Meshless Methods For Semilinear Elliptic Boundary Value Problems

The semilinear elliptic boundary value problems(SEBVPs) appear widely in physics, mechanics, chemical engineering, astronomy and so on. Because the SEBVPs often have multiple solutions, which reveal a lack of stability, it is difficult to compute the multiple solutions numerically. This thesis describes the radial basis functions(RBF) meshless methods of the SEBVPs and the visualization of the solutions. According to three different cases of the solutions of the SEBVPs, this thesis presents RBF meshless methods respectively and analyzes the multiplicity and visualization of the solutions.In chapter two, it introduces the coupling method of RBF meshless method and Newton's method to solve the SEBVP, which has a single solution. Then the numerical example is given. Compared with other algorithms, the results show that the method is easy to programme, has a high precision and do not need the mesh in the area.In chapter three, it introduces the coupling of meshless method with RBF collocation method and scaling iterative algorithm(SIA) to solve the positive solution of the SEBVPs and the singularly perturbed SEBVPs on different areas. We discover it is feasible to solve the positive solution of the SEBVPs and the singularly perturbed SEBVPs by the coupling method. The method is easy to implement, has a high precision and is superior to the coupling method of BEM, FEM and FDM. In chapter four, it introduces RBF collocation meshless method combined with search newton method to solve the sign-changing solution of the SEBVPs and visualizes the solution. The numerical examples show that our algorithm has the properties of easy programming, high precision and meshfree in the area. It is also suitable for the irregular area. So it is an efficient method for the SEBVP.