A Meshless Method Based on Radial Basis and Spline Interpolation for 2-D and 3-D Inhomogeneous Biharmonic BVPs

Abstract

This paper presents a meshless method which, at the first step, utilises the radial basis functions collocation scheme to approximate the unknown function at specific nodal points. The difficulty of these biharmonic-type problems is the multiple boundary conditions, as well as high derivatives terms. The inhomogeneous biharmonic equation is replaced by two Poisson equations of an intermediate function where Neumann's boundary conditions is of second derivatives. It uses the imposed-kernel technique (IKT) to overcome multiple boundary conditions where Neumann's boundary conditions is of first derivatives. In the next step, our meshless approach uses two-and three-dimensional cubic spline interpolation to get smooth, approximate solutions for the 2-D and 3-D inhomogeneous biharmonic-type problem, respectively. Numerical experiments are included to demonstrate the reliability and efficiency of this method.

Adaptive Numerical Method for Approximation of Traffic Flow Equations

“…(1) Mesh-free methods based on weak forms, such as the element-free Galerkin method (EFG) [28] (2) Mesh-free methods based on strong forms, such as collocation method based on radial basis functions (RBFs) [29,30] (3) Mesh-free methods based on a combination of weak forms and collocation method [31,32] In the written works, several methods without weak mesh are stated which are as follows:…”

Meshless formulation to two‐dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence

“…Compared with the finite element method (FEM), finite volume method (FVM) and boundary element method (BEM) [1,2], meshfree methods are applied to set up system of algebraic equations for entire problem domain with no need to meshing of the domain discretization in order to use a set of points scattered inside the domain of the problem such as sets of points on the boundaries of the domain to show the domain of the problem and its boundaries. Moreover, some meshless methods are on the basis of collocation approaches (strong forms) like the meshless collocation method based on radial basis functions (RBFs) [3][4][5][6][7][8][9] and some other kinds of meshfree methods regarding weak forms and hybrid of collocation approach and weak forms, like element free Galerkin (EFG) [10,11] and meshless local Petrov-Galerkin (MLPG) [12][13][14]. Another approach has been utilized for approximated solution of differential equations is spectral collocation method like the spectral meshless radial point interpolation (SMRPI) method [15][16][17] the point interpolation method by means of the RBFs is employed to form shape functions.…”

Local radial basis function interpolation method to simulate 2D fractional‐time convection‐diffusion‐reaction equations with error analysis

“…A. Shirzadi et al considered the same problem in both weak convection coefficients and convection dominant cases and applied meshless local Petrov‐Galerkin (MLPG) method to obtain better results but MLPG is time consuming as the result of determining shape parameter and calculating integration around each nodal point, the readers are referred to the Refs. to survey meshless methods. Z. J. Fu et al have considered the problem (1)–(4) and proposed Laplace transformed boundary particle method to solve it effectively.…”