Doctoral Dissertations

Abstract

"In recent decades, nonlocal models have been proved to be very effective in the study of complex processes and multiscale phenomena arising in many fields, such as quantum mechanics, geophysics, and cardiac electrophysiology. The fractional Laplacian(βˆ’Ξ”)𝛼/2 can be reviewed as nonlocal generalization of the classical Laplacian which has been widely used for the description of memory and hereditary properties of various material and process. However, the nonlocality property of fractional Laplacian introduces challenges in mathematical analysis and computation. Compared to the classical Laplacian, existing numerical methods for the fractional Laplacian still remain limited. The objectives of this research are to develop new numerical methods to solve nonlocal models with fractional Laplacian and apply them to study seismic wave modeling in both homogeneous and heterogeneous media.

To this end, we have developed two classes of methods: meshfree pseudospectral method and operator factorization methods. Compared to the current state-of-the-art methods, both of them can achieve higher accuracy with less computational complexity. The operator factorization methods provide a general framework, allowing one to achieve better accuracy with high-degree Lagrange basis functions. The meshfree pseudospectral methods based on global radial basis functions can solve both classical and fractional Laplacians in a single scheme which are the first compatible methods for these two distinct operators. Numerical experiments have demonstrated the effectiveness of our methods on various nonlocal problems. Moreover, we present an extensive study of the variable-order Laplacian operator (βˆ’Ξ”)𝛼(x)/2 by using meshfree methods both analytically and numerically. Finally, we apply our numerical methods to solve seismic wave modeling and study the nonlocal effects of fractional wave equation"--Abstract, p. iv

Advisor(s)

Zhang, Yanzhi

Committee Member(s)

Bohner, Martin, 1966-
Emdadi, Arezoo
He, Xiaoming
Seleson, Pablo

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Computational and Applied Mathematics

Publisher

Missouri University of Science and Technology

Publication Date

Spring 2022

Pagination

xvi, 208 pages

Note about bibliography

Includes_bibliographical_references_(pages 195-207)

Rights

Β© 2022 Yixuan Wu, All Rights Reserved

Document Type

Dissertation - Open Access

File Type

text

Language

English

Thesis Number

T 12235

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