A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain d...

In this paper we present the theoretical framework needed to justify the use of a kernelbased collocation method (meshfree approximation method) to estimate the solution of highdimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on...

This paper solves the Laplace equation ∆u = 0 on domains Ω ⊂ R by meshless collocation on scattered points of the boundary ∂Ω. In contrast to the Method of Fundamental Solutions, there are no singularities and no artificial boundaries, since we use new singularity–free positive definite kernels which are harmonic in both arguments. In contrast to many other techniques, e.g. the Boundary Point M...

Though the technique introduced by E. Kansa [7, 8] is very successful in engineering applications, there were no proven results so far on the unsymmetric meshless collocation method for solving PDE boundary value problems in strong form. While the original method cannot be proven to be fail–safe in general, we prove asymptotic feasibility for a generalized variant using separated trial and test...

In this paper, attempt is made to solve a few problems using the Polynomial Point Collocation Method (PPCM), the Radial Point Collocation Method (RPCM), Smoothed Particle Hydrodynamics (SPH), and the Finite Point Method (FPM). A few observations on the accuracy of these methods are recorded. All the simulations in this paper are three dimensional linear elastostatic simulations, without account...

The numerical method for solving the nonlinear eigenvalue problem has been developed by using the collocation Element-Free Galerkin Method (EFGM) and its performance has been numerically investigated. The results of computations show that the approximate solution of the nonlinear eigenvalue problem can be obtained stably by using the developed method. Therefore, it can be concluded that the dev...

This paper presents an efficient meshless approach for solving electrostatic problems. This novel approach is based on combination of radial basis functions-based meshless unsymmetric collocation method with projection domain decomposition method. Under this new method, we just need to solve a Steklov-Poincaré interface equation and the original problem is solved by computing a series of indepe...

The set Λ consists of infinitely many linear real–valued functionals λ that usually take the form of point evaluations of functions or derivatives at points inside a domain or on some boundary or interface layer. If several differential or boundary operators are involved, we simply put everything into a single set Λ of functionals of various types. We call (1) a generalized interpolation proble...

In this paper we report some recent advances regarding applications using the method of finite spheres; a truly meshfree numerical technique developed for the solution of boundary value problems on geometrically complex domains. First, we present the development of a preprocessor for the generation of nodal points on twodimensional computational domains. Then, the development of a specialized v...