Boundary integral equation methods for the elastic and thermoelastic waves in three dimensions

In this paper, we consider the boundary integral equation (BIE) method for solving the exterior Neumann boundary value problems of elastic and thermoelastic waves in three dimensions based on the Fredholm integral equations of the first kind. The innovative contribution of this work lies in the proposal of the new regularized formulations for the hyper-singular boundary integral operators (BIO) associated with the time-harmonic elastic and thermoelastic wave equations. With the help of the new regularized formulations, we only need to compute the integrals with weak singularities at most in the corresponding variational forms of the boundary integral equations. The accuracy of the regularized formulations is demonstrated through numerical examples using the Galerkin boundary element method (BEM).Comment: 24 pages, 6 figure

Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity

Retarded stochastic differential equations (SDEs) constitute a large collection of systems arising in various real-life applications. Most of the existing results make crucial use of dissipative conditions. Dealing with "pure delay" systems in which both the drift and the diffusion coefficients depend only on the arguments with delays, the existing results become not applicable. This work uses a variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. This paper establishes existence and uniqueness of stationary distributions for retarded SDEs that need not satisfy dissipative conditions. The retarded SDEs considered in this paper also cover SDEs of neutral type and SDEs driven by L\'{e}vy processes that might not admit finite second moments.Comment: page 2

Exponential Mixing for Retarded Stochastic Differential Equations

In this paper, we discuss exponential mixing property for Markovian semigroups generated by segment processes associated with several class of retarded Stochastic Differential Equations (SDEs) which cover SDEs with constant/variable/distributed time-lags. In particular, we investigate the exponential mixing property for (a) non-autonomous retarded SDEs by the Arzel\`{a}--Ascoli tightness characterization of the space \C equipped with the uniform topology (b) neutral SDEs with continuous sample paths by a generalized Razumikhin-type argument and a stability-in-distribution approach and (c) jump-diffusion retarded SDEs by the Kurtz criterion of tightness for the space \D endowed with the Skorohod topology.Comment: 20 page

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.0288