Date of Award

Summer 8-2010

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Dr. C. S. Chen

Committee Chair Department

Mathematics

Committee Member 2

Dr. Joseph Kolibal

Committee Member 2 Department

Mathematics

Committee Member 3

Dr. Jiu Ding

Committee Member 3 Department

Mathematics

Committee Member 4

Dr. HaiyanTian

Committee Member 4 Department

Mathematics

Committee Member 5

Dr. Chaoyang Zhang

Committee Member 5 Department

Mathematics

Abstract

Meshless methods are relatively new numerical methods which have gained popularity in computational and engineering sciences during the last two decades. This dissertation develops two new localized meshless methods for solving a variety partial differential equations.

Recently, some localized meshless methods have been introduced in order to handle large-scale problems, or to avoid ill-conditioned problems involving global radial basis function approximations. This dissertation explains two new localized meshelss methods, each derived from the global Method of Approximate Particular Solutions (MAPS). One method, the Localized Method of Approximate Particular Solutions (LMAPS), is used for elliptic and parabolic partial differential equations (PDEs) using a global sparse linear system of equations. The second method, the Explicit Localized Method of Approximate Particular Solutions (ELMAPS), is constructed for solving parabolic types of partial differential equations by inverting a finite number of small linear systems. For both methods, the only information that is needed in constructing the approximating solution to PDEs, consists of the local nodes that fall within the domain of influence of the data. Since the methods are completely mesh free, they can be used for irregularly shaped domains. Both methods are tested and compared with existing global and local meshless methods. The results illustrate the accuracy and efficiency of our proposed methods.

Included in

Mathematics Commons

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