Date of Award

Summer 2017

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Ching-Shyang Chen

Committee Chair Department

Mathematics

Committee Member 2

Haiyan Tian

Committee Member 2 Department

Mathematics

Committee Member 3

Huiqing Zhu

Committee Member 3 Department

Mathematics

Committee Member 4

Zhifu Xie

Committee Member 4 Department

Mathematics

Abstract

Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their order becomes large. The multiple scale technique is applied to circumvent the difficulty of ill-conditioning.

The derived polynomial particular solutions are also applied in the localized method of particular solutions to solve large-scale problems. Many numerical experiments have been performed to show the effectiveness of the particular solutions on this algorithm.

As another part of the dissertation, a modified method of particular solutions (MPS) has been used for solving nonlinear Poisson-type problems defined on different geometries. Polyharmonic splines are used as the basis functions so that no shape parameter is needed in the solution process. The MPS is also applied to compute the sizes of critical domains of different shapes for a quenching problem. These sizes are compared with the sizes of critical domains obtained from some other numerical methods. Numerical examples are presented to show the efficiency and accuracy of the method.

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