Direct Monte Carlo simulation of internal energies of departure for binary mixtures of geological interest are gathered and compared to those calculated using a linear mixing rule. Simulation results for gas–oil, oil–oil, and oil–water mixtures show that the linear mixing rule used in the Gibbs–Helmholtz Constrained (GHC) equation of state framework gives accurate approximations of binary mixtu...

The Helmholtz equation arises in the study of electromagnetic radiation, optics, acoustics, etc. In spherical coordinates, its general solution can be written as a spherical harmonic series which satisfies the radiation condition at infinity, ensuring that the wave is outgoing. The boundary condition at infinity is hard to enforce with a finite element method since a suitable approximation need...

A three-dimensional Cartesian parabolic-equation model with a higher-order approximation to the square-root Helmholtz operator is presented for simulating underwater sound propagation in ocean waveguides. The higher-order approximation includes cross terms with the free-space square-root Helmholtz operator and the medium phase speed anomaly. It can be implemented with a split-step Fourier algor...

Our purpose of this paper is to study isolated singular solutions semilinear Helmholtz equation \begin{document}$ -\Delta u-u = Q|u|^{p-1}u \quad{\rm in}\ \ \mathbb{R}^N\setminus\{0\},\ \qquad\lim\limits_{|x|\to0}u(x) +\infty, $\end{document} where N\geq 2 $\end{document}, p>1 and the potential Q: \mathbb{R}^N\to (0,+\infty) a Hölder continuous function satisfying extra decaying conditions a...