In this thesis, the method of approximate particular solutions(MAPS) and localized method of approximate particular solutions(LMAPS) with polynomial basis, and radial basis functions are proposed and applied on the optimal control problems(OCPs) governed by partial differential equations(PDEs). This study proceeds in several steps. First, polynomial basis and radial basis functions are used to globally approximate solutions for the PDEs which have been combined into a single matrix system numerically from the optimality conditions of the OCPs. Secondly, polynomial and radial basis functions are used to locally approximate particular solutions for the same matrix system numerically. We use these approaches to two types of problems, a smooth and singular problem. The first example numerically experiments on a square domain and the second example on an L-shaped disc domain. These approaches are tested and compared. The results show our proposed method for solving optimal control problems governed by partial differential equations works.

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Localized Method Of Approximate Particular Solutions For Solving Optimal Control Problems Governed By PDES

In the past, dealing with fourth-order partial differential equations using the Local Method was not reliable due to difficulties in solving them directly. An approach such as splitting these equations into two Poisson differential equations was adopted to alleviate such challenges. However, this has a limitation since it is only applicable to Dirichlet and Laplace boundary conditions. In this paper, we solve fourth-order PDEs directly using the LMAPS. The improvement on the accuracy of this Method was as a result of the proposed distribution of boundary conditions to alternating boundary points. And, also the use of suitable shape parameter; calculated using LOOCV(Leave-One-Out-Cross-Validation) Algorithm [23]. The effectiveness of this Method was evident when we compared the results from two numerical examples.

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Localized Method of Approximate Particular Solutions for Solving Fourth-Order PDEs

LOCALIZED MESHLESS METHODS WITH RADIAL BASIS FUNCTIONS FOR EIGENVALUE PROBLEMS

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Amy M Kern

Papers

Localized Meshless Methods With Radial Basis Functions For Eigenvalue Problems