The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [31], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This p...

The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in bounded domain. It demonstrated by combining shifted points outside domain and determining coefficients linear sum satisfy boundary condition on finite boundary. In this paper, existence MFS Neumann problems modified Helmholtz disk domains rigorously demonstrated. We...

We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ((+k 2)u = 0 on a Lipschitz domain for all wave number k 2C with Im k 0. Following the approach for the case of smooth boundary 3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothne...

A new analytical characterization of balls in the Euclidean space ℝm is obtained. Previous results this kind involved either harmonic functions or solutions to modified Helmholtz equation (both have positive fundamental solutions), whereas are used here. This achieved at expense a restriction imposed on size admissible domains—a feature absent inverse mean value properties known previously.