Award Date

May 2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Committee Member

Xin Li

Second Committee Member

Rohan Dalpatadu

Third Committee Member

Douglas Burke

Fourth Committee Member

Woosoon Yim

Number of Pages

178

Abstract

In this dissertation we propose and examine numerical methods for solving the boundary value problems of partial differential equations (PDEs) by meshless methods. First we aim at getting approximate particular solutions up of a nonhomogeneous equation by radial basis methods. For instance, the collocation method by radial basis functions (RBFs) for finding particular solutions is very popular in the literature. Now the particular solutions of certain important PDEs by RBF approximation are available with the order of convergence to the exact solutions provided. Here we explore and examine the numerical performances of these particular solutions in various examples. We then consider and solve the following boundary value problems of the homogeneous equation by the methods of fundamental solutions (MFS). Choose a fictitious domain and choose some collocation points on boundary of domain and some source points on boundary of fictitious domain Then we have an approximate solution vn of the homogeneous equations by MFS. Hence, u(x) = up(x) + vn(x) is considered as the numerical solution of our original problem. In this dissertation, we present various examples to show the efficiencies of the above mentioned methods, especially for Poisson’s, Helmholtz, and biharmonic equations of Dirichlet, or Newmann, or Robin (Mixed) boundary conditions, with numerical results provided correspondingly in tables and graphs.

Disciplines

Mathematics

Language

English


Included in

Mathematics Commons

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