Date of Award
Fall 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
Zhifu Xie
Committee Member 3 Department
Mathematics
Committee Member 4
Huiqing Zhu
Committee Member 4 Department
Mathematics
Abstract
In the numerical solution of partial differential equations (PDEs), there is a need for solving large scale problems. The Radial Basis Function Differential Quadrature (RBFDQ) method and local RBF-DQ method are applied for the solutions of boundary value problems in annular domains governed by the Poisson equation, inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. By choosing the collocation points properly, linear systems can be obtained so that the coefficient matrices have block circulant structures. The resulting systems can be efficiently solved using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). For the local RBFDQ method, the MDAs used are modified to account for the sparsity of the arrays involved in the discretization. An adjusted Fasshauer estimate is used to obtain a good shape parameter value in the applied radial basis functions (RBFs) for the global RBF-DQ method while the leave-one-out cross validation (LOOCV) algorithm is employed for the local RBF-DQ method using a sample of local influence domains. A modification of the kdtree algorithm is used to select the nearest centers for each local domain. In several numerical experiments, it is shown that the proposed algorithms are capable of solving large scale problems while maintaining high accuracy.
Copyright
2017, Daniel Watson
Recommended Citation
Watson, Daniel, "Radial Basis Function Differential Quadrature Method for the Numerical Solution of Partial Differential Equations" (2017). Dissertations. 1468.
https://aquila.usm.edu/dissertations/1468