Existence and multiplicity of solutions for anisotropic elliptic equation

In this article we study the nonlinear problem $$\left\{ \begin{array}{lr} -\sum_{i=1}^{N}\partial_{x_{i}}a_{i}(x,\partial_{x_{i}}u)+ b(x)~|u|^{P_{+}^{+}-2}u =\lambda f(x,u) \quad in \quad \Omega\\ u=0 \qquad on \qquad \partial\Omega \end{array} \right.$$ Using the variational method, under appropriate assumptions on $f$, we obtain a result on existence and multiplicity of solutions.

Differential inequality and non-oscillation of fourth order differential equation

The oscillatory theory of fourth order differential equations has not yet been developed well enough. The results are known only for the case when the coefficients of differential equations are power functions. This fact can be explained by the absence of simple effective methods for studying such higher order equations. In this paper, the authors investigate the oscillatory properties of a class of fourth order differential equations by the variational method. The presented variational method allows to consider any arbitrary functions as coefficients, and our main results depend on their boundary behavior in neighborhoods of zero and infinity. Moreover, this variational method is based on the validity of a certain weighted differential inequality of Hardy type, which is of independent interest. The authors of the article also find two-sided estimates of the least constant for this inequality, which are especially important for their applications to the main results on the oscillatory properties of these differential equations.

Application of Sasaki's numerical variational technique to the analysis of height and wind fields over Indian region

An objective analysis method based on Sasaki's numerical variational analysis technique has been taken up for the analysis of geopotential height and wind over the Indian region. The univariate optimum interpolation (UOI) method is used to generate the initial or input fields. These fields are then adjusted by the variational method. A study of this method over Indian and adjoining region for 850, 700, 500, 300 and 200 hPa levels is made from 4 to 8 July 1979 and the analyses obtained using this method are compared with the FGGE analyses.

Dynamic Analysis of a Rotating Double-Tapered FGM Beam with Various Shear Models Using a Constrained Variational Method

A constrained variation modeling method for free vibration analysis of rotating double tapered functionally graded beams with different shear deformation beam theories is proposed in this paper. The material properties of the beam are supposed to continuously vary in the width direction with power-law exponent for different indexes. The mathematical formulation is developed based on the geometrically exact beam theory for each beam segment, the admissible functions denoting motion quantities are then expressed by a series of Chebyshev orthogonal polynomials. The governing equations are eventually derived using the constrained variational method to involve the continuity conditions of adjacent segments. Different shear deformation beam theories have been incorporated in the formulations, and the nonlinear effect of bending–stretching coupling vibration together with the Coriolis effect is taken into account. Comparison of dimensionless natural frequencies is performed with the existing literature to ensure the accuracy and reliability of the proposed method. Comparative discussions are performed on the vibration behaviors of the double tapered rotating functionally graded beam with first-order shear deformation beam theory and other higher-order shear deformation beam theories. The effect of material property graduation, power-law index, rotation speed, hub radius, slenderness ratio, and taper ratios is scrutinized via parametric studies, respectively.

Data assimilation by the variational method with results for the Indian region

ABSTRACT. The variational method for data assimilation as implemented in the operational scheme at ECMWF is briefly presented. The performance of the variational scheme (3D-Var) with respect to tropical cyclones and the Asian summer monsoon is investigated and compared to the Optimum Interpolation scheme. It is found that the analysis of near-surface winds has improved significantly particularly in the vicinity of tropical storms and depressions. The better analyses have led to improvements in the short range forecasts (day 1 to day 3) of such systems. The summer monsoon appears slightly stronger in the 3D-Var analyses, giving enhanced forecast precipitation over the Western Ghats and over large parts of northern India. Only in the latter of these two areas does this verify with observations. The forecasts for India of geopotential, wind and temperature have improved significantly at all forecast ranges, as verified against own analyses. These results are based on 28 cases in two separate 2-week periods.

The Fekete-Szego problem by a variational method

The article is devoted to the well-known Fekete and Szego problem. The paper investigate the problem in sufficient detail using some new observations by the classical method of internal variations, developed at the Tomsk School of Complex Analysis. One particular case is considered. We carried out complete qualitative analysis of the functional-differential equation relative boundary mapping. We completely solved the problem for the real parameter.

Variational approach to the Schrödinger equation with a delta-function potential

We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.