Pseudospectral Methods for the Fractional Laplacian on R

Abstract

In this thesis, first, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over R, we map the unbounded domain into a finite one, and represent the resulting function as a trigonometric series. Therefore, a key ingredient is the computation of the fractional Laplacian of an elementary trigonometric function. As an application of the method, we do the simulation of Fisher¿s equation with the fractionalLaplacian in the monostable case.In addition, using complex variable techniques, we compute explicitly, in terms of the 2F1 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the Higgins functions, the Christov functions, and their sine-like and cosine-like versions. After discussing the numerical difficulties in the implementation of the proposed formulas, we develop another method that gives exact results, by using variable precision arithmetic.Finally, we discuss some other numerical approximations of the fractional Laplacian using a fast convolution technique. While the resulting techniques are less accurate, they are extremely fast; furthermore, the results can be improved by the use of Richardson's extrapolation.