#### Date of Award

Spring 5-8-2015

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Committee Chair

Haiyan Tian

#### Committee Chair Department

Mathematics

#### Committee Member 2

Jiu Ding

#### Committee Member 2 Department

Mathematics

#### Committee Member 3

James Lambers

#### Committee Member 3 Department

Mathematics

#### Committee Member 4

Huiqing Zhu

#### Committee Member 4 Department

Mathematics

#### Abstract

A time-dependent method is coupled with the Method of Approximate Particular Solutions (MAPS) of Delta-shaped basis functions, the Method of Fundamental Solutions (MFS), and the Method of Approximate Fundamental Solutions (MAFS) to solve a second order nonlinear elliptic partial differential equation (PDE) on regular and irregular shaped domains. The nonlinear PDE boundary value problem is first transformed into a time-dependent quasilinear problem by introducing a fictitious time. Forward Euler integration is then used to ultimately convert the problem into a sequence of time-dependent linear nonhomogeneous modified Helmholtz boundary value problems on which the superposition principle is applied to split the numerical solution at each time step into a homogeneous solution and an approximate particular solution. The Crank-Nicholson method is also examined as an option for the numerical integration as opposed to the forward Euler method. A Delta-shaped basis function, which can handle scattered data in various domains, is used to provide an approximation of the source function at each time step and allows for a derivation of an approximate particular solution of the associated nonhomogeneous equation using the MAPS. The corresponding homogeneous boundary value problem is solved using MFS or MAFS. Numerical results support the accuracy and validity of these computational methods. The proposed numerical methods are additionally applied in nonlinear thermal explosion to determine the steady state critical condition in explosive regimes.

#### Recommended Citation

Jones, Corey Leon, "Time Integration Methods of Fundamental Solutions and Approximate Fundamental Solutions for Nonlinear Elliptic Partial Differential Equations" (2015). *Dissertations*. 70.

http://aquila.usm.edu/dissertations/70

#### Included in

Analysis Commons, Engineering Commons, Non-linear Dynamics Commons, Numerical Analysis and Computation Commons, Partial Differential Equations Commons