Fractional Laplace operator and Meijer G-function

2015/09/28 by Bartłomiej Dyda, Alexey Kuznetsov, Dyda, Bartłomiej +3 · 3 citations
Mathematics · #Mathematical functions and polynomials #Fractional Differential Equations Solutions #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1509.08529

Abstract

We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x|², or generalized hypergeometric functions of -|x|², multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1-|x|²)₊^alpha/2 (-Delta)^alpha/2 with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper "Eigenvalues of the fractional Laplace operator in the unit ball".

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• Electrostatic computations for statistical mechanics and random matrix applications
• A mixed formulation for the fractional Poisson problem
• Nonlocal anisotropic Riesz interactions with a physical confinement

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