In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized Schrödinger operator.

In this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.

In this work we utilize the boundary integral equation and the Dual Reciprocity Boundary Element Method (DRBEM) for the solution of the steady state convection-diffusion-reaction equations with variable convective coefficients in two-dimension. The DRBEM is a numerical method to transform the domain integrals into the boundary only integrals by using the fundamental solution of Helmholtz equation. Some examples are calculated to confirm the accuracy of the approach. The results obtained by the analytic solutions are in good agreement with ones provided by the DRBEM technique.

We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn . The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.

The paper is to solve the problem of the response of stratified half-space subjected to triangularly distributed time-harmonic loadings on an axial symmetric area in the stratified half-space. Since the distributed area can be very small, the solution can be employed to deal with the problems of elastodynamics using the fundamental solution method. The advantage of the presented solution over Green function is that no singularity will occur in the presented solution. Some numerical results are given and some conclusions have been made. The presented solution is very efficient in computation.

In this paper, an improved singular boundary method (SBM), viewed as one kind of modified method of fundamental solution (MFS), is firstly applied for the numerical analysis of two-dimensional (2D) Stokes flow problems. The key issue of the SBM is the determination of the origin intensity factor used to remove the singularity of the fundamental solution and its derivatives. The new contribution of this study is that the origin intensity factors for the velocity, traction and pressure are derived, and based on that, the SBM formulations for 2D Stokes flow problems are presented. Several examples are provided to verify the correctness and robustness of the presented method. The numerical results clearly demonstrate the potentials of the present SBM for solving 2D Stokes flow problems.

This study proposes a new formulation of singular boundary method (SBM) to solve the 2D potential problems, while retaining its original merits being free of integration and mesh, easy-to-program, accurate and mathematically simple without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). The key idea of the SBM is to introduce the concept of the origin intensity factor to isolate the singularity of fundamental solution so that the source points can be placed directly on the physical boundary. This paper presents a new approach to derive the analytical solution of the origin intensity factor based on the proposed subtracting and adding-back techniques. And the troublesome sample nodes in the ordinary SBM are avoided and the sample solution is also not necessary for the Neumann boundary condition. Three benchmark problems are tested to demonstrate the feasibility and accuracy of the new formulation through detailed comparisons with the boundary element method (BEM), MFS, regularized meshless method (RMM) and boundary distributed source (BDS) method.

In this paper, a $W^{-1,N'}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on $\mathbb R^N$, or on a regular bounded open set of $\mathbb R^N$. The proof is based partially on the Strauss inequality [Strauss, Partial Differential Equations: Proc. Symp. Pure Math. 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [J. Eur. Math. Soc. 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and to prove an existence result for the stationary Navier-Stokes equation when the viscosity tensor is only in L1.

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R2, is given.