Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collocation methods have, under many conditions, eigenvalues with positive real part, preventing the use of such methods for time-dependent problems. We explore this difficulty at theoretical and practical levels. Theoretically, we prove that differentiation matrices for conditionally positive definite RBFs are...

In this article, we consider the discretization of the time-dependent Schrödinger equation using radial basis functions (RBF). We formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions. Since we can show that time-stability of the discretization is not guaranteed for an RBF-collocation method, we propose to employ a Galerkin ansatz instead. For ...

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three RBFs, including Gaussian, multiquadric (MQ), and inverse (IMQ) without the shape parameter, are adopted in study. With consideration of many outside domain, distance RBF is always greater than zero, such that can remove parameter...

In this paper, we propose the spectral collocation method based on radial basis functions to solve the fractional Bagley-Torvik equation under uncertainty, in the fuzzy Caputo's H-differentiability sense with order ($1< nu < 2$). We define the fuzzy Caputo's H-differentiability sense with order $nu$ ($1< nu < 2$), and employ the collocation RBF method for upper and lower approximate solutions. ...

In this paper, we discuss multiscale radial basis function collocation methods for solving certain elliptic partial differential equations on the unit sphere. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. Two variants of the collocation method are considered (sometimes ...

This paper introduces an effective H-adaptive upgrade to solution of the transport phenomena by the novel Local Radial Basis Function Collocation Method (LRBFCM). The transport variable is represented on overlapping 5-noded influence-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the variable are calculated from the ...

Several analytical and numerical methods exist to solve the orbit propagation of the two-body problem. Analytic solutions are mainly implemented for the unperturbed/classical two-body problem. Numerical methods can handle both the unperturbed and the perturbed two-body problem. The literature is rich with numerical methods addressing orbit propagation problems such as, Gauss-Jackson, Higher ord...

Symmetric multiscale collocation methods with radial basis functions allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. In particular, the condition number an...