The Method Of Particular Solutions For Solving Nonlinear Elliptic Equations

In the field of science and engineering,partial differential equations(PDEs)play a very important role in the process of transforming physical phenomena into mathematical models.Unfortunately,most of the partial differential equations(PDEs)have no analytical solution.We can only use numerical methods to find numerical solutions.Therefore,it is great importance to obtain a very accurate numerical solution.In the solution of linear partial differential equations(PDEs),one of the most used meshless method is radial basis functions(RBFs)collocation method,also known as Kansa method.This method has been very mature,and it is recognized as a very important method for solving the partial differential equations without grids.However,when solving nonlinear elliptic equations,the numerical solution obtained by the Kansa method is not accuracy enough.This thesis presents a new meshless method for solving boundary value problems of nonlinear elliptic equations.According to the idea,the given function is no longer directly used as a basis function.Instead,is used the particular solution in the differential equation as a basis function to approximate the numerical solution.This meshless method is an indirect method.Such indirect RBF collocation method is called as the method of particular solutions.The main contents of this thesis are as follows:We introduce the method of particular solution and the steps of method of particular solution in solving the boundary value problem of the linear elliptic equations.Introducing the method of particular solution based on the particular solution of radial basis function(RBF)multiquadric(MQ),based on higher order polynomial basis functions,based on the particular solution of multiquadric(MQ)augmented higher order polynomial basis functions.Due to the instability of high-order polynomials,the condition number of the system matrix will be high.The system matrix is a serious illconditioned matrix that can have a very large impact on the accuracy of the numerical solution.Therefore we need to pre-condition the matrix.We introduced multi-scale techniques.We introduce the process of solving nonlinear elliptic equations using the method of particular solutions(MPS)of the above three functions as a basis function.In general,solution of nonlinear equations requires linearization of the problem and feasible iterative methods.In this section,we will introduce two iterative methods for solving nonlinear problems.One is the MATLAB nonlinear solver,and the other is the well-known Picard iterative method.We introduce the error of numerical solution.We use the above three special solutions to solve nonlinear elliptic equations with Dirichlet boundary and Neumann boundary conditions on irregular boundaries.We compare the error performances of these three methods on these problems.Finally,we will summarize and analyze the results.