Taylor Series Based Domain Collocation Meshless Method for Problems with Multiple Boundary Conditions including Point Boundary Conditions

Abstract

Many sophisticated real world science and engineering problems after formulation simply reduce to a problem of finding a solution of partial differential equations (PDEs) with relevant boundary conditions over a domain. Numerical methods like FEM, FDM and BEM are most used and popular methods to solve these real-world PDEs. However, in last few decades considerable amount of research has been dedicated to develop meshless methods which don’t involve tedious and time consuming process of generating high quality mesh for the domain. Many of these meshless methods have difficulty in handling point boundary conditions which are quite frequent in engineering applications. Hence, in this paper, a Taylor series based domain collocation PDE solution methodology is proposed. The proposed methodology is well suited to handle multiple boundary conditions including point boundary conditions. The main idea of the method is to formulate a function which satisfies all the boundary conditions and then generalize the function to a family of functions by using Taylor series. Since the family of functions already satisfies the boundary conditions, the PDE solution can be determined by finding the values of unknown Taylor coefficients for which the residual of the PDE over the domain is closest to zero. Using domain collocation method, the linear PDE problem transforms into a linear regression problem. The proposed method is extended by using multi-point Taylor series to solve problems with point boundary conditions. The proposed method has been successfully applied to solve homogenous/non-homogenous Helmholtz and Poisson’s PDEs in the paper. The proposed methodology has also been shown to solve complex PDEs efficiently with less number of degrees of freedom (DOFs) as compared to Taylor meshless method (TMM). The proposed method is illustrated for both problems with Dirichlet and Neumann boundary conditions. Moreover, the method has been also presented to solve a problem where the boundary is defined using a set of points instead of an analytical function.