9,360 research outputs found
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
Fractional Calculus in Wave Propagation Problems
Fractional calculus, in allowing integrals and derivatives of any positive
order (the term "fractional" kept only for historical reasons), can be
considered a branch of mathematical physics which mainly deals with
integro-differential equations, where integrals are of convolution form with
weakly singular kernels of power law type. In recent decades fractional
calculus has won more and more interest in applications in several fields of
applied sciences. In this lecture we devote our attention to wave propagation
problems in linear viscoelastic media. Our purpose is to outline the role of
fractional calculus in providing simplest evolution processes which are
intermediate between diffusion and wave propagation. The present treatment
mainly reflects the research activity and style of the author in the related
scientific areas during the last decades.Comment: 33 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1008.134
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
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