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.     

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.     

Numerical solution of boundary value problems by using an optimized two-step block method

Numerical Algorithms - This paper aims at the application of an optimized two-step hybrid block method for solving boundary value problems with different types of boundary conditions. The proposed...     

A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems

^{We consider the construction of a special family of Runge–Kutta(RK) collocation methods based on intra-step nodal points ofChebyshev–Gauss–Lobatto type, with A-stability andstiffly accurate characteristics. This feature with its inherentimplicitness makes them suitable for solving stiff initial-valueproblems. In fact, the two simplest cases consist in the well-knowntrapezoidal rule and the fourth-order Runge–Kutta–LobattoIIIA method. We will present here the coefficients up to eighthorder, but we provide the formulas to obtain methods of higherorder. When the number of stages is odd, we have considereda new strategy for changing the step size based on the use ofa pair of methods: the given RK method and a linear multistepone. Some numerical experiments are considered in order to checkthe behaviour of the methods when applied to a variety of initial-valueproblems.}     

X-ray microanalysis of human cementum.

An energy dispersive x-ray microanalysis study was performed throughout the total length of cementum on five impacted human teeth. Mineral content of calcium, phosphorous, and magnesium were determined with an electron probe from the cemento-enamel junction to the root apex on the external surface of the cementum. The concentration profiles for calcium, phosphorous, and magnesium were compared by using Ca/P and Mg/Ca atomic percent ratio. Our findings demonstrated that the Ca/P ratio at the cemento-enamel junction showed the highest values (1.8-2.2). However, the area corresponding to the acellular extrinsic fiber cementum (AEFC) usually located on the coronal one-third of the root surface showed a Ca/P media value of 1.65. Nevertheless, on the area representing the fulcrum of the root there is an abrupt change in the Ca/P ratio, which decreases to 1.3. Our results revealed that Mg(2+) distribution throughout the length of human cementum reached its maximum Mg/Ca ratio value of 1.3-1.4 at.% around the fulcrum of the root and an average value of 0.03%. A remarkable finding was that the Mg/Ca ratio pattern distribution showed that in the region where the Ca/P ratio showed a decreasing tendency, the Mg/Ca ratio reached its maximum value, showing a negative correlation. In conclusion, this study has established that clear compositional differences exist between AEFC and cellular mixed stratified cementum varieties and adds new knowledge about Mg(2+) distribution and suggests its provocative role regulating human cementum metabolism.     

A New Eighth-order A-stable Method for Solving Differential Systems Arising in Chemical Reactions

Implicit Runge–Kutta methods are successful algorithms for the numerical solution of stiff differential equations, as they usually appear in chemical reactions. This article describes the construction of a particular implicit method based on internal stages obtained from certain Chebyshev collocation points. The resulting method has algebraic order 8 and A-stability characteristic. An embedding technique using the Runge–Kutta method and a linear multistep one is provided in order to change the step size. Numerical experiments illustrate the behaviour of the new method, showing that it may reach great accuracy and be competitive with other well-known codes.     

On the frequency choice in trigonometrically fitted methods

The choice of frequency in trigonometrically fitted methods is a fundamental question, especially if long-term prediction is considered. For linear oscillators, the frequency of the method is the same as the frequency of the solution of the differential equation. However, for nonlinear problems the frequency of the method is, in general, different from the frequency of the true solution. We present some experiments showing how the frequency depends strongly on certain values.     

An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems

In this paper, we present two families of second-order and third-order explicit methods for numerical integration of initial-value problems of ordinary differential equations. Firstly, a family of second-order methods with two free parameters is derived by considering a suitable rational approximation to the theoretical solution of the problem at some grid points. Imposing that the principal term of the local truncation error of this family vanishes, we obtain an expression for one of the parameters in terms of the other. With this approach, a new one-parameter family of third-order methods is obtained. By selecting any 3(2) pair of second and third order methods, they can be implemented as an embedded type method, thus leading to a variable step-size formulation. We have considered one 3(2) pair of second and third order methods and made a comparison of numerical results with several ode solvers which are currently used in practice. The comparison of numerical results shows that the embedded 3(2) pair outperforms the methods considered for comparison.