J. Díaz Bejarano

Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

Abstract An improved version of a Krylov-Bogoliubov method that gives the approximate solution of the non-linear cubic oscillator x + c 1 x + c 3 x 3 + ef(x, x dot ) = 0 in terms of Jacobi elliptic functions is described. Compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual Krylov-Bogoliubov method (which gives the approximate solution in terms of circular functions). These expressions are especially simple for quasilinear (c3 = 0) and quasi-pure-cubic (c1 = 0) oscillators. Two types of cubic oscillators have been used as examples: the linear damped oscillator f(x, x dot ) = x dot , and the van der Pol oscillator f(x, x dot ) = (α − βx 2 ) x dot . The approximate solutions of these quasilinear and quasi-pure-cubic oscillators are simple and accurate. The influence of the non-linearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied.

Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions

Abstract It is shown that the Krylov–Bogoliubov methods that give the approximate oscillatory solution of the equation in terms of Jacobi elliptic functions are applicable not only when c 1>0 and c 3>0, but also when c 1>0 and c 3 0. In particular, the most precise of these methods, the Christopher-Brocklehurst method, is discussed in detail. Its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters.

Generalized fourier series for the study of limit cycles

The approximate solution, to first order, of non-linear differential equations is studied using the method of harmonic balance with generalized Fourier series and Jacobian elliptic functions. As an interesting use of the series, very good analytic approximations to the limit cycles of Lienards ordinary differential equation (ODE), X + g(X) = f(X)X, are presented. Specifically, it is shown that, contrary to an opinion given in a well-known textbook on non-linear oscillations, g(X) not only modifies the period but influences the topology. In the generalized van der Pol equation with f(X) = e(1−X2) and g(X) = AX + 2BX3 for ϵ < 0·1, the presence of zero, one, or three limit cycles is found to depend on the value of AB.

Anharmonic asymmetric oscillator: A classical and quantum treatment

The nonrelativistic anharmonic–asymmetrical–oscillator (AAO) is studied in full. The classical equation of motion is solved using Jacobi elliptic functions which have both real and imaginary periods. The imaginary period is connected with the imaginary part of the energy in quantum mechanics. The resonances (Siegert states) are calculated in the JWKB approximation.

A rapid method for determination of the isotopic composition of uranium samples by alpha spectrometry

Abstract A simple method of analyzing alpha spectra from natural and enriched or depleted uranium samples is developed. The procedure is non-iterative, and takes into consideration low-energy tail and branching-ratio corrections to accurately calculate the area corresponding to each uranium isotope ( 234 U, 235 U, 236 U, 238 U) in the spectrum, and then the isotopic composition of the sample. A BASIC computer program, called ENURA, has been developed to perform all the necessary calculations to give the results together with their uncertainties. Several samples were prepared with different uranium concentrations made from standard solutions with known compositions, and the method was checked against the experimental measurements from these samples. Other series of uranium spectra were theoretically constructed using a given line shape in order to cover the required range of enriched or depleted uranium.

Natural isotopic separation of uranium in the Guadiana basin

In the course of measurements of uranium contamination in rivers of the Guadiana basin we have found the logical contamination near the uranium mines and a234U/238U ratio near 1 in the neighbourhood of the sources. With distance we find not only the expected decreasing contamination but also higher values of the activity ratio234U/238U. This observation seems to contradict the usual explanation of preferential leaching of234U as origin of the general234U enrichment of ground water.

Generalized exponential, circular, and hyperbolic functions for nonlinear wave equations

Wave functions are presented in the form of generalized exponentials that are solutions of some of the most usual linear and nonlinear wave equations. The solutions are given in terms of the elliptic functions of Jacobi and presented in a form as similar as possible to the usual circular functions. Some simple theorems are demonstrated to present the solutions as the simplest possible extension of the usual exponentials.

Constraints on uncertainties and their application to the emission probabilities of alpha-particles

Abstract It often happens that the mean values of certain quantities are subject to constraints as, for example, when a sum is known exactly although the individual contributions have been measured independently and their corresponding uncertainties assigned. In this paper, the influence of a constraint on the final expression of the results is studied in detail, and is illustrated in the alpha-particle emission probabilities for several nuclides from nuclear data tables. A simple rule emerges: If the mean values must add to 100% and if one of the variances is greater than half the sum of all the variances, then the precision in the expression of the results can be improved.

Solution of the anharmonic quartic potential oscillator problem

The problem of the one-dimensional generalized anharmonic quartic potential oscillator is studied in full. Solutions of the classical equations of motion for the different types of potential are given in terms of the Jacobi elliptic functions, in both classically allowed and forbidden regions of each potential for physically interesting initial conditions. The solutions and some of their properties are also given by means of bilinear transformations of the corresponding anharmonic symmetric oscillator solutions for similar regions or initial conditions.

Evaporation of nonzero rest mass particles from a black hole

Analytic expressions for the transmission coefficient and the emission and the absorption rates for scalar particles with mass and a chargeless, nonrotating black hole are calculated by using Jacobian elliptic functions and integrals in the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation.