The paper deals with the numerical solution of initial-boundary value problems for diffusion equation with variable coefficients by using a local weak formulation and a meshless approximation of spatial variations of the field variable. The time variation is treated either by the Laplace transform technique or by the linear Lagrange interpolation in the time stepping approach. Advanced formulat...

While the generation of meshes has always posed challenges for computational scientists, the problem has become more acute in recent years. Increased computational power has enabled scientists to tackle problems of increasing size and complexity. While algorithms have seen great advances, mesh generation has lagged behind, creating a computational bottleneck. For industry and government looking...

Here are three short stories on meshless methods using kernel techniques: • Any well–posed linear problem in the native space NΦ of a symmetric (strictly) positive definite kernel Φ can be successfully solved by symmetric meshless collocation. This applies to a large variety of standard linear PDE problems. • Relaxing interpolation conditions by allowing some small absolute error can significan...

Meshless methods have been extensively popularized in literature in recent years, due to their flexibility in solving boundary value problems. The meshless local Petrov-Galerkin(MLPG) method for solving the bending problem of the thin plate is presented and discussed in the present paper. The method uses the moving least-squares approximation to interpolate the solution variables, and employs a...

Radial Basis Functions (RBF) interpolation theory is briefly introduced at the “application level” including some basic principles and computational issues. The RBF interpolation is convenient for un-ordered data sets in n-dimensional space, in general. This approach is convenient especially for a higher dimension N 2 conversion to ordered data set, e.g. using tessellation, is computationally v...

This paper examines the numerical solution of the nonlinear coupled Burger’s equations with various values of viscosity by local meshless methods. The local radial basis functions collocation method (LRBFCM) belongs to the class of truly meshless methods which do not need any underlying mesh but work on a set of uniform or random nodes only, without any a priori node to node connectivity. The n...

We use the first-order shear deformation theory and a meshless method based on radial basis functions in a pseudospectral framework for predicting the free vibration behavior of thick orthotropic, monoclinic and hexagonal plates. The shape parameter is obtained by an optimization procedure. The three translational and two rotational degrees of freedom of a point of the laminate are independentl...

We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the me...

Partial differential equations (PDEs) on surfaces arise in a variety of application areas including biological systems, medical imaging, fluid dynamics, mathematical physics, image processing and computer graphics. In this paper, we propose a radial basis function (RBF) discretization of the closest point method. The corresponding localized meshless method may be used to approximate diffusion o...