Research On Ridge Basis Function Collocation Method

In meshless methods, the study of the numerical solutions of PDEs with radial basis function interpolation has yielded a number of substantial results. In contrast, the study with the ridge basis function interpolation is much less investigated both in the theory of numerical methods and in the practical applications of the theory. Meshless collocation methods for the numerical solutions of PDEs are increasingly adopted due to their advantages including efficiency and flexibility, and radial basis functions are popularly employed to represent the solutions of PDEs. Motivated by the advantages of ridge basis function representation of a given function, such as the connection to neural network, fast convergence as the number of terms is increased, better approximation effects and various applications in engineering problems, and hence a meshless method is developed based on the collocation method and ridge basis function interpolation in this thesis, where ridge basis function is used as an approximating function to structure the meshless method, and the collocation method is taken as the discrete method.This thesis has made some beneficial probe about ridge basis function collocation method's theory and application. And the main work can be summarized as follows.Firstly, ridge basis function collocation method for elliptic equations has been developed, and the existence and uniqueness of the solution to the method is given. According to numerical example and analysis, it is seen that the method of this thesis is feasible, and influence factors of accuracy for numerical solutions are analyzed. This method is a truly meshless technique without mesh discretization:it neither needs the computation of integrals, nor requires a partition of the region and its boundary. When compared to radial basis function collocation method, the finite difference method and the finite element method, the method of this thesis achieves higher accuracy, with least required computation time.Secondly, the method is applied to parabolic equations to examine its appropriateness. Here, the discussion of the explicit scheme and implicit scheme of parabolic equations is provided, and the existence and uniqueness of the solution to the explicit scheme and strong conditions of the explicit scheme are established and discussed. Moreover, meshless method has been applied for numerical solutions of 1 and 2-dimensional equations.Lastly, the meshless method for convection-dominated diffusion equations is developed. This method can effectively reduce the numerical oscillations by using implicit scheme of the equations, and the existence and uniqueness of the solution of weak conditions is given. According to numerical example and analysis, it is seen that the implicit scheme is feasible. When compared to the finite element method, the implicit scheme achieves higher accuracy, with least required computation time. Therefore, the method of this thesis can achieve satisfactory results.In particular, regarding the importance of radial basis functions, a new ridge basis function is proposed, where the positive definite of the new ridge basis function is proved. Moreover, a novel meshless method for elliptic equations and convection-diffusion equations is developed by utilizing the collocation method and the new ridge basis function interpolation. According to numerical example and analysis, it is seen that the new ridge basis function is feasible. And the meshless method with our new ridge basis function achieves a high accuracy. Therefore, the new ridge basis function proposed here is meaningful for developing meshless methods with ridge basis functions.