9,360 research outputs found

    Finite Domain Anomalous Spreading Consistent with First and Second Law

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    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

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    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

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    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 α(1,2)\alpha\in(1,2) 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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