An exact frequency equation in closed form for Timoshenko beam clampled at both ends

The author has discovered several errors which are not typographical in the frequency equations for a Timoshenko beam clamped at both ends by Huang who presented the frequency equations and normal mode equations for all six common types of simple, finite beams in closed form for the first time. The exact frequency equations in closed form for Timoshenko beams clamped at both ends are derived based on his analysis. And then in order to justify the amended solutions of Huang, two versions of the closed form exact method and the Ritz method are applied. The frequency equations by the previous researcher present frequencies for only the flexural modes, while the closed form exact method and the Ritz method give ones for the thickness–shear modes as well as the bending modes. The purpose of the present study is to reveal the errors, correct them, and give some numerical results.     

An exact solution of a regularized Mekkenko-Migdal equation in two dimensions for small Wilson loops

A regularized form of the Makeenko-Migdal equation is introduced. For the particular case of two dimensions this equation can be solved for small, simple loops. The solution reveals the third-order phase transition of Gross and Witten.     

An exact solution of Einstein's equation with a dilaton field

An exact solution of Einstein gravity coupled to a dilaton field is found. The solution is conformally flat and is invariant under Lorentz transformations. The singularities and conformal structure of the metric are examined.     

An exact solution to the multi-dimensional line transfer equation

The theory of generalized analytic functions is used to obtain an exact closed form analytical solution to a transfer problem for spectral line radiation in a multi-dimensional atmosphere. The multi-dimensional full-space and half-space Green's functions so obtained are quite general and may be used, along with the corresponding orthogonality relationships, to obtain solutions to any general multi-dimensional radiative transfer problem involving model two-level atoms. An application of the method using perturbation techniques is illustrated.     

An exact solution of the Korteweg-de Vries equation with dissipation

By means of the method proposed in the papers [1, 2] we look for solutions of the Korteweg—de Vries equation with dissipation. A new solution is found and expressed by means of the Weierstrass P-function.Partially supported by NSF Grant No. INT 73.20002 A01 formerly GF-41958.     

Representation of the Helmholtz equation solution in the form of a series based on backscattering multiplicity

Abstract

The representation of the Helmholtz equation solution in the form of a series based on backscattering multiplicity is considered. New methods for calculating wave fields propagating in inhomogeneous media, which have been developed on the basis of this series, are presented. The results that have been obtained using these new methods are briefly described.     

Representation of the Helmholtz equation solution in the form of a series based on backscattering multiplicity

The representation of the Helmholtz equation solution in the form of a series based on backscattering multiplicity is considered. New methods for calculating wave fields propagating in inhomogeneous media, which have been developed on the basis of this series, are presented. The results that have been obtained using these new methods are briefly described.     

An exact solution of Laplace's equation for cuspidal geometry

An exact solution of Laplace's equation is obtained for a system of conducting electrodes with cuspidal symmetry. The significance of this result in predicting and verifying the equilibrium configuration of a rotationally symmetric conducting fluid subject to electrostatic stress is discussed.This work was supported in part by the Division of Materials Research, National Science Foundation, Grant No. DMR-8108829     

An exact solution for the linearized multifrequency radiation diffusion equation

An exact solution, based on Fourier and Laplace (FL) transforms, is developed for a linearization of the system modeling the multifrequency radiation diffusion and matter energy balance equations. The model uses an ideal gas equation of state. Opacities are proportional to the inverse of the cube of the frequency, thereby simulating free-free transitions. The solution is obtained in terms of integrals over the FL coefficients of the initial conditions and explicit sources. Results are presented for two special cases. (1) No sources, initially cold radiation field, and a localized matter energy profile. (2) Initially cold matter and radiation fields and a source of matter energy extending over finite space and time intervals.     

An exact solution of Fisher equation and its stability

In this paper, the Fisher equation is analysed. One of its travelling wave solution is obtained by comparing it with KdV--Burgers (KdVB) equation. Its amplitude, width and speed are investigated. The instability for the higher order disturbances to the solution of the Fisher equation is also studied.     

A fast method for the solution of the Helmholtz equation

In this paper, we consider the numerical solution of the Helmholtz equation, arising from the study of the wave equation in the frequency domain. The approach proposed here differs from those recently considered in the literature, in that it is based on a decomposition that is exact when considered analytically, so the only degradation in computational performance is due to discretization and roundoff errors. In particular, we make use of a multiplicative decomposition of the solution of the Helmholtz equation into an analytical plane wave and a multiplier, which is the solution of a complex-valued advection–diffusion–reaction equation. The use of fast multigrid methods for the solution of this equation is investigated. Numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies.     

A theorem for quantum operator correspondence to the solution of the Helmholtz equation

We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., V(x1, x2, x3)|x1, x2, x3 x1, x2, x3| d3 x = V(X1, X2, X3) = e-λ2/4: V(X1, X2, X3) :,where V(x1, x2, x3) is the solution to the Helmholtz equation ?2V + λ2V = 0, the symbol : : denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac’s radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.     

An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux

In the steady Couette flow of a granular gas the sign of the heat flux gradient is governed by the competition between viscous heating and inelastic cooling. We show from the Boltzmann equation for inelastic Maxwell particles that a special class of states exists where the viscous heating and the inelastic cooling exactly compensate each other at every point, resulting in a uniform heat flux. In this state the (reduced) shear rate is enslaved to the coefficient of restitution α, so that the only free parameter is the (reduced) thermal gradient ϵ. It turns out that the reduced moments of order k are polynomials of degree k−2 in ϵ, with coefficients that are nonlinear functions of α. In particular, the rheological properties (k = 2) are independent of ϵ and coincide exactly with those of the simple shear flow. The heat flux (k = 3) is linear in the thermal gradient (generalized Fourier’s law), but with an effective thermal conductivity differing from the Navier–Stokes one. In addition, a heat flux component parallel to the flow velocity and normal to the thermal gradient exists. The theoretical predictions are validated by comparison with direct Monte Carlo simulations for the same model.     

Interactions of traveling fronts: An exact solution of a nonlinear diffusion equation

An exact solution which describes the coalescence of two traveling fronts of the same sense into a front connecting two stable constant states is found in terms of the direct method for a simple nonlinear diffusion equation. Head-on collisions of two fronts of opposite sense are also examined numerically.     

Improved transmission conditions for a one-dimensional domain decomposition method applied to the solution of the Helmholtz equation

The scattering problem of a time-harmonic electromagnetic wave from a perfect electric conductor (PEC) coated with materials is considered, and solved by coupling a finite element method with an integral equation prescribed on the outer boundary of the computational domain. To reduce the numerical complexity, a one-dimensional domain decomposition method (DDM) is employed: the computational domain is partitioned into concentric subdomains (SDs), and Robin transmission conditions (TCs) are prescribed on the interfaces. For some configurations and/or materials, the convergence of the corresponding DDM algorithm happens to be slow. A possible remedy is to enhance the efficiency of the TCs by approximating the exact ones more accurately. To this end, we first consider the simplified 2D problem of a circular PEC cylinder with an homogeneous coating and up to four SDs with circular interfaces, thus allowing to obtain the exact TCs in closed-form. Approximate local or non-local TCs are derived from these exact ones, and numerical examples demonstrate their superiority over the standard Robin TCs. Then, the case of an elliptical PEC cylinder with one interface in free-space is investigated. Also, the issues pertaining to the uniqueness of the solutions and convergence of the algorithm are addressed.     

Far-field solution of the Helmholtz equation for double heterostructure diode lasers

On the basis of the Helmholtz equation the far-field distribution is derived for double heterostructure lasers. The results show that the far-field distribution in the direction normal to the junction plane approaches a Lorentzian function, but parallel to the junction it may be approximated by a Gaussian function. The far-field intensity patterns have analogous elliptic form. It is also shown, for the first time, that the separability condition is not strictly valid for the far-field of a laser diode. Only in the vicinity of the optical axis the field can be expressed as a product of two separate functions, each of which depends only on one of the two transverse coordinates parallel and perpendicular to the diode junction.     

An exact solution for a rotating body with negligible mass

If it becomes possible to test general relativity by laboratory experiments on rotation, the ratios of mass to angular momentum per unit mass are likely to be extremely small. Solutions for a rotating body with low m/a are therefore of interest. Here I discuss Papapetrous exact solution, which has zero mass and arbitrary angular momentum.     

An exact solution to the relativistic equation of motion of acharged particle driven by a linearly polarized electromagneticwave

An exact analytic solution is found for a basic electromagnetic wave-charged particle interaction by solving the nonlinear equations of motion. The particle position, velocity, and corresponding time are found to be explicit functions of the total phase of the wave. Particle position and velocity are thus implicit functions of time. Applications include describing the motion of a free electron driven by an intense laser beam