Date of Award

Fall 12-2014

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Ching-Shyang Chen

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 meshless method for solving partial differential equations (PDEs) which combines the method of fundamental solutions (MFS) and the method of particular solutions (MPS) is formulated and tested. The hybrid method finds a numerical approximation by solving only one system of equations as opposed to the two-stage method of fundamental solutions and method of particular solutions. This new approach, denoted MFS-MPS, one-stage MFS-MPS, or hybrid method, can be applied to a wide variety of PDEs including PDEs with variable coefficients. The MFS-MPS can simplify Helmholtz-type differential operators to Laplacian-type differential operators providing flexibility and simplification to calculating particular solutions and fundamental solutions. The hybrid method is tested using different Radial Basis Functions as well as different configurations of interpolation points and source points. Its accuracy is compared with the Kansa method and the MPS. The relatively new leave-one-out cross validation (LOOCV) technique is employed to find sub-optimal shape parameters and source parameters.

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