The numerical solution of boundary value problems for ordinary differential equations on infinite intervals is considered. The infinite interval is cut at a finite, large enough point and additional boundary conditions are posed there. For the solution of the resulting problem, /(-stable symmetric collocation methods are employed. Using the behavior of the solution of the "infinite" problem, me...

In this paper we investigate the application of radial basis functions to solve variational problems as they appear in the weak formulation of elliptic partial diierential equations. Two algorithms based on a multilevel scheme are introduced. Their numerical advantages and disadvantages are discussed and demonstrated on an example. x1. Introduction The classical Galerkin approach for the soluti...

Meshfree methods are viewed as next generation computational techniques. With evident limitations of conventional grid based methods, like FEM, in dealing with problems of fracture mechanics, large deformation, and simulation of manufacturing processes, meshfree methods have gained much attention by researchers. A number of meshfree methods have been proposed till now for analyzing complex prob...

This paper is concerned with the hp-adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle–partition of unity method. The proposed refinement scheme automatically constructs new discretization points (or particles), the meshfree analogue of an adaptive h-refinement, and local approximation spaces with better local resolution, a p-refinem...

Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and nume...

Lane-Emden differential equations of order fractional has been studied. Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analytic methods has been introduced.

The point collocation method of finite spheres (PCMFS) is used to model the hyperelastic response of soft biological tissue in real time within the framework of virtual surgery simulation. The proper orthogonal decomposition (POD) model order reduction (MOR) technique was used to achieve reduced-order model of the problem, minimizing computational cost. The PCMFS is a physics-based meshfree num...

In this paper we discuss the collocation method for a large class of Fredholm linear integro-differential equations. It will be shown that, when a certain higher order interpolation operation is added to the collocation solution of this equation, the new approximations will, under suitable assumptions, admit a multiterm error expansion in even powers of the step-size h. Based on this expansion,...

We introduce a single patch collocation method in order to compute solutions of initial value problems of partial differential equations whose spatial domains are 3spheres. Besides the main ideas, we discuss issues related to our implementation and analyze numerical test applications. Our main interest lies in cosmological solutions of Einstein’s field equations. Motivated by this, we also elab...

Graphics Processing Units (GPUs), originally developed for computer games, now provide computational power for scientific applications. A study on the comparison of computational speed-up and efficiency of a GPU with a CPU for the Finite Pointset Method (FPM), which is a numerical tool in Computational Fluid Dynamics (CFD) is presented. As FPM is based on the point cloud, it is so expensive whe...