Wavelet Galerkin method for fractional elliptic differential equations

Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically solving fractional elliptic differential equations (FEDEs). The technique issues are solved and the detailed algorithm descriptions are provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin method we propose for FEDEs has the striking benefit of efficiency, since the condition numbers of the corresponding stiffness matrixes are small and uniformly bounded; and the Toeplitz structure of the matrix still can be used to reduce cost. Numerical results and comparison with the ordinary Galerkin methods are presented to demonstrate the advantages of the wavelet Galerkin method we provide.Comment: 20 pages, 0 figure

Finite Domain Anomalous Spreading Consistent with First and Second Law

After reviewing the problematic behavior of some previously suggested finite interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space-time fractional differential equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space-time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.Comment: 14 figure

Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects

We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.Comment: 36 pages, 3 figures (5 files eps). Invited lecture by R. Gorenflo at the 373. WE-Heraeus-Seminar on Anomalous Transport: Experimental Results and Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), 12-16 July 2006; Chairmen: R. Klages, G. Radons and I.M. Sokolo