Rate of convergence of the method of fundamental solutions and hyperinterpolation for modified Helmholtz equations on the unit ball

Approximate solutions of boundary value problems of homogeneous modified Helmholtz equations on the unit ball are explicitly constructed by the method of fundamental solutions (MFS) with the order of approximation provided. Hyperinterpolation is used to find particular solutions of non-homogeneous equations, and the rate of convergence of solving boundary value problems of non-homogeneous equations is derived. Numerical examples are shown to demonstrate the efficiency of the methods.      

Multiple periodic oscillations in a nonlinear suspension bridge system

In this paper, we study periodic oscillations in a suspension bridge system governed by the coupled nonlinear wave and beam equations describing oscillations in the supporting cable and roadbed under periodic external forces. By applying a variational reduction method, it is proved that the suspension bridge system has at least three periodic oscillations.     

Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions

By extending Wendlands meshless Galerkin methods using RBFs, we develop mixed methods for solving fourth-order elliptic and parabolic problems by using RBFs. Similar error estimates as classical mixed finite element methods are proved. AMS subject classification 35G15, 65N12     

Estimates at infinity for positive solutions to a class of p-Laplacian problems in RN

In this paper we consider a class of logistic-type problems for the p-Laplacian in the whole space. Using minimization we prove existence of a positive solution and its behavior at infinity. We also consider questions of uniqueness and sharp estimates at infinity.     

A new FDTD scheme for Maxwell’s equations in Kerr-type nonlinear media

Numerical Algorithms - In this paper, we develop a totally new direct finite difference solver for solving the Maxwell’s equations in Kerr-type nonlinear media. The direct method is free of...     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities

In this note we provide simple and short proofs for a class of inequalities of Caffarelli-Kohn-Nirenberg type with sharp constants. Our approach suggests some definitions of weighted Sobolev spaces and their embedding into weighted L² spaces. These may be useful in studying solvability of problems involving new singular PDEs.