Doctoral Dissertations

Siwei Duo

Keywords and Phrases

Fractional Laplacian; Fractional Schrodinger equation; Montgomery's identity; Plane wave dynamics; Weighted interpolation; Weighted trapezoidal

Abstract

"Nonlocal models have recently become a powerful tool for studying complex systems with long-range interactions or memory effects, which cannot be described properly by the traditional differential equations. So far, different nonlocal (or fractional differential) models have been proposed, among which models with the fractional Laplacian have been well applied. The fractional Laplacian (-Δ)α/2 represents the infinitesimal generator of a symmetric α-stable Lévy process. It has been used to describe anomalous diffusion, turbulent flows, stochastic dynamics, finance, and many other phenomena. However, the nonlocality of the fractional Laplacian introduces considerable challenges in its mathematical modeling, numerical simulations, and mathematical analysis.

To advance the understanding of the fractional Laplacian, two novel and accurate finite difference methods -- the weighted trapezoidal method and the weighted linear interpolation method are developed for discretizing the fractional Laplacian. Numerical analysis is provided for the error estimates, and fast algorithms are developed for their efficient implementation. Compared to the current state-of-the-art methods, these two methods have higher accuracy but less computational complexity. As an application, the solution behaviors of the fractional Schördinger equation are investigated to understand the nonlocal effects of the fractional Laplacian. First, the eigenvalues and eigenfunctions of the fractional Schrödinger equation in an infinite potential well are studied, and the results provide insights into an open problem in the fractional quantum mechanics. Second, three Fourier spectral methods are developed and compared in solving the fractional nonlinear Schördinger equation (NLS), among which the SSFS method is more effective in the study of the plane wave dynamics. Sufficient conditions are provided to avoid the numerical instability of the SSFS method. In contrast to the standard NLS, the plane wave dynamics of the fractional NLS are more chaotic due to the long-range interactions"--Abstract, page iii.

Advisor(s)

Zhang, Yanzhi

Committee Member(s)

He, Xiaoming
Le, Vy Khoi
Singler, John R.
Vojta, Thomas

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Mathematics

Publisher

Missouri University of Science and Technology

Publication Date

Summer 2017

Pagination

xii, 220 pages

Note about bibliography

Includes bibliographic references (pages 211-219).

Rights

© 2017 Siwei Duo, All rights reserved.

Document Type

Dissertation - Open Access

File Type

text

Language

English

Thesis Number

T 11478

Electronic OCLC #

1104294596

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Included in

Mathematics Commons

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