Time Integration Methods of Fundamental Solutions and Approximate Fundamental Solutions for Nonlinear Elliptic Partial Differential Equations

Author

Corey Leon Jones, University of Southern MississippiFollow

Date of Award

Spring 5-1-2015

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.

Copyright

2015, Corey Leon Jones

Recommended Citation

Jones, Corey Leon, "Time Integration Methods of Fundamental Solutions and Approximate Fundamental Solutions for Nonlinear Elliptic Partial Differential Equations" (2015). Dissertations. 70.
https://aquila.usm.edu/dissertations/70