Polynomial Particular Solutions For Solving A Class Of Higher Order Elliptic Equations

Elliptic partial differential equations are widely used in many fields,such as mathematical physics,mechanics,civil engineering and so on.For example,in elastic mechanics,electromagnetics,building structures,aerospace structures and other issues,elliptic partial differential equations can be used to construct mathematical models.However,the exact solutions of partial differential equations can be obtained only in very few cases.Therefore,a more popular numerical method is used to explore the numerical solutions of partial differential equations.The traditional numerical methods,such as the finite element method and the finite difference method,are time-consuming and laborious,especially for complex three-dimensional geometric structures.Therefore,in the past two decades,the meshless method has effectively solved many partial differential equations due to its wide applicability,such as dealing with complex domains,avoiding triangulation and remeshing.Among them,the special solution is an effective RBF collocation meshless method,and the given RBF special solution is used as a basis function to approximate the numerical solution.However,the accuracy and stability of the solution depend on the selection of RBF shape parameters.Therefore,in order to improve the accuracy and stability of the solution,the polynomial basis function is used to replace the special solution of the radial basis function ―the polynomial special solution.In this paper,this method is used to study a class of high-order elliptic partial differential equations,which not only has advantages in feasibility and computational cost,but also has advantages in solving accuracy.The structure of this paper is as follows: In chapter one,the research background of solving elliptic PDEs is briefly introduced,and then the development history of meshless methods and several common numerical methods for solving PDEs are described.In the second chapter,the related concepts such as radial basis function are given,and then the method of particular solutions,method of polynomial particular solutions and multi-scale technique are introduced.Finally,the main research work of this paper is summarized.In the third chapter,the singularly perturbed non-local boundary value problem of higher order elliptic PDEs and a class of parabolic PDEs dependent on time t are studied.In order to test the effectiveness of the algorithm,numerical examples with different orders and different parameters are solved and analyzed.In this way,the problem of solving higher order elliptic PDEs with special method is studied.The results show that this method has the characteristics of high precision,good convergence and easy implementation.In the fourth chapter,the research status of eigen-value problems for high-order elliptic partial differential equations is introduced firstly.In order to prove the effectiveness of the algorithm,the method of polynomial particular solutions is applied to numerical experiments with different orders and parameters,and precision order tests are carried out on the number of configuration points and computational domains that may affect the calculation accuracy,the numerical simulation results show that the method of polynomial particular solutions has the advantages of high convergence and high precision.In the fifth chapter,firstly,the main research work of this paper is comprehensively summarized,and then the further work arrangement and prospect are made.