On the fractional derivatives of radial basis functions

Transcript

1 On the frctionl derivtives of rdil bsis functions Mrym Mohmmdi Robert Schbck b Deprtment of Mthemticl Sciences Isfhn University of Technology Isfhn Irn b Institut für Numerische und Angewndte Mthemtik Universität Göttingen Lotzestrße D Göttingen Germny Abstrct The pper provides the frctionl integrls nd derivtives of the Riemnn-Liouville nd puto type for the five kinds of rdil bsis functions RBFs including the powers Gussin multiqudric Mtern nd thin-plte splines in one dimension It llows to use high order numericl methods for solving frctionl differentil equtions The results re tested by solving two frctionl differentil equtions The first one is frctionl ODE which is solved by the RBF colloction method nd the second one is frctionl PDE which is solved by the method of lines bsed on the sptil tril spces spnned by the Lgrnge bsis ssocited to the RBFs Keywords: Riemnn-Liouville frctionl integrl Riemnn-Liouville frctionl derivtive puto frctionl derivtive Rdil bsis functions 1 Introduction Frctionl clculus hs gined considerble populrity nd importnce due to its ttrctive pplictions s new modelling tool in vriety of scientific nd engineering fields such s viscoelsticity [13] hydrology [1] finnce [5 3] nd system control [] These frctionl models described in the form of frctionl differentil equtions tend to be much more pproprite for the description of memory nd hereditry properties of vrious mterils nd processes thn the trditionl integer-order models In the lst decde number of numericl methods hve been developed to solve frctionl differentil equtions Most of them rely on the finite difference method to discretize both the frctionl-order spce nd time derivtive [ ] Some numericl schemes using low-order finite elements [ 4] mtri trnsfer technique [10 11] nd spectrl methods [14 16] hve lso been proposed Unlike trditionl numericl methods for solving prtil differentil equtions meshless methods need no mesh genertion which is the mjor problem in finite difference finite element nd spectrl methods [0 6] Rdil bsis function methods re truly meshless nd simple enough to llow modelling of rther high dimensionl problems [ ] These methods cn be very efficient numericl schemes to discretize non-locl opertors like frctionl differentil opertors In this pper we provide the required formuls for the frctionl integrls nd derivtives of Riemnn- Liouville nd puto type for RBFs in one dimension The rest of the pper is orgnized s follows In section we give some importnt definitions nd theorems which re needed throughout the remining sections of the pper The corresponding formuls of the frctionl integrls nd derivtives of Riemnn- Liouville nd puto type for the five kinds of RBFs re given in section 3 The results re pplied to solve two frctionl differentil equtions in section 4 The lst section is devoted to brief conclusion Preliminries In this section we outline some importnt definitions theorems nd known properties of some specil functions used throughout the remining sections of the pper [ ] In ll cses denotes orresponding uthor Emil ddresses: m_mohmmdi@mthiutcir Mrym Mohmmdi schbck@mthuni-goettingende Robert Schbck

2 non-integer positive order of differentition nd integrtion Definition 1 The left-sided Riemnn-Liouville frctionl integrl of order of function f is defined s I f 1 τ 1 fτdτ > Γ Definition The right-sided Riemnn-Liouville frctionl integrl of order of function f is defined s I b f 1 b τ 1 fτdt where m Definition 4 The right-sided Riemnn-Liouville frctionl derivtive of order of function f is defined s where m d m D b f 1m Γm d m b τ m 1 fτdτ < b Definition 5 The Riesz spce frctionl derivtive of order of function f t on finite intervl b is defined s f t c D + D b f t where 1 cos π 1 D 1 d m f t Γm d m τ m 1 fτ tdτ d m b D b f t 1m Γm d m τ m 1 fτ tdτ Definition 6 The left-sided puto frctionl derivtive of order of function f is defined s where m D 1 f Γm τ m 1 f m τdτ > Definition 7 The right-sided puto frctionl derivtive of order of function f is defined s where m D 1 m b f Γm b τ m 1 f m τdτ < b The definitions bove hold for functions f with specil properties depending on the situtions It is cler tht D f D m [ I m f ] D b f 1 m D m [ I m b f ] D [ f m I f ] m D [ f 1 m m I b f ] m b

3 Theorem 8 For β > 1 nd > we hve Proof I β 1 Γ I β Γβ + 1 Γ + β + 1 +β τ 1 τ β dτ 1 Γ where u τ Now with the chnge of vrible z I β +β Γ 1 0 z 1 1 z β dz Theorem 9 For β > 1 nd > we hve Proof D β 0 u we get u 1 u β du +β Γβ + 1 B β + 1 Γ Γ + β + 1 +β Γβ + 1 Γβ + 1 β D β D m [ I m β] Γβ + 1 [ Γm + β + 1 Dm m +β] Γβ + 1 Γβ + 1 β Theorem 10 For β > 1 nd > we hve Proof D β D β Γβ + 1 Γβ + 1 β m I β m Γβ + 1 Γβ m + 1 I m β m Γβ + 1 Γβ + 1 β Theorem 11 The following reltions between the Riemnn-Liouville nd the puto frctionl derivtives hold [8]: D D b m 1 f D f m 1 f D b f We now list some known properties of some specil functions f k Γk + 1 k 1 k f k b Γk + 1 b k Γ 1 π ΓΓ + 1 for ll b β; F 1 1 β; + 1; F 1 b; c; 1 c b F 1 c c b; c; 3 d d ν K ν ν K ν 1 n D m λ n n 1 + n Γλ + 1 Γλ m + 1 λ m m N 3

4 where Γ b β; F pq 1 p ; b 1 b q ; K ν nd n denote the Gmm function lower incomplete Bet function Hypergeometric series modified Bessel function of the second kind nd Pochhmmer symbol respectively 3 Frctionl derivtives of RBFs in one dimension Since RBFs re usully evluted on Eucliden distnces we hve to evlute D ϕr D ϕ y for ll y R where D cn be one of the nottions used for frctionl integrls nd derivtives in section nd ϕr is one of the RBFs listed in Tble 1 [5] The following theorems show tht finding the frctionl integrls nd derivtives of ϕ cn led to those of RBFs in one dimension Theorem 1 For ll y R nd > we hve where ξ sign y Proof We get I ϕ y 1 Γ where u τ y Then Moreover I ϕy 1 Γ where u y τ Then I ϕ y ξ ξ yi ϕ y Then 4 nd 5 give the result τ 1 ϕτ ydτ 1 Γ y y y u 1 ϕudu I ϕ y yi ϕ y 4 τ 1 ϕy τdτ 1 Γ y y y u 1 ϕudu I ϕy 1 y I ϕ y 5 Remrk 1 Similrly one cn show tht for ll y R nd we hve where ξ sign y Proof We get D ϕ y ξ ξ yd ϕ y D ϕ y D m [ I m ϕ y] D m [ yi m ϕ y] yd ϕ y 6 Moreover D ϕy D m [ I m ϕy ] D m [ 1 m y I m ϕ y ] 1 m D m [ y I m ϕ y ] 1 y D ϕ y 7 Then 6 nd 7 give the result 4

5 Remrk Similrly one cn show tht for ll y R nd we hve where ξ sign y Proof We get Moreover D ϕ y ξ D ξ y ϕ y D ϕ y I m [ϕ y m ] I m [ϕ m y] y I m ϕ m y y D ϕ y 8 D ϕy Then 8 nd 9 give the result I m [ϕy m ] 1 m I m [ϕ m y ] 1 m y I m ϕ m y 1 y D ϕy 9 Remrk 3 Similrly one cn show tht for ll y R nd < b D b ϕ y ξ D ϕ y ξb y where ξ sign y In the sequel we evlute the Riemnn-Liouville frctionl integrl nd derivtive nd lso the puto frctionl derivtive of ϕ corresponding to the five kinds of RBFs listed in Tble 1 31 Powers For ϕ β the following results hold Theorem 15 For > 0 we hve 0I β Γβ + 1 Γ + β + 1 +β β > 1 0D β Γβ + 1 Γβ + 1 β β > 1 D 0 β Γβ + 1 Γβ + 1 β β > 1 Proof Theorems 8 9 nd 10 for 0 give directly the results Theorem 16 For 0 n N nd > we hve I n D n D n n! n! n n k k n k!γ + k + 1 n n k k n k!γk + 1 n m n! m m n k k n m k!γm + k + 1 5

6 Proof The Tylor epnsion of n bout the point gives n n n! n k n k!k! k Now ccording to the linerity of the Riemnn-Liouville frctionl integrl nd derivtive we hve I n n n! n k n k!k! I k n! n n n! n k Γk + 1 +k n k!k! Γ + k + 1 n k k n k!γ + k + 1 D n n n! n k n k!k! D k n! n n n! n k Γk + 1 k n k!k! Γk + 1 n k k n k!γk + 1 D n I m n m n m n! m m n! n m! I m n m n k k n k!γm + k + 1 Remrk 4 Similrly one cn show tht for 0 we hve 0I e 0D e 0D e Γ1 + F 1 1; 1 + Γ1 F 1 1; 1 Γ1 F 1 1; 1 Proof The Tylor epnsion bout the point 0 gives e 1 n n n! n + ; ; ; 6

7 Therefore 0I e 0D e 0D e 1 n n n! 0 I n Γ1 + 1 n n n! Γn + 1 Γn n+ n 1 n 1 + n n! Γ1 + Γ1 + F 1 1; ; Γ1 1 n n n! 0 D n 1 n n n! 1 n 1 n n 1+ n + n n! Γn + 1 Γn + 1 n n 1 n 1 n n! Γ1 Γ1 F 1 1; 1 ; 1 n 1 n n 1 n n n! 1 n n D n! 0 n Γ1 F 1 1; 1 ; Theorem 18 For 0 nd > we hve Proof I e D e D e I e D e D e 1 n n! n n! m m 1 n n! n n! 1 n n n! I n n n n k k n k!γ + k n n! n n! 1 n n! n n! 1 n n n! D n 1 n n! n n! n k k n k!γk + 1 n m 1 n n n! n! n n k k n m k!γm + k + 1 n n k k n k!γ + k n n n! n! n n k k n k!γ + k + 1 n n k k n k!γk n n D n! n 1 n n n! n! m m n m m m 1 n n! n n! n m n k k n k!γk + 1 n k k n m k!γm + k + 1 n k k n m k!γm + k + 1 7

8 Remrk 5 Similrly one cn show tht for 0 we hve 0I 1 + β 0D 1 + β b n k b k n k!γk + 1 n m Γ1 + F β ; 1 + Γ1 F 1 1 β ; 1 b n k b k n m k!γm + k ; ; 0D 1 + β Γ1 F 1 1 β ; 1 ; Proof The Tylor epnsion bout the point 0 gives Therefore 0I 1 + β 1 + β β n n n! 0 I n Γ1 + β n n n! n 1 n β n n n! 1 n β n n 1 + n n! Γ1 + Γ1 + F β ; ; Γn + 1 Γn n+ 1 n 1 n β n n 1+ n + n n! 0D 1 + β Γ1 1 n β n n 0D n! n 1 n β n n n! 1 n β n n 1 n n! Γ1 Γ1 F β ; 1 ; Γn + 1 Γn + 1 n 1 n 1 n β n n 1 n n n! D β 1 n β n n n! D 0 n Γ1 F β ; 1 ; 8

9 Theorem 0 For 0 nd > we hve I 1 + β Γ β + 1 n! n n!γ β n + 1 n n n k k n k!γ + k + 1 D 1 + β Γ β n! n k k + 1 n n!γ β n + 1 n k!γk + 1 D 1 + β Γ β n m + 1 m m n! n n!γ β n + 1 Proof I 1 + β β n n n! I n Γ β + 1 Γ β n + n! 1n n! Γ β + 1 n! n n n!γ β n + 1 n k k n m k!γm + k + 1 n k k n k!γ + k + 1 n n k k n k!γ + k + 1 D 1 + β β n n n! D n Γ β + 1 Γ β n + n! 1n n! Γ β + 1 n! n n n!γ β n + 1 n k k n k!γk + 1 n n k k n k!γk + 1 D 1 + β β n n D n! n Γ β + 1 n m Γ β n + 1n n! n! m m Γ β + 1 m m Remrk 6 Similrly one cn show tht for < b I b 1 + β Γ β + 1b n! n n!γ β n + 1 n! n n!γ β n + 1 n n n k k n m + k!γm + k + 1 n m n k k n m k!γm + k + 1 b n k b k n k!γ + k + 1 D b 1 + β Γ β n! b n k b k + 1b n n!γ β n + 1 n k!γk + 1 D b 1 + β 1 m Γ β n m + 1b m b m n! n n!γ β n + 1 b n k b k n m k!γm + k + 1 9

10 34 Thin-plte splines For ϕ n ln n N the following results hold Theorem 1 For > 0 we hve 0I n Γn + 1 ln Γn n [ln + Ψn + 1 Ψn ] 0D n Γn + 1 ln Γn + 1 n [ln + Ψn m + 1 Ψn + 1 m 1 r 1 +m!γn m + 1 rm r!γn m + r + 1 ] r1 0D n Γn + 1 ln Γn + 1 n [ln + Ψn m + 1 Ψn + 1 m 1 r 1 +m!γn m + 1 rm r!γn m + r + 1 ] where Ψ is the logrithmic derivtive of the Gmm function Proof I n ln 1 Γ 1 d dn r1 τ 1 τ n lnτdτ 1 d dn I n 1 Γ τ 1 τ n dτ Then it suffices to find the derivtive of the Riemnn-Liouville frctionl integrl of the powers RBF n with respect to n Therefore 0I n ln 1 d Γn + 1 dn Γ + n + 1 +n which in turn gives Γ 0I n n + 1 +n + Γn + 1 +n ln Γn Γ n Γn + 1 +n ln Γn Now by substituting we hve 0I n ln Γ n + 1 Ψn + 1Γn + 1 Γ n Ψn Γn Γn + 1 Γn n [ln + Ψn + 1 Ψn ] 10 Since it is well known tht for the thin-plte splines nd their derivtives t 0 the limiting vlue 0 is considered voiding the singulrity ccording to theorem 11 we hve Now for the puto frctionl derivtive we hve 0D n ln 0 D n ln 0D n m ln 0 I n ln m 10

11 But for 0 n ln m m m r r0 d m r dr n dm r d r ln Γn + 1 Γn m + 1 n m ln + Γn + 1 Γn m + 1 n m ln + m m d m r dr n r dm r d r ln r1 m m Γn + 1 n m+r r Γn m + r + 1 r1 m Γn + 1 Γn m + 1 n m ln + m!γn + 1 n m r1 1 r 1 r 1! r 1 r 1 rm r!γn m + r + 1 Then D 0 n Γn + 1 ln Γn m I m n m ln m +m!γn + 1 r1 1 r 1 rm r!γn m + r + 1 0I m n m Γn + 1 Γn + 1 n [ln + Ψn m + 1 Ψn + 1 m 1 r 1 +m!γn m + 1 rm r!γn m + r + 1 ] r1 Theorem For 0 nd > we hve I n ln n+ F 1 + n + 1; + 1; ln Ψ + n + 1 Γ1 + + n + 1 k Ψ + n + k + 1 k + + k k k! D n ln Γn + 1n m m F 1 m n + 1; m + 1; Γ1 + m ln Ψn + 1 m 1 r 1 + m! Γn m + 1 rm r!γn m + r + 1 r1 m n k Ψn + k + 1 Γn m + 1 m + k k k k! Proof We know tht Then by using 3 we get I β β Γ1 + F 11 β; + 1; I β +β Γ1 + F 1 + β + 1; + 1; 11 11

12 Thus Moreover nd so d dn d dn I n ln 1 d +n dn F 1 + n + 1; + 1; Γ1 + d n F 1 + n + 1; + 1; Γ1 + dn Γ1 + n lnf 1 + n + 1; + 1; + n d dn F 1 + n + 1; + 1; F 1 + n + 1; + 1; d k + n + 1 k k dn + 1 k k k! F 1 + n + 1; + 1; k k + 1 k k k! k k + 1 k k k! Γ + n Γ + n + k + 1Γ + n + 1 Γ + n + 1Γ + n + k + 1 Γ + n + 1 Γ + n + k + 1 Ψ + n + k + 1 Ψ + n + 1 k k + 1 k k k! + n + 1 k Ψ + n + k + 1 Ψ + n n + 1 k Ψ + n + k + 1 k + k k Ψ + n + 1F 1 + n + 1; + 1; k! Then by substituting the reltion bove into 1 nd simplifying the epressions we obtin I n ln n+ F 1 + n + 1; + 1; ln Ψ + n + 1 Γ1 + + n + 1 k Ψ + n + k + 1 k + + k k 13 k! Now for the puto frctionl derivtive we hve D n m ln I n ln m Then by using 13 nd 11 nd fter simplifying the epressions we hve D n Γn + 1 ln Γn m + 1 I m n m ln m 1 r 1 +m!γn + 1 I m rm r!γn m + r + 1 n m r1 Γn + 1n m m F 1 m n + 1; m + 1; Γm + 1 ln Ψn + 1 m 1 r 1 + m! Γn m + 1 rm r!γn m + r + 1 r1 m n k Ψn + k + 1 Γn m + 1 m + k k k k! 1

13 Remrk 7 Similrly one cn show tht for < b I b n ln n+ b b F 1 + n + 1; + 1; b ln Ψ + n + 1 Γ1 + b + n + 1 k Ψ + n + k + 1b k + + kb k k! D n ln b 1m Γn + 1 n b m b m F 1 m n + 1; m + 1; b Γm + 1 b ln Ψn + 1 m 1 r 1 + m! Γn m + 1 rm r!γn m + r + 1 r1 m n k Ψn + k + 1 Γn m + 1 m + k k b k k! 35 Mtern For ϕ ν K ν with non-integer ν > 0 the following results hold Theorem 3 For > we hve I ν π K ν sin πν Proof We know tht ν Γ1 νγ1 + F 3 1 1; 1 ν ; 4 ν Γν + 1 +ν F 1 ν + 1 πγ + ν + 1 ; + ν + 1 ; + ν + 1; 4 ν 1 k Γ k + 1Γ ν + k + 14 k k! F 1k ; k + ; + ν+1 k ν Γ 1 4 k k!ν + k + 1Γν + k + 1 F 1 ν + k ; ν + k + ; 4 I ν K ν π sinπν I ν J ν I ν J ν where k ν 4 J ν k!γν + k + 1 Now I ν J ν 1 Γ τ 1 τ ν J ν τdτ 1 Γ 1 Γ ν 4 k k!γν + k + 1 τ 1 τ ν+k dτ τ 1 τ ν τ ν k τ 4 k!γν + k + 1 dτ 13

14 By the chnge of vrible u τ we hve Therefore 1 τ 1 τ ν+k dτ +ν+k 1 u 1 u ν+k du I ν J ν Now by using 1 nd we get I ν J ν +ν+k Γν + k + 1Γ Γ + ν + k + 1 bν + k + 1 ; +ν+k Γν + k + 1Γ ν 4 k k!γγν + k + 1 Γ + ν + k + 1 bν + k + 1 ; Γν + k +ν+k ν 1 4 k k!γν + kγ + ν + k + 1 ν Γν + k + 1 +ν+k πγ + ν + k + 1k! ν+1 1 ν Γ +ν ν π ν+1 1 ν Γ bν + k + 1 ; +ν+k ν 4 k k!γγν + k + 1 k 4 k k!ν + k + 1Γν + k + 1 F 1ν + k ; ν + k + ; Γν + k + 1 k Γ + ν + k + 1k! +ν ν Γν + 1 πγ + ν + 1 ν+1 1 ν Γ k 4 k k!ν + k + 1Γν + k + 1 F 1ν + k ; ν + k + ; ν + 1 k k + ν + 1 k k! k 4 k k!ν + k + 1Γν + k + 1 F 1ν + k ; ν + k + ; +ν ν Γν + 1 ν + 1 k k πγ + ν ν + 1 k + ν + 1 k 4k k! ν+1 1 k ν Γ 4 k k!ν + k + 1Γν + k + 1 F 1ν + k ; ν + k + ; ν Γν + 1 +ν F 1 ν + 1 πγ + ν + 1 ; + ν ν + 1; 4 ν+1 1 ν Γ k 4 k k!ν + k + 1Γν + k + 1 F 1ν + k ; ν + k + ; 14

15 Moreover I ν J ν 1 Γ ν τ 1 τ ν J ν τdτ 1 Γ ν Γ4 k k!γ ν + k + 1 ν +k Γ4 k k!γ ν + k + 1 τ 1 τ k dτ τ 1 τ ν τ ν Γk + 1Γ Γ + k + 1 bk + 1 ; Γk + 1 k 4 k k!γ ν + k + 1Γ + k + 1 ν +k bk + 1 ; 4 k k!γγ ν + k + 1 ν Γ1 νγ1 + ν 1 Γ ν Γ1 νγ1 + ν 1 Γ 1 k k 1 ν k 1 + k 4 k k! k 4 k k + 1Γ ν + k + 1k! F 1k ; k + ; ν Γ1 νγ1 + F 3 1 ν 1 Γ 1 k 1 k k 1 ν k 1+ k + k 4k k! k 4 k k + 1Γ ν + k + 1k! F 1k ; k + ; 1; 1 ν ; 4 k 4 k k + 1Γ ν + k + 1k! F 1k ; k + ; k τ 4 k!γ ν + k + 1 dτ Thus I ν K ν π sin πν ν Γ1 νγ1 + F 3 1 1; 1 ν ; 4 ν Γν + 1 +ν F 1 ν + 1 πγ + ν + 1 ; + ν + 1 ; + ν + 1; 4 ν 1 k Γ k + 1Γ ν + k + 14 k k! F 1k ; k + ; + ν+1 k ν Γ 1 4 k k!ν + k + 1Γν + k + 1 F 1 ν + k ; ν + k + ; 4 Theorem 4 For > we hve D ν K ν 1 m I m ν m K ν m Proof By definition we hve D ν K ν I m ν K ν m 1 m I m ν m K ν m 15

16 Remrk 8 For the specil cse 0 nd > 0 we hve 0I ν π ν K ν sin πν Γ1 νγ1 + F 3 1 1; 1 ν + 1 ν Γν + 1 +ν F 1 ν + 1 πγ + ν + 1 ; + ν + 1 ; + ; 4 + ν + 1; 4 0D ν K ν 1 m 0I m ν m K ν m Remrk 9 Similrly one cn show tht for < b I b ν K ν 1 π sin πν ν Γ1 νγ1 + F 3 1 1; 1 ν ; 4 ν Γν + 1 +ν F 1 ν + 1 πγ + ν + 1 ; + ν + 1 ; + ν + 1; 4 ν 1 b b k Γ k + 1Γ ν + k + 14 k k! F 1k ; k + ; b + bν+1 b k ν Γ 1 4 k k!ν + k + 1Γν + k + 1 F 1 ν + k ; ν + k + ; 4 4 Appliction D ν K ν 1 D ν K ν b In this section we pply the results of the previous section to solve two frctionl differentil equtions The first one is frctionl ODE which is solved by the RBF colloction method nd the second one is frctionl PDE which is solved by the method of lines bsed on the sptil tril spces spnned by the Lgrnge bsis ssocited to the RBFs In both cses we work with scled RBFs ie y ϕ c where the RBF scle c controls their fltness The infinite sums ppering in the previous formuls re truncted once the terms re smller in mgnitude thn mchine precision 41 Test problem 1 onsider the following frctionl ODE []: 0D t 3/ ut + ut ft t 0 T ] u0 u 0 0 Let t i 1 i n be the equidistnt discretiztion points in the intervl [0 T ] such tht t 1 0 nd t n T Then the pproimte solution cn be written s ut n λ j ϕ t t j j0 16

17 where t j s re known s centers The unknown prmeters λ j re to be determined by the colloction method Therefore we get the following equtions for the ODE n λ j 0 D 3/ t ϕ t i t j + j0 n λ j ϕ t i t j ft i i n 1 14 j0 nd the following equtions for the initil conditions n λ j ϕ t 1 t j 0 15 j0 n λ j ϕ t 1 t j 0 16 j0 Then led to the following system of equtions: 0 D 3/ t ϕ + ϕ ϕ 1 [λ] F 0 ϕ 1 0 The necessry mtrices nd vectors re ϕ ϕ t i t j i n 11 j n 0D 3/ t ϕ 0D 3/ t ϕ t i t j ϕ 1 ϕ t 1 t j 1 j n ϕ 1 ϕ t 1 t j 1 j n λ λ j 1 j n T F ft i i n 1 T i n 11 j n Now we tke 501 points in the intervl 0 t 50 nd work with the powers RBF ϕ 3 with c 10 4 The numericl solutions re plotted with different right-hnd side functions ft 1 ft te t nd ft e t sin0t in Figures 1 nd 3 respectively The results re in greement with the results of [] 4 Test problem onsider the following Riesz frctionl differentil eqution [8]: ut t K u t [0 π] t 0 T ] u 0 u 0 u0 t uπ t 0 17 where u is for emple solute concentrtion nd K represents the dispersion coefficient Let i 1 i n be the equidistnt discretiztion points in the intervl [0 π] such tht 1 0 nd n π Now we construct the Lgrnge bsis L 1 L n of the spn of the functions ϕ j 1 j n vi solving the system L ϕa 1 where L L j 1 j n ϕ ϕ j 1 j n A ϕ i j 1 i n1 j n If L is differentil opertor nd if the RBF ϕ is sufficiently smooth to to llow ppliction of L the required derivtives LL j of the Lgrnge bsis L j come vi solving LL LϕA 1 17

18 Due to the stndrd Lgrnge conditions the zero boundry conditions t 0 0 nd π re stisfied if we use the spn of the functions L L n 1 s our tril spce Then the pproimte solution cn be written s with the unknown vector n 1 u t β j tl j j βt β j t j n 1 We now write the PDE t point i i n 1 s follows: n 1 n 1 β jtl j i K β j t L j i j The initil conditions lso provide j β j 0 u 0 j j n 1 Thus we get the following system of ODEs β t K L βt with the initil conditions β0 U 0 where L L j i i n 1 j n 1 U 0 u 0 i i n 1 T Now consider problem 17 with the prmeters 18 K 05 T 04 nd u 0 π The numericl solution is plotted by using the Gussin RBF with c 1 nd tking 101 discretiztion points in Figure 4 In the second eperiment we use prmeters 15 K 05 T 05 nd u 0 sin4 The numericl solution is plotted by using the Gussin RBF with c 1 nd tking 101 discretiztion points in Figure 5 The results re in greement with the results of [8] It should be noted tht using the multiqudric RBF with β 1 nd c 1 lso gives the sme results 5 onclusion The Riemnn-Liouville frctionl integrl nd derivtive nd lso the puto frctionl derivtive of the five kinds of RBFs including the powers Gussin multiqudric Mtern nd thin-plte splines in one dimension re obtined These formuls llow to use new frctionl vritions of numericl methods bsed on RBFs Two emples of such techniques re given The first one is frctionl ODE which is solved by the RBF colloction method nd the second one is frctionl PDE which is solved by the method of lines bsed on the sptil tril spces spnned by the Lgrnge bsis ssocited to RBFs References [1] DA Benson SW Whetcrft nd MM Meerschert Appliction of frctionl dvection dispersion eqution Wter Resour Res 36:

19 [] GJ Fi nd JP Roop Lest squre finite-element solution of frctionl order two-point boundry vlue problem omput Mth Appl 48: [3] Frnke nd R Schbck onvergence order estimtes of meshless colloction methods using rdil bsis functions Adv in omp Mth 8: [4] Frnke nd R Schbck Solving prtil differentil equtions by colloction using rdil bsis functions Appl Mth omp 93: [5] R Gorenflo F Minrdi E Scls nd M Rberto Frctionl clculus nd continuous-time finnce III The diffusion limit Mthemticl finnce Trends in Mth Birkhäuser Bsel [6] GH Go nd ZZ Sun A compct difference scheme for the frctionl sub-diffusion equtions omput Phys 30: [7] J Hung N Ningming nd Y Tng A second order finite difference-spectrl method for spce frctionl diffusion eqution [8] M Ishtev R Scherer nd L Boydjiev On the puto opertor of frctionl clculus nd - Lguerre functions Mth Sci Res J 9: [9] Y Hon R Schbck nd X Zhou An dptive greedy lgorithm for solving lrge rdil bsis function colloction problem Numer Algorithms 3: [10] M Ilic F Liu I Turner nd V Anh Numericl pproimtion of frctionl-in-spce diffusion eqution I Frct lc Appl Anl 8: [11] M Ilic F Liu I Turner nd V Anh Numericl pproimtion of frctionl-in-spce diffusion eqution II -with nonhomogeneous boundry conditions Frct lc Appl Anl 9: [1] EJ Kns scttered dt pproimtion scheme with pplictions to computtionl fluid-dynmics I: Solutions to prbolic hyperbolic nd elliptic prtil differentil equtions omput Mth Appl 19: [13] R Koeller Appliction of frctionl clculus to the theory of viscoelsticity J Appl Mech 51: [14] XJ Li nd J Xu A spce-time spectrl method for the time frctionl differentil eqution SIAM J Numer Anl 47: [15] YM Lin nd J Xu Finite difference/spectrl pproimtions for the time-frctionl diffusion eqution J omput Phys 5: [16] Li F Zeng nd F Liu Spectrl pproimtion to the frctionl integrl nd derivtives Frct lc Appl Anl 15: [17] AM Mthi nd HJ Hubold Specil functions for pplied sciences Springer Science 008 [18] [19] MM Meerschert nd Tdjern Finite difference pproimtions for frctionl dvectiondiffusion equtions J omput Appl Mth 17: [0] R Mokhtri nd M Mohmmdi Numericl solution of GRLW eqution using Sinc-colloction method omput Phys omm 181: [1] R Mokhtri nd M Mohseni A meshless method for solving mkdv eqution omput Phys omm 183:

20 [] I Podlubny Frctionl Differentil Equtions An Introduction to Frctionl Derivtives Frctionl Differentil Equtions Some Methods of Their Solution nd Some of Their Applictions Acdemic Press 1999 [3] M Rberto E Scls nd F Minrdi Witing-times nd returns in high-frequency finncil dt: An empiricl study Phys A 314: [4] JP Roop omputtionl spects of FEM pproimtion of frctionl dvection dispersion equtions on bounded domins in R J omput Appl Mth 193: [5] R Schbck MATLAB Progrmming for Kernel-Bsed Methods Preprint Göttingen 009 [6] R Schbck nd H Wendlnd Kernel techniques: From mchine lerning to meshless methods Act Numeric 15: [7] R Schere SL Kll L Boydjievc nd B Al-Sqbi Numericl tretment of frctionl het equtions Appl Numer Mth 58: [8] Q Yng F Liu nd I Turner Numericl methods for frctionl prtil differentil equtions with Riesz spce frctionl derivtives Appl Mth Modelling 34: [9] Y Zhng A finite difference method for frctionl prtil differentil eqution Appl omput Mth 15:

21 Tble 1: Definition of some types of RBFs where β ν nd n re RBF prmeters Nme Definition Gussin ep r / Multiqudric 1 + r / β/ Powers Mtern/Sobolev Thin-plte splines r β K ν rr ν r n lnr u t Figure 1: Solution of the test problem 1 for ft u t Figure : Solution of the test problem 1 for ft te t 1

22 u t Figure 3: Solution of the test problem 1 for ft e t sin0t u Figure 4: Solution of the test problem with 18 K 05 nd T u Figure 5: Solution of the test problem with 15 K 05 nd T 05