Solving Partial Differential Equations Using Meshless Methods Based On Polynomial Basis Functions

In this paper,two meshless methods based on polynomial basis functions are utilized to solve partial differential problems.The availability of the closed-form particular solution for a given differential equation based on a chosen basis function is crucial for solving partial differential equations using the method of approximate particular solutions(MAPS).In general,the derivation of such a closed-form particular solution is by no means trivial,particularly for higher order and 3D partial differential equations.In this paper,we apply the MAPS based on the polynomial functions to solve the forth order partial differential equations through the use of particular solutions of Helmholtz-type equations.We give a simple algebraic procedure to avoid the direct derivation of the closed-form particular solutions for fourth order partial differential equations.Since the closed-form particular solutions for the second order partial differential equations with constant coefficients are known,the proposed solution procedure is simple and direct.The polynomial basis functions are well-known for yielding illconditioned systems when their order becomes large.The multiple scale technique is applied to circumvent the difficulty of ill-conditioning problem.We also applied the localized method of approximate particular solutions(LMAPS)based on the polynomial basis functions for solving axisymmetric problems.Since only the low order polynomial basis functions are used,no preconditioning treatment is required and the solution is quite stable.Five numerical examples are given to demonstrate the effectiveness of our proposed approach.