On rapidly oscillating solutions of a nonlinear elliptic equation

The aim of this article is to examine the solutions of the boundary value problem of the nonlinear elliptic equation ε 2△u = f(u). We describe the asymptotic behavior as ε tends to zero of the solutions on a spherical crown C of RN , (N ≥ 2) in a direct non-classical formulation which suggests easy proofs. We propose to look for interesting solutions in the case where the condition at the edge of the crown is a constant function. Our results are formulated in classical mathematics.Their proofs use the stroboscopic method which is a tool of the nonstandard asymptotic theory of differential equations.

Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems.

Solving Oscillating Problems Using Modifying Runge-Kutta Methods

This paper develop conventional Runge-Kutta methods of order four and order five to solve ordinary differential equations with oscillating solutions. The new modified Runge-Kutta methods (MRK) contain the invalidation of phase lag, phase lag’s derivatives, and amplification error. Numerical tests from their outcomes show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientific literature.

A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner (Falkner, 1936. Phil. Mag. S. 7, 621). This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

Non-Oscillating solutions of a generalized system of ODEs with derivative terms

We consider a system of ODEs of mixed order with derivative terms appearing in the non-linear function and show the existence of a solution which does not oscillate for such system. We applied the fixed point technique to show that under certain conditions there exists at least one solution to the system which is not only non-oscillating, but also asymptotically constant.

Particle production in modified gravity in the early and present day universe

Gravitational equations of motion in modified theories of gravity have oscillating solutions, both in the early and in the present day universe. Particle production by such oscillations is analyzed and possible observational consequences are considered. This phenomenon has impact on energy spectrum of cosmic rays and abundance of dark matter particles.

Mechanism of autonomous synchronization of the circadian KaiABC rhythm

ABSTRACTThe cyanobacterial circadian clock can be reconstituted by mixing three proteins, KaiA, KaiB, and KaiC, in vitro. In this protein mixture, oscillations of the phosphorylation level of KaiC molecules are synchronized to show the coherent oscillations of the ensemble of many molecules. However, the mechanism of this synchronization remains elusive. In this paper, we explain a theoretical model that considers the multifold feedback relations among the structure and reactions of KaiC. The simulated KaiC hexamers show stochastic switch-like transitions at the level of single molecules, which are synchronized in the ensemble through the sequestration of KaiA into the KaiC-KaiB-KaiA complexes. The proposed mechanism quantitatively reproduces the synchronization that was observed by mixing two oscillating solutions in different phases. The model results suggest that biochemical assays with varying concentrations of KaiA or KaiB can be used to test this hypothesis.