Ridge Basis Function Meshless Collocation Method For Partial Differential Equation

Recently, the meshless method is an active topic in the areas of computational science and approximation theory. Based on the approximate nodes, this method can eliminate meshes completely or partly, and need not initiatory plot along with rebuild of meshes, also can handle the disadvantage of the approximate function of the finite method. Over past decades, meshless methods have applied in many different areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of partial differential equations.Helmholtz problems and convection diffussion problems have been used widely in many fields, such as physics, mechanics, engineering, and so on. Hence, the research on their numerical solutions not only has theoretical values, but also has practical significances.The main results in this paper as following:1. The collocation method based on Gauss ridge basis function is used to solving Helmholtz problems. Through the study of different types of Helmholtz problems, the feasibility and adaptability of the method is certified;2. The characteristic ridge basis meshless method for convection-dominated diffussion equations is studied. The numerical experiment show that the method has a higher accuracy and is easy to program and calculate comparing with the characteristic finite element method (CFEM).