Numerical approximation of the fractional Laplacian on R using orthogonal families

In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the F12 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the complex Higgins functions, the complex Christov functions, and their sine-like and cosine-like versions. Then, aft...

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Bibliographic Details
Published in:Applied numerical mathematics Vol. 158; pp. 164 - 193
Main Authors: Cayama, Jorge, Cuesta, Carlota M., de la Hoz, Francisco
Format: Journal Article
Language:English
Published: Elsevier B.V 01.12.2020
Subjects:
Arbitrary-precision arithmetic
Condition numbers
Differentiation matrices
Fractional Laplacian
Fractional nonlinear Schrödinger equation
Pseudospectral methods
Fractional nonlinear Schrödinger equation
Pseudospectral methods
Fractional Laplacian
Differentiation matrices
Condition numbers
Arbitrary-precision arithmetic
ISSN:0168-9274, 1873-5460
Online Access:Get full text
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Summary:In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the F12 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the complex Higgins functions, the complex Christov functions, and their sine-like and cosine-like versions. Then, after studying the asymptotic behavior of the resulting expressions, we discuss the numerical difficulties in their implementation, and develop a method using arbitrary-precision arithmetic that computes them accurately. We also explain how to create the differentiation matrices associated to the complex Higgins functions and to the complex Christov functions, and study their condition numbers. In this regard, we show how arbitrary-precision arithmetic is the natural tool to deal with ill-conditioned systems. Finally, we simulate numerically the fractional nonlinear Schrödinger equation using the developed tools.
ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2020.07.024