We have developed a three level implicit method for solution of the HELMHOLTZ EQUATION. Using the cubic spline in space and finite difference in time directions. The approach has been modified to drive Numerov type finite difference method. The method yield the tri-diagonal linear system of algebraic EQUATIONs which can be solved by using a tri-diagonal solver. Stability and error estimation of the presented method are analyzed. The obtained results satisfied the ability and efficiency of the method.

This paper is devoted to applying the sixth-order compact finite difference approach to the HELMHOLTZ EQUATION. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the compact finite difference method. Through this way, the computational costs of the method with large numbers of nodes are greatly reduced. The efficiency and accuracy of the scheme are investigated by solving some illustrative examples, having the exact solutions.

Finite and boundary element methods have been used by many authors to solve mathematical physics problems. However, the coupling of these two methods happens to be more efficient as it combines their merits. In this paper, the mathematical analysis of the coupling of finite and boundary element methods for the HELMHOLTZ EQUATION is presented.

The high costs of meshing, required problem dependent fundamental solutions, singularity, modeling all over the domain and …,are the most important weaknesses in the common numerical mesh methods for solving continuum mechanics problems. In this study, aiming for eliminating some of these shortcomings, one of the well-known Radial Basis Functions (RBF) methods, Multiquadric (MQ), was developed for analyzing 2D seismic waves in reservoirs of rigid dams. In this regard, the HELMHOLTZ EQUATION and the governing complex boundary conditions are reproduced using MQ function in the frequency domain. The results showed that using the real and complex forms of the MQ function, the computational time will be optimized for frequencies smaller and larger than the natural frequency of the reservoir, respectively. Also, to determine the most important factor in accuracy and convergence of MQ method, the optimal shape parameter, firstly the inefficiency of some of the previously introduced methods was shown, then a new high-speed algorithm was presented. The results of this research show that the optimal shape parameter can be formulated in terms of the frequencies of earthquake loads. This advantage simplifies considerably application of MQ method in this particular problem and reduces its computational time. The high accuracy of the present method compared to the exact solutions was shown in two different examples with and without considering effects of sediment absorption. The achieved high accuracy is due to a continuous estimation function defined all over the domain and using an exact algorithm for finding the optimal shape parameter.

The HELMHOLTZ free energy and EQUATION of the state of an fcc crystal are calculated, where the interaction between the molecules is hard sphere-Yukawa potential. Here the perturbational density functional method is used. This method is introduced by Ebner and co-workers. In this method the density functional Taylor expansion is applied for the crystal configuration up to second order. And for the uniform parts an exact expression is used. The results are compared with those obtained by Monte Carlo computer simulation. The agreement is good.

Magneto hydrodynamic waves, propagating along spicules, may become unstable and the expected instability is of Kelvin-HELMHOLTZ type. Such instability can trigger the onset of wave turbulence leading to an effective plasma heating and particle acceleration. In present study, two-dimensional magneto hydrodynamic simulations performed on a Cartesian grid is presented in spicules with different densities, moving at various speeds depending on their environment. Simulations being applied in this study show the onset of Kelvin-HELMHOLTZ type instability and transition to turbulent flow in spicules. Development of Kelvin-HELMHOLTZ instability leads to momentum and energy transport, dissipation, and mixing of fluids. When magnetic fields are involved, field amplification is also possible to take place.

Purpose: Saturated lights appear brighter than white lights of the same luminance. This is the HELMHOLTZ–Kohlrausch (H–K) effect, and the phenomenon can be estimated by modeling achromatic luminance and saturation to total brightness. Current H–K effect models are different between women and men and are also more variable in women, which may be due to hormonal changes across the menstrual cycle (MC). Methods: Total brightness (B) and achromatic luminance (L) were measured across blue, green, yellow-green, yellow, and red hues. These data were measured along with salivary hormone levels for nine cycling women and seven oral contraceptive (OC) users at points representing the menstrual, peri-ovulation, and luteal phases. Results: Simple brightness/luminance (B/L) ratio estimates of the H–K effect did not differ by OC use or MC phase, but B/L ratios were higher for the red stimulus in cycling women than OC users during the luteal phase. Estrogen, progesterone, and their interaction predicted 18% of the variation in brightness for cycling women. For OC users, only estrogen could be fit to brightness models where it accounted for 5% of brightness variance. Conclusion: These findings first provide clear support for separating cycling women from OC users, particularly when examining long-wavelength mechanisms. Next, the interaction of OC use and MC phase on B/L ratios for the red stimulus adds to a rich history of long-wavelength mechanisms. Lastly, the current result amends previous brightness models with multiple hormone terms for cycling women but not OC users.

The unsteady HELMHOLTZ type EQUATION of anisotropic functionally graded materials (FGMs) is considered in this study. The study is to find numerical solutions to initial boundary value problems governed by the EQUATION. A combined Laplace and boundary element method is used to solve the problems. The analysis derives a boundary-only integral EQUATION that is used to compute the numerical solutions. The analysis also results in another class of anisotropic FGMs of applications. Some problems are considered. The numerical solutions obtained are accurate and consistent.

Radial basis functions (RBFs) are a powerful method for obtaining the numerical solution of high-dimensional problems. They are often referred to as a meshfree method and the spectrally accurate can be achieved by them. In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion. We develop our approach in two-dimensional spaces for solving HELMHOLTZ EQUATIONs. In this paper, the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. Numerical examples are presented to demonstrate the effectiveness and robustness of the proposed method for solving two-dimensional HELMHOLTZ EQUATIONs.