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The semi-analytical modeling of acoustic wave resonators is discussed in this study. A polynomial approach to wave propagation is used to predict the electromechanical behavior of piezoelectric resonators in a circular ring with hexagonal elastic symmetry. We present a Legendre polynomial-based 3D semi-analytical technique capable of accounting for finite lateral dimensions. Using delta functions, the boundary conditions are immediately integrated into the wave equations. The Legendre polynomial method's validity is confirmed by comparing polynomial results with finite element method (FEM) results from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we employ a comparative study of the orthogonal collocation variational iteration method (OCVIM) and variational iteration method (VIM) for the solution of the one dimensional nonlinear Burger's equations through the use of certain orthogonal polynomials (Legendre and first kind Chebyshev polynomials of order one). The legendre's polynomial as a trial functions are more preferable than the Chebyshev polynomial as applicable to the one dimensional nonlinear Burger's equations. Here, the proposed methods were implemented vial a trial function for the initial approximate estimation through the orthogonal collocation approach. The complete solution sets (approximate solution) are thus obtained using the variational iteration method (VIM) in the approximation of the analytic solutions for the proposed problems. Numerical experimentations were carried out for the accuracy, reliability and convergence of the method for the proposed problems. Interesting results have been obtained with the aid of MAPLE 18 via the laid down algorithms which clearly show that the OCVIM is a superior numerical solver than the traditional variational iteration method (VIM). The results also show that the orthogonal collocation variational iteration method is an accurate and reliable solver for the proposed problems with excellent rate of convergence as compared with those available in literature. [ABSTRACT FROM AUTHOR]

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for unconditional basis in L2 terms of the norms of the elements of these systems. [ABSTRACT FROM AUTHOR]

While several papers deal with effects of truncation techniques on the total radiance there is a lack of such studies concerning polarized radiance. We present our first results obtained in studying the impact of phase matrix truncation on the accuracy of polarized radiance. Like for the total radiance, the single scattering (TMS) correction is crucial and efficient in case of the polarized radiance. We show that there is a systematic error on polarized radiance that increases with truncation factor and optical depth. This error is not induced by the restitution of the phase matrix angular dependency as a function of the number of terms in the Legendre polynomials expansion. Given a truncation factor, this error reaches a maximum asymptotic value for optical depth corresponding to the saturation of polarization. For the cloud droplet that we use in our study, the maximum error is 4.5% at λ=412nm. It raises the question of whether the condition for scale invariance of the scalar radiative transfer equation (RTE) can apply to vector RTE. [ABSTRACT FROM AUTHOR]

A theoretical study on the shape instability of a slightly deformed bubble encapsulated by a viscoelastic membrane in an ultrasound field is performed. The membrane effects of the inplane stress and the bending moment are incorporated into the traction jump condition at the bubble surface. The spherical motion of the bubble is numerically obtained by solving the Rayleigh-Plesset equation with the elastic stress. The deflection therefrom is linearized and expanded with respect to the Legendre polynomial. Two amplitudes for each shape mode are introduced because the membrane has mobility not only in the radial direction but also in the tangential direction. A simple expression for the natural frequency of shape mode is derived. Stability diagrams for the higherorder shape mode are mapped out in the phase space of driving amplitude versus driving frequency. The most unstable driving frequency is found to satisfy an integer multiple relationship with twice of the higher-order natural frequency. This finding is justified by a fact that the system with a boundary layer approximation is simplified into Mathieu's equation. Liquid viscosity plays an important role in the shape stability due to the vorticity generation on the deformed membrane. [ABSTRACT FROM AUTHOR]

I compute linear, adiabatic pulsation frequencies for axisymmetric and nonaxisymmetric low order p modes to examine the rotational splitting. The calculations are not restricted to low rotation rates or to a single Legendre polynomial representation of the horizontal variation of the eigenfunction. At low rotation, the splitting is proportional to the rotation rate and the azimuthal order of the mode (m), as one would expect. At higher rotation rates the splitting ceases to be linear as higher order terms begin to influence the splitting. This first manifests itself as a difference in the magnitude of the splitting between positive and negative values of m. Still higher rotation rates change the spacing between adjacent values of m. Differential rotation affects rotational splitting such that values of m less than l show the largest effects. This should provide information about the interior angular momentum distribution if all mode splittings are present and can be identified. [ABSTRACT FROM AUTHOR]

In this paper, we show some constructive a priori error estimates for H02-projection into a space of polynomials on a one dimensional interval. Here, ’constructive’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. By using the property of Legendre polynomials, we try to determine such constants as small as possible. Particularly, we will show the optimal constant could be enclosed in a very narrow interval. Then an application of the results to finite element H02-projection in one dimension is presented. This kind of estimates will play an important role in the numerical verification of solutions for nonlinear fourth order elliptic problems. [ABSTRACT FROM AUTHOR]

The problem of the nonlinear harmonic response in dielectric relaxation of polyelectrolytes is considered in the context of coupled (rotation-translation) kinetic equations. Perturbation theory is used to derive the complex dielectric increment for an assembly of non-polar rodlike macroions surrounded by small counterions and acted on by a strong dc bias electric field superimposed on a weak ac field. The approach starts from a Smoluchowski equation in configuration space of molecular orientations (∂) and displacements (x) first used for Kerr effect relaxation by Szabo et al. [J. Chem. Phys. 85, 7472 (1986)]. By solving this equation for the stationary regime, we calculate the expectation value of where P1 and H1 stand for the first Legendre and Hermite polynomials, characterizing the first harmonic component of the electric polarization of the macroion. The results so obtained in the form of analytic expressions are illustrated by Cole-Cole-like diagrams and three-dimensional dispersion and absorption plots in order to show the influence of the coupling parameter a measuring the importance of the rotational motion over the translational one. [ABSTRACT FROM AUTHOR]

The velocity electron distribution function is developed in terms of Legendre polynomials. It is shown that only a few even-order polynomials contribute to ionic electron excitation, usually the 2 first ones. From the measurement of line polarization it is then possible to separate the relative importance of each contribution. To illustrate this theory, some K[sub α] lines are studied. They correspond to He-like, Li-like and Be-like systems. They are very close in wavelength but well separated. Except for the He-like lines, the other lines are emitted by autoionizing levels. Compared to He-like lines, they are relatively weak for low nuclear charge Z but are intense for high nuclear charge. The spatial and temporal analysis of the polarization of these lines is very promising to detect the presence of nonthermal anisotropic electron in hot plasmas. [ABSTRACT FROM AUTHOR]