Fourier methods for oscillatory differential problems with a constant high frequency

The numerical solution of highly oscillatory initial value problems of second order with a unique high frequency is considered. New methods based on Fourier approximations are proposed. These methods can integrate the problems with reasonable stepsizes not dependent on the size of the frequency

Energy-preserving methods for Poisson systems

AbstractWe present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type that are able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see Brugnano et al. [10] and references therein). It is seen that the proposed approach allows us to obtain the methods recently derived in Cohen and Hairer (2011) [17], giving an alternative derivation of such methods and a new proof of their order. Sufficient conditions that ensure the existence of a unique solution of the implicit equations defining the formulae are given. A study of the implementation of the methods is provided. In particular, order and preservation properties when the involved integrals are approximated by means of a quadrature formula, are derived

Sixth-order symmetric and symplectic exponentially fitted Runge–Kutta methods of the Gauss type

AbstractThe construction of exponentially fitted Runge–Kutta (EFRK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is considered. Based on the symplecticness, symmetry, and exponential fitting properties, two new three-stage RK integrators of the Gauss type with fixed or variable nodes, are obtained. The new exponentially fitted RK Gauss type methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(λt),exp(−λt)}, λ∈C, and in particular {sin(ωt),cos(ωt)} when λ=iω, ω∈R. The algebraic order of the new integrators is also analyzed, obtaining that they are of sixth-order like the classical three-stage RK Gauss method. Some numerical experiments show that the new methods are more efficient than the symplectic RK Gauss methods (either standard or else exponentially fitted) proposed in the scientific literature

An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

In this note a study of the convergence properties of some starters (Formula presented.) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter (Formula presented.) a set of values (Formula presented.) that lead to the q-convergence in the sense that Newton’s sequence (Formula presented.) generated from (Formula presented.) is well defined, converges to the exact solution (Formula presented.) of KE and further (Formula presented.) holds for all (Formula presented.). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s (Formula presented.)-test with (Formula presented.). Also since in KE the convergence rate of Newton’s method tends to zero as (Formula presented.), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with (Formula presented.) for all (Formula presented.) and a fixed (Formula presented.). Some remarks on the use of this theorem to derive a priori estimates of the error (Formula presented.) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained

Numerical methods for non conservative perturbations of conservative problems

In this paper the numerical integration of non conservative perturbations of differential systems that possess a first integral, as for example slowly dissipative Hamiltonian systems, is considered. Numerical methods that are able to reproduce appropriately the evolution of the first integral are proposed. These algorithms are based on a combination of standard numerical integration methods and certain projection techniques. Some conditions under which known conservative methods reproduce that desirable evolution in the invariant are analysed. Finally, some numerical experiments in which we compare the behaviour of the new proposed methods, the averaged vector field method AVF proposed by Quispel and McLaren and standard RK methods of orders 3 and 5 are presented. The results confirm the theory and show a good qualitative and quantitative performance of the new projection methods

Efficacy of direct current generated by multiple-electrode arrays on F3II mammary carcinoma: experiment and mathematical modeling

BACKGROUND: The modified Gompertz equation has been proposed to fit experimental data for direct current treated tumors when multiple-straight needle electrodes are individually inserted into the base perpendicular to the tumor long axis. The aim of this work is to evaluate the efficacy of direct current generated by multiple-electrode arrays on F3II mammary carcinoma that grow in the male and female BALB/c/Cenp mice, when multiple-straight needle electrodes and multiple-pairs of electrodes are inserted in the tumor. METHODS: A longitudinal and retrospective preclinical study was carried out. Male and female BALB/c/Cenp mice, the modified Gompertz equation, intensities (2, 6 and 10 mA) and exposure times (10 and 20 min) of direct current, and three geometries of multiple-electrodes (one formed by collinear electrodes and two by pair-electrodes) were used. Tumor volume and mice weight were measured. In addition, the mean tumor doubling time, tumor regression percentage, tumor growth delay, direct current overall effectiveness and mice survival were calculated. RESULTS: The greatest growth retardation, mean doubling time, regression percentage and growth delay of the primary F3II mammary carcinoma in male and female mice were observed when the geometry of multiple-pairs of electrodes was arranged in the tumor at 45, 135, 225 and 325o and the longest exposure time. In addition, highest direct current overall effectiveness (above 66%) was observed for this EChT scheme. CONCLUSIONS: It is concluded that electrochemical therapy may be potentially addressed to highly aggressive and metastic primary F3II murine mammary carcinoma and the modified Gompertz equation may be used to fit data of this direct current treated carcinoma. Additionally, electrochemical therapy effectiveness depends on the exposure time, geometry of multiple-electrodes and ratio between the direct current intensity applied and the polarization current induced in the tumor

Mathematical modeling and forecasting of COVID-19: experience in Santiago de Cuba province

In the province of Santiago de Cuba, Cuba, the COVID-19 epidemic has a limited progression that shows an early small-number peak of infections. Most published mathematical models fit data with high numbers of confirmed cases. In contrast, small numbers of cases make it difficult to predict the course of the epidemic. We present two known models adapted to capture the noisy dynamics of COVID-19 in the Santiago de Cuba province. Parameters of both models were estimated using the approximate-Bayesian-computation framework with dedicated error laws. One parameter of each model was updated on key dates of travel restrictions. Both models approximately predicted the infection peak and the end of the COVID-19 epidemic in Santiago de Cuba. The first model predicted 57 reported cases and 16 unreported cases. Additionally, it estimated six initially exposed persons. The second model forecasted 51 confirmed cases at the end of the epidemic. In conclusion, an opportune epidemiological investigation, along with the low number of initially exposed individuals, might partly explain the favorable evolution of the COVID-19 epidemic in Santiago de Cuba. With the available data, the simplest model predicted the epidemic evolution with greater precision, and the more complex model helped to explain the epidemic phenomenology

Explicit Runge–Kutta methods for the numerical solution of linear inhomogeneous IVPs

Runge–Kutta methods for the numerical solution of inhomogeneous linear initial value problems with constant coefficients are considered. A general procedure to construct explicit s-stage RK methods with order s, for the class of IVP under consideration, depending on the nodes ci,i=1,…,s is presented. This procedure only requires the solution of successive linear equations in the elements of the matrix A and avoids the solution of non linear equations. We obtain several RK schemes with number of stages s=5,…,8 and maximal order p=s for the class of problems under consideration. Finally, some numerical experiments to test the behaviour of the new RK schemes are presented

Global error estimation based on the tolerance proportionality for some adaptive Runge–Kutta codes

AbstractModern codes for the numerical solution of Initial Value Problems (IVPs) in ODEs are based in adaptive methods that, for a user supplied tolerance δ, attempt to advance the integration selecting the size of each step so that some measure of the local error is ≃δ. Although this policy does not ensure that the global errors are under the prescribed tolerance, after the early studies of Stetter [Considerations concerning a theory for ODE-solvers, in: R. Burlisch, R.D. Grigorieff, J. Schröder (Eds.), Numerical Treatment of Differential Equations, Proceedings of Oberwolfach, 1976, Lecture Notes in Mathematics, vol. 631, Springer, Berlin, 1978, pp. 188–200; Tolerance proportionality in ODE codes, in: R. März (Ed.), Proceedings of the Second Conference on Numerical Treatment of Ordinary Differential Equations, Humbold University, Berlin, 1980, pp. 109–123] and the extensions of Higham [Global error versus tolerance for explicit Runge–Kutta methods, IMA J. Numer. Anal. 11 (1991) 457–480; The tolerance proportionality of adaptive ODE solvers, J. Comput. Appl. Math. 45 (1993) 227–236; The reliability of standard local error control algorithms for initial value ordinary differential equations, in: Proceedings: The Quality of Numerical Software: Assessment and Enhancement, IFIP Series, Springer, Berlin, 1997], it has been proved that in many existing explicit Runge–Kutta codes the global errors behave asymptotically as some rational power of δ. This step-size policy, for a given IVP, determines at each grid point tn a new step-size hn+1=h(tn;δ) so that h(t;δ) is a continuous function of t.In this paper a study of the tolerance proportionality property under a discontinuous step-size policy that does not allow to change the size of the step if the step-size ratio between two consecutive steps is close to unity is carried out. This theory is applied to obtain global error estimations in a few problems that have been solved with the code Gauss2 [S. Gonzalez-Pinto, R. Rojas-Bello, Gauss2, a Fortran 90 code for second order initial value problems, 〈http://www.netlib.org/ode/〉], based on an adaptive two stage Runge–Kutta–Gauss method with this discontinuous step-size policy

Approximate trigonometric series expansions of some bounded solutions in the Tsien problem

INTHE last decades, studies on the dynamics of a spacecraft that is moving in the gravitational field of a celestial body and subjected to some low-thrust propulsion system have attracted the attention of many researchers in astrodynamics. Here, we consider a particular case, the so-called Tsien problem [1], in which the spacecraft moves in a circular Keplerian orbit (often referred to as the parking orbit), and after a given time t0 0, it is subjected to a constant outward radial acceleration. By using the integrals of energy and angular momentum, this problem can be reduced to a two-dimensional scenario. Denoting by r and ? the polar coordinates that give the position of the spacecraft in the plane of the orbit [1], the differential equations that define their motion are ..