A new definition of the fractional Laplacian

Transcription

1 A new efinition of the fractional Laplacian W. Chen imula Reearch Laboratory, P. O. Box. 34, NO-325 Lyaker, Norway (9 eptember 2002) Keywor: fractional Laplacian, fractional erivative, Riez potential, Green econ ientity, hyper-ingularity, bounary conition. Introuction The fractional Laplacian an the fractional erivative are two ifferent mathematical concept (amko et al, 987). Both are efine through a ingular convolution integral, but the former i guarantee to be the poitive efinition via the Riez potential a the tanar Laplace operator, while the latter via the Riemann-Liouville integral i not. It i note that the fractional Laplacian can not be interprete by the fractional erivative in the ene of either Riemann-Liouville or Caputo. Both the fractional Laplacian an the fractional erivative have foun application in many complicate engineering problem. In particular, the fractional Laplacian attract new attention in recent year owing to it unique capability ecribing anomalou iffuion problem (Hanyga, 200). It i, however, note that the tanar efinition of the fractional Laplacian lea to a hyper-ingular convolution integral an i alo obcure about how to implement the bounary conition. Thi purpoe of thi note i to introuce a new efinition of the fractional Laplacian to overcome thee major rawback. Thi tuy i carrie out with the ongoing project of mathematical an numerical moeling of meical ultraoun wave propagation ponore by the imula Reearch Laboratory in Norway.

2 2. Riez potential an fractional Laplacian The fractional Laplacian i commonly coniere the invere of the Riez potential (Gorenflo an Mainari, 998). The Riez potential I of orer of n imenion rea (ZÄHLE, 997; amko et al, 987) I [( ) 2 Γ( 2) Γ Ω π Ω, 0<<2, () 2 where Γ enote the Euler gamma function, an Ω i n-imenion integral omain. The fractional Laplacian can be efine by (e.g., ee Gorenflo an Mainari, 998). 2 ( ) I. (2) Thu, the above fractional Laplacian i alo often calle the Riez fractional erivative. The above efinition (2) of the fractional Laplacian can be actually retate a [ x ]. (3) 2 2 ( ) I ( ) It i known that the raial Laplacian operator ha the expreion 2 +, (4) 2 r r r which r. (3) can then be reuce to 2 ( ) Γ[ ( 2 + ) π 2 Γ( 2 2) x ( 2 + ) Γ[ ( 2 + ) π 2 Γ( 2 2) Ω 2+ ξ Ω Ω + Ω. (5)

3 It i note that (5) encounter the etrimental iue uch a the hyper-ingularity. An alternative way i thu preente below to efine the fractional Laplacian without the perplexing iue in the Riez fractional erivative (3) 2 2 ( ) I [ ] Γ[ ( 2 + ) π 2 Γ( 2 2) Ω 2+ Ω. (6) The Green econ ientity i ueful to implify (6) an can be tate a Ω v v ξ vω u v, (7) Ω where repreent the urface of the omain, an n i the unit outwar normal. Let 2+ v, (8) an bounary conition R, (9) x Γ D x Γ N Q, (0) where Γ D an Γ N are the urface part correponing to the Dirichlet bounary an the Neumann bounary, an uing the Green econ ientity, the efinition (6) i then reuce to

4 2 ( ) h h 2 ( ) + D n Ω + Ω N +, () where h π Γ( 2 2) ( 2 + ) Γ[ ( 2 + ). (2) It i een from () that the preente fractional Laplacian efinition i thu coniere the Riez fractional erivative (the tanar fractional Laplacian) augmente with the bounary integral, which i a parallel to the fractional erivative in the Caputo ene relative to that in the Riemann-Lioville ene. Our efinition alo ha inherent the regularization of the hyper-ingularity. The above two efinition ( ) 2 an ( ) 2 involve in only ymmetric fractional Laplacian. To clearly illutrate the baic iea of thi tuy without lo of generality, we only conier the iotropic meia in thi paper. For the traitional efinition of the aniotropic fractional Laplacian ee Feller (97) an Hanyga (200). By analogy with the new efinition (6) an (), it will be traightforwar to have the correponing new expreion of the aniotropic fractional Laplacian. Albeit a long hitory, the reearch on the pace fractional Laplacian till appear poor in the literature (Gorenflo an Mainari, 998). In recent year, ome interet arie from anomalou iffuion problem. The reaer are avie to fin more etaile ecription of the fractional Laplacian from amko et al (987), Zalavky (994), Gorenflo an Mainari (998), Hanyga (200) an reference therein.

5 3. FEM icretization formulation Let the FEM icretization of a Laplacian operator be expree a v 2 p Kp, (3) where p v repreent the preure value vector at the icrete noe, an K i the poitive efinite FEM icretization matrix. The correponing FEM formulation of the /2 orer fractional Laplacian i then obtaine by v ( p) K p, (4) By uing a uperpoition analyi (Bathe an Wilon, 976), Chen (2002) erive a FEM formulation of the power law attenuation imilar to (4) in form. Thi remin u that the FEM moal analyi approach eman no extra effort to olve the preent fractional Laplacian moel (4) which only involve the common moal parameter uch a eigenvalue an eigenvector of matrix K. Reference: Bathe, K. an Wilon, E. L (976), Numerical Metho in Finite Element Analyi, Prenticle-Hall, New Jerey. Gorenflo, R. an Mainari, F. (998), Ranom walk moel for pace-fractional iffuion procee, Fractional Calculu & Applie Analyi,, Hanyga, A. (200), Multi-imenional olution of time-fractional iffuion-wave equation, to appear in Proc. R. oc. Lonon A. amko,. G., Kilba, A. A., Marichev, O. I. (987), Fractional Integral an Derivative: Theory an Application (Goron an Breach cience Publiher). ZÄHLE, M. (997), Fractional ifferentiation in the elf-affine cae. V - The local egree of ifferentiability, Math. Nachr. 85, Zalavky, G. M. (994), Fractional kinetic equation for Hamiltonian chao, Phyica D, 76, 0-22.