Existence of solutions for a damped nonlinear impulsive problem with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for second‐order nonlinear damped impulsive differential equations with Dirichlet boundary condition. By critical point theory, we obtain some existence theorems of solutions for the nonlinear problem. We extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

ERROR ESTIMATION FOR NUMERICAL METHODS USING THE ULTRA WEAK VARIATIONAL FORMULATION IN MODEL OF NEAR FIELD SCATTERING PROBLEM

In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.     

Motion of a vortex filament with axial flow in the half space

We consider a nonlinear third order dispersive equation which models the motion of a vortex filament immersed in an incompressible and inviscid fluid occupying the three dimensional half space. We prove the unique solvability of initial–boundary value problems as an attempt to analyze the motion of a tornado.     

A new model for the analysis of reinforced concrete members with a coupled HdBNM/FEM

As a truly boundary-type meshless method, the hybrid boundary node method (HdBNM) does not require ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. In this paper, the HdBNM is coupled with the finite element method (FEM) for predicting the mechanical behaviors of reinforced concrete. The steel bars are considered as body forces in the concrete. A bond model is presented to simulate the bond-slip between the concrete and steels using fictitious spring elements. The computational scale and cost for meshing can be further reduced. Numerical examples, in 2D and 3D cases, demonstrate the efficiency of the proposed approach.     

An accurate application of the integral method applied to the diffusion of oxygen in absorbing tissue

Accurate integral methods are applied to a one dimensional moving boundary problem describing the diffusion of oxygen in absorbing tissue. These methods have been well studied for classic Stefan problems but this situation is unusual because there is no condition which contains the velocity of the moving boundary explicitly. This paper begins by giving a short time solution and then discusses some of the previous integral methods found in the literature. The main drawbacks of these solutions are that they cannot be solved from $t=0$ and also cannot determine the end behaviour. This is due to the non-uniform initial profile which integral methods typically fail to capture. The use of a novel transformation removes this non-uniformity and, on applying optimal integral methods to the resulting system, leads to simple and yet very accurate approximate solutions that overcome the deficiencies of previous methods.     

Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions

In this paper, we propose complete radiation boundary conditions (CRBCs) for solutions of the convected Helmholtz equation with a uniform mean flow in a waveguide. We first study CRBCs for the Helmholtz equation in a waveguide. Noting that the convected Helmholtz equation is associated with the Helmholtz equation via the Prandtl–Glauert transformation, CRBCs for the convected Helmholtz equation is derived from CRBCs for the Helmholtz equation. We analyse well-posedness and convergence of approximate solutions satisfying CRBCs for the convected Helmholtz equation. In addition, simple numerical experiments will be presented to confirm the theoretical results.     

On the dynamics of radially symmetric granulomas

A granuloma is a collection of macrophages that contains bacteria or other foreign substances that the body?s immune response is unable to eliminate. In this paper we present a simple mathematical model of radially symmetric granuloma dynamics. The model consists of a coupled system of two semi-linear parabolic equations for the macrophage density, and the bacterial density. The boundary of the granuloma is free. This simple framework makes it possible to conduct a mathematical analysis of the system dynamics. In particular, we show that the model system has a unique solution, and that, depending on the biological parameters; the bacterial load either disappears over time or persists. We use numerical methods to establish the existence of stationary solutions and examine how a stationary solution changes with the reproductive rate of the bacteria. These simulations show that the structure of the granuloma breaks down as the reproductive rate of the bacteria increases.     

Debt–equity swap with finite time horizon—variational inequality approach

This paper concerns the finite-horizon optimal reorganization problem under debt–equity swap. The model of equity is formulated as a parabolic variational inequality, or equivalently, a free boundary problem, where the free boundary corresponds to the optimal reorganization boundary. The existence and uniqueness of the solution are proven and the behavior of the free boundary, such as smoothness, monotonicity and boundedness, is studied. To the best of our knowledge, this is the first complete set of results on debt–equity swap for finite maturity obtained using PDE techniques.