0), B. Jin, R. Lazarov, Z. Zo, Error estimates for a semidisrete finite element metod for frational order paraboli eqations, SIAM J. mer. Anal., () (03), F. Zeng, C. Li, F. Li, I. Trner, Te se of finite differene/element approaes for solving te time-frational sbdiffsion eqation, SIAM J. Si. Compt., 3 (6) (03), A976- A F. Zeng, C. Li, F. Li, I. Trner, merial algoritms for time-frational sbdiffsion eqation wit seond-order aray, SIAM J. Si. Compt., 37 (0), A-A78.. F. Cen, Q. X, J.S. Hestaven, A mlti-domain spetral metod for time-frational differential eqations, J. Compt. Pys., 93 (03), M. Zeng, F.Li, I.Trner, V. An, A novel ig order spae-time spetral metod for te time frational fokker--plank eqation, SIAM J. Si. Compt., 37 (0), A70-A7.. F. Li, P. Zang, I. Trner, K. Brrage, V. An, A new frational finite volme metod for 8

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