Recently ([21]), a new meshfree approximation method for Darcy’s problem has been introduced and analyzed. This method is based on a symmetric collocation approach using radial basis functions producing solutions with an analytically divergence-free velocity part. However, the error analysis provided in [21] works only for smooth solutions, where the smoothness is intrinsically linked to the sm...

In this paper we propose a meshfree technique for the numerical solution of the two dimensional Burger’s equation. Collocation method using the Radial Basis Functions (RBFs) is coupled with first order accurate finite difference approximation. Different types of RBFs are used for this purpose. Performance of the proposed method is successfully tested in terms of various error norms. In the case...

Abstract In this paper, a new model for the below-boiling point evaporation process with meshfree collocation method is developed. order to capture phase change process, two different approaches are proposed: multi-phase and single-phase. First, approach considered, where novel mass transfer assumes that diffusion driven by vapor concentration gradient in air near interface primary driving forc...

This work presents a global radial basis collocation combining with the quasiNewton iteration method for solving semilinear elliptic partial differential equations. A convergence analysis for such a meshfree discretization has been established. The main result is that there exists an exponential convergence rate with respect to the number and the shape of the radial basis functions. In addition...

This paper is an extension of previous work where we laid the foundation for the kernel-based collocation solution of stochastic partial differential equations (SPDEs), but dealt only with the simpler problem of right-hand-side Gaussian noises. In the present paper we show that kernel-based collocation methods can be used to approximate the solutions of high-dimensional elliptic partial differe...

We present a meshless technique which can be seen as an alternative to the Method of Fundamental Solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artifical boundary needed, and there is a f...

In this article, we present a novel meshfree framework for fluid flow simulations on arbitrarily curved surfaces. First, introduce new Lagrangian to model Meshfree points or particles, which are used discretize the domain, move in sense along given surface. This is done without discretizing bulk around surface, parametrizing and background mesh. A key novelty that introduced handling of with ev...

Meshfree Collocation Method is used to solve linear two-dimensional problems. This method differs from weak form methods such as Galerkin method and no cellular networking and no numerical  integration. Therefore, this method has no constraints such as the integration accuracy and the integration CPU time, and equations can be isolated directly from the strong form of governing PDE. The fundame...

In the theoretical part of this paper, we introduce a simplified proof technique for error bounds and convergence of a variation of E. Kansa’s well-known unsymmetric meshless collocation method. For a numerical implementation of the convergent variation, a previously proposed greedy technique is coupled with linear optimization. This algorithm allows a fully adaptive on-the-fly data-dependent m...

In this paper, we introduce a radial basis function collocation method for computing solutions to the time-dependent radiative transfer equation. For these computations, we use finite differences to discretize the time coordinate, a discrete ordinate method to discretize the directional variable, and an expansion in radial basis functions to approximate the spatial dependence of the solution. W...