Theoretical investigation and numerical computation on meshless collocation method for solving partial differential equations
Student thesis: Doctoral Thesis
Related Research Unit(s)
In the last decades, the use of radial basis functions (RBFs) has proven to be efficient and robust in multivariate interpolation and solving partial differential equations (PDEs). In this thesis we focus on the stability analysis using meshless collocation method by RBFs for solving partial differential equations. The original umsymmetric meshless collocation method was firstly introduced by E. Kansa in 1986. Hon and his collaborators later extended the method to solve various nonlinear initial and boundary value problems. In the first part of the thesis, we investigate the stability and convergence of unsymmetric meshless collocation methods. Some theoretical results are obtained based on the work of R. Schaback who gave a general framework for obtaining error bounds and convergence of a large class of unsymmetric meshless numerical methods in solving well-posed linear operator equations. For simplicity, we consider in this thesis the standard Poisson boundary value problem (PBVP). Using the works of F. J. Ward, H. Wendland and R. Arcangeli et al., we give in this thesis a stability condition of meshless collocation methods in solving the PBVP. Based on the theoretical stability result, in the second part of the thesis, we devise a meshless computational algorithm for solving a real application problem arisen from financial option pricing model. This involves the numerical techniques for solving a partial integro-differential governing equation under some initial and boundary condition problem with unknown free boundary. Numerical examples are given to verify the effectiveness, accuracy, and robustness of the meshless collocation method.
- Differential equations, Partial, Numerical solutions, Collocation methods