Decomposition of pseudo-radioactive chemical products with a mathematical approach

The aim of this paper is to study the decomposition of pseudo–radioactive products that follow a dynamics determined by a trigonometric factor. In particular for maps of the form $e^{\cos (\pi t)}$ is proved that an asymptotic sampling recomposition property, generalizing the classical Shannon–Whittaker–Kotel’nikov Theorem, works.     

A strategy for selecting the frequency in trigonometrically-fitted methods based on the minimization of the local truncation errors and the total energy error

Trigonometrically-fitted methods have been largely used for solving second-order differential problems, and particularly for solving the radial Schrödinger equation (see for instance Alolyan and Simos in J Math Chem 50:782–804, 2012; Simos in J Math Chem 34:39–58, 2003, 44:447–466, 2008; Vigo-Aguiar and Simos in J Math Chem 29:177–189, 2001, 32:257–270, 2002 and the references therein contained). It is well-known that for periodic or oscillatory problems, trigonometrically fitted methods are more efficient than non-fitted methods. A large number of different approaches have been considered in the scientific literature to obtain analytical approximations to the frequency of oscillation in case of periodic solutions, which are valid for a large range of amplitudes of oscillation. However, these techniques have been limited to obtaining only one or two iterates because of the great amount of algebra involved. In this paper we consider the use of a trigonometrically fitted method to obtain numerical approximations for the solutions. This yields very acceptable results provided that the approximation of the parameter of the method is done with great accuracy. Many trigonometrically fitted methods have been reported in the literature, but there is no decisive way to obtain the optimal frequency value. We present a strategy for the choice of the parameter value in the adapted method, based on the minimization of the sum of the total energy error and the local truncation errors in the solution and in the derivative. We include an example solved numerically that confirms the good performance of the strategy adopted.     

Family of Twelve Steps Exponential Fitting Symmetric Multistep Methods for the Numerical Solution of the Schrödinger Equation

In the present paper we present a family of twelve steps symmetric multistep methods. The explicit part of new family of methods is applied to the scattering problems of the radial Schrödinger equation. This application shows the efficiency of the new family of methods.     

A numerical ODE solver that preserves the fixed points and their stability

In this paper, we provide a one-step predictor-corrector method for numerically solving first-order differential initial-value problems with two fixed points. The method preserves the stability behaviour of the fixed points, which results in an efficient integrator for this kind of problem. Some numerical examples are provided to show the good performance of the method.     

Symmetric Eighth Algebraic Order Methods with Minimal Phase-Lag for the Numerical Solution of the Schrödinger Equation

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.