91 research outputs found
Riesz potential versus fractional Laplacian
This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian
Fractional central differences and derivatives
Journal of Vibration and Control, 14(9–10): 1255–1266, 2008Fractional central differences and derivatives are studied in this article. These are generalisations to real orders of the ordinary positive (even and odd) integer order differences and derivatives, and also
coincide with the well known Riesz potentials. The coherence of these definitions is studied by applying
the definitions to functions with Fourier transformable functions. Some properties of these derivatives are presented and particular cases studied
Rational approximation to the fractional Laplacian operator in reaction-diffusion problems
This paper provides a new numerical strategy to solve fractional in space
reaction-diffusion equations on bounded domains under homogeneous Dirichlet
boundary conditions. Using the matrix transform method the fractional Laplacian
operator is replaced by a matrix which, in general, is dense. The approach here
presented is based on the approximation of this matrix by the product of two
suitable banded matrices. This leads to a semi-linear initial value problem in
which the matrices involved are sparse. Numerical results are presented to
verify the effectiveness of the proposed solution strategy
Fractional central differences and derivatives
Journal of Vibration and Control, Vol. 14, Nº 9-10Fractional central differences and derivatives are studied in this article. These are generalisations to real orders of the ordinary positive (even and odd) integer order differences and derivatives, and also
coincide with the well known Riesz potentials. The coherence of these definitions is studied by applying the definitions to functions with Fourier transformable functions. Some properties of these derivatives are presented and particular cases studied
Bilateral tempered fractional derivatives
Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it cannot be considered as a derivative.publishersversionpublishe
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