The Fractional Laplacian
By C. Pozrikidis. 2016
319 citations recorded by the source corpus.
The fractional Laplacian in one dimension Random walkers with constant steps Ordinary diffusion Random jumpers Central limit theorem and stable distributions Power-law probability jump lengths A principal-value integral Wires and springs The fractional Laplacian Fourier transform Effect of fractional order Numerical computation of the fractional Laplacian Green's function of the fractional Laplace equation Fractional Poisson equation in a restricted domain Green's function of unsteady fractional diffusion Numerical discretization in one dimension Computation of a principal-value integral Fractional Laplacian differentiation matrix Fractional Poisson equation Evolution under fractional diffusion Differentiation by spectral expansion Further concepts in one dimension Fractional first derivative Properties of the fractional first derivative Laplacian potential Fractional derivatives from finite-difference stencils Fractional third derivative Fractional fourth derivative Periodic functions Sine, cosines, and the complete Fourier series Cosine Fourier series Sine Fourier series Green's functions Integral representation of the periodic Laplacian Numerical discretization Periodic differentiation matrix Differentiation by spectral expansion Embedding of the fractional Poisson equation The fractional Laplacian in three dimensions Stipulation on the Fourier transform Integral representation Fractional gradient Laplacian potential Green's function of the fractional Laplace equation The Riesz potential Triply periodic Green's function Fractional Poisson equation Evolution under fractional diffusion Periodic functions and arbitrary domains Fractional Stokes flow The fractional Laplacian in two dimensions Stipulation on the Fourier transform Integral representation Fractional gradient Laplacian potential Green's function of the fractional Laplace equation The Riesz potential Doubly periodic Green's function Fractional Poisson equation Evolution due to fractional diffusion Periodic functions and arbitrary domains Appendix A: Selected definite integrals Appendix B: The Gamma function Appendix C: The Gaussian distribution Appendix D: The fractional Laplacian in arbitrary dimensions Appendix E: Fractional derivatives Appendix F: Aitken extrapolation of an infinite sum References Index
Subfields: Computational Theory and Mathematics
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Citations in the corpus
- fPINNs: Fractional Physics-Informed Neural Networks (2018, methodology): “The first term on the right-hand-side is a fractional Laplacian, which is defined in the sense of directional derivatives [48, 49]:”
- A Simple Solver for the Fractional Laplacian in Multiple Dimensions (2018, methodology): “to a natural deterioration of the rate of convergence of our simple approach to an observed rate of Ophq. 1.2 Related work A discretization scheme similar to that presented here appears in Pozrikidis [38], though without discussion of accuracy or the importance of windowing for singularity subtracti”
- Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians (2019, background): “Due to the nonlocal feature of (−∆) α 2 , we need to assume u ∈ C 2m+[α],{α}+ǫ loc (R ) with arbitrarily small ǫ > 0 (merely u ∈ C is not enough) to guarantee that (−∆) α 2 u ∈ C(R) (see [12, 35]), and hence u is a classical solution to equation (1.”
- What is the fractional Laplacian? A comparative review with new results (2018, methodology): “Overview During the past few decades, scientists have been exploring fractional calculus as a tool for developing more sophisticated mathematical models that can accurately describe complex anomalous systems [1, 2, 3, 4].”
- Numerical approximation of fractional powers of elliptic operators (2013, background): “For a detailed discussion about this setting and other possible ways to define fractional powers of the Laplacian (and more general elliptic operators) we refer to [4, 14, 16].”