In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using...

A combined compact difference scheme is proposed for linear second-order partial differential equations with mixed derivative. The scheme is based on a nine-point stencil at interior with sixth-order accurate local truncation error. Fourier analysis is used to analyze the spectral resolution of the proposed scheme. Numerical tests demonstrate at least sixth-order convergence rate with Dirichlet...

The Incompressible Smoothed Particle Hydrodynamic (ISPH) is one of the particle methods and commonly used to solve some complicated physical problems including free surface flow problems. The study regarding the boundary treatment has become an active research area in the mesh-free or particle method recently for measuring the accurate and robust pressure near the boundary. The penetrations of ...

We consider a competition-diffusion system for two competing species; the density of the first species satisfies a parabolic equation together with a inhomogeneous Dirichlet boundary condition whereas the second one either satisfies a parabolic equation with a homogeneous Neumann boundary condition, or an ordinary differential equation. Under the situation where the two species spatially segreg...

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane. These boundary value problems arise in a study of timeharmonic acoustic scattering of an incident eld by a sound-soft, in nite rough surface where the total eld vanishes (the Dirichlet problem) or by an in nite, impedance rough surface where the total ...

We consider the boundary value problem ∆u + |x| 2α u p = 0, α > 0, in the unit ball B with homogeneous Dirichlet boundary condition and p a large exponent. We find a condition which ensures the existence of a positive solution up concentrating outside the origin at k symmetric points as p goes to +∞. The same techniques lead also to a more general result on general domains. In particular, we ha...

Our domain G = (0,L) is an interval of length L. The boundary ∂G = {0,L} are the two endpoints. We consider here as an example the case (DD) of Dirichlet boundary conditions: Dirichlet conditions at x = 0 and x = L. For other boundary conditions (NN), (DN), (ND) one can proceed similarly. In one dimension the Laplace operator is just the second derivative with respect to x: ∆u(x, t) = uxx(x, t)...

In this paper, we introduce the dissipative spectral methods (DSM) for the first order linear hyperbolic equations in one dimension. Specifically, we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition. The error estimates of the methods are shown to be quasioptimal for variable-coefficients equations. Numerical results are giv...