Abstract
We briefly introduced the difficulties in solving partial differential problems in Physics by using some traditional mesh-based methods, such as finite difference method, finite element method and finite volume method. Although those methods could be easily applied to some low dimensional problem, they have a comparably hard time to use over irregular domain and to expand to higher dimensional problem. Then we brought up the main character of this thesis — radial basis function (RBF) and the RBF collocation, i.e. Kansa’s method. After we illustrate some different types RBFs and their properties, we moved forward to a basic one dimension hanging string problem by applying Kansa’s method and multiquadric(MQ)function. In the process of the hanging string problem, we tried to clearly demonstrate the main idea of the RBF collocation method while showing the virtues of the method we used and also did the error estimations for some discussions of the stability problem of the RBF collocation method. As one of virtues of RBF collocation method is comparably easy to implemented into higher dimensional problem. We then applied the same kernel MQ function and the same Kansa’s method into the two dimensional problem of hanging string problem — the hanging net problems with different boundary conditions. Then we continued the discussion of two dimensional problems by solving for a classic electrostatic problem — two parallel plates inside a conducting cylinder. With our knowledge of physics, this problem is Poisson equation with irregular domain and non-homogeneous boundary conditions. During the processes, we summarized the main steps for solving lower-dimension partial differential equations and showed the virtues of Kansa’s methods.