On oscillation of higher-order advanced trinomial differential equations

We study the oscillatory property of the higher-order trinomial differential equation with advanced effects $$ x^{(n)}(t)+p(t)x'(t)+q(t)x \bigl(\sigma (t) \bigr)=0,\quad \sigma (t) \geq t. $$ x ( n ) ( t ) + p ( t ) x ′ ( t ) + q ( t ) x ( σ ( t ) ) = 0 , σ ( t ) ≥ t . Suppose that all solutions of the corresponding ($n-1$ n − 1 )th-order two-term differential equation $$ y^{(n-1)}(t)+p(t)y(t)=0 $$ y ( n − 1 ) ( t ) + p ( t ) y ( t ) = 0 are non-oscillatory. In order to supplement the research in the theory of oscillation proposed by (Džurina et al. in Electron. J. Differ. Equ. 2015:70, 2015), two types of clearly confirmable criteria for oscillatory behavior of the investigated equation are obtained. Some examples are offered to describe our main results.

A novel fourth-order WENO interpolation technique

Context. Several numerical problems require the interpolation of discrete data that present at the same time (i) complex smooth structures and (ii) various types of discontinuities. The radiative transfer in solar and stellar atmospheres is a typical example of such a problem. This calls for high-order well-behaved techniques that are able to interpolate both smooth and discontinuous data. Aims. This article expands on different nonlinear interpolation techniques capable of guaranteeing high-order accuracy and handling discontinuities in an accurate and non-oscillatory fashion. The final aim is to propose new techniques which could be suitable for applications in the context of numerical radiative transfer. Methods. We have proposed and tested two different techniques. Essentially non-oscillatory (ENO) techniques generate several candidate interpolations based on different substencils. The smoothest candidate interpolation is determined from a measure for the local smoothness, thereby enabling the essentially non-oscillatory property. Weighted ENO (WENO) techniques use a convex combination of all candidate substencils to obtain high-order accuracy in smooth regions while keeping the essentially non-oscillatory property. In particular, we have outlined and tested a novel well-performing fourth-order WENO interpolation technique for both uniform and nonuniform grids. Results. Numerical tests prove that the fourth-order WENO interpolation guarantees fourth-order accuracy in smooth regions of the interpolated functions. In the presence of discontinuities, the fourth-order WENO interpolation enables the non-oscillatory property, avoiding oscillations. Unlike Bézier and monotonic high-order Hermite interpolations, it does not degenerate to a linear interpolation near smooth extrema of the interpolated function. Conclusion. The novel fourth-order WENO interpolation guarantees high accuracy in smooth regions, while effectively handling discontinuities. This interpolation technique might be particularly suitable for several problems, including a number of radiative transfer applications such as multidimensional problems, multigrid methods, and formal solutions.

A Runge Kutta Discontinuous Galerkin Method for Lagrangian Compressible Euler Equations in Two-Dimensions

This paper presents a new Lagrangian type scheme for solving the Euler equations of compressible gas dynamics. In this new scheme the system of equations is discretized by Runge-Kutta Discontinuous Galerkin (RKDG) method, and the mesh moves with the fluid flow. The scheme is conservative for the mass, momentum and total energy and maintains second-order accuracy. The scheme avoids solving the geometrical part and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the non-oscillatory property of the scheme.

A Stable Tracking Control Method for an Autonomous Welding Mobile Robot

This paper proposes an Autonomous Welding Mobile Robot (AWMR) used in shipbuilding and large spherical tank welding and its control method. In this method, both the moving and rotating velocity of the mobile platform and the cross slider were controlled simultaneously. The stability of control method was proved through the use of a Liapunov function. Although any set of positive parameters makes the system stable, a condition on the parameters for the system being critically damped for a small disturbance is obtained through linearizing the system's differential equation. In the simulation, we analyzed line and circle-line tracking problem. And the tracking responses with and without velocity limited is also presented. It is shown that the output responses represent non-oscillatory property and effectiveness of the tracking method.

Conditional Intrinsic Voltage Oscillations in Mature Vertebrate Neurons Undergo Specific Changes in Culture

Although intrinsic neuronal properties in invertebrates are well known to undergo specific adaptive changes in culture, long-term adaptation of similar properties in mature vertebrate neurons remain poorly understood. To investigate this, we used an organotypic slice preparation from the spinal cord of adult turtles maintainable for several weeks in culture conditions. N-methyl-d-aspartate (NMDA)-induced-tetrodotoxin (TTX)-resistant voltage oscillations in motoneurons were ∼10 times faster in culture than in acute preparations. Oscillations in culture were abolished by NMDA receptor antagonists or by high extracellular Mg2+ concentrations. However, in contrast with results from motoneurons in the acute slice, NMDA-induced oscillations in culture did not depend on CaV1.3 channel activation as they still remained after nifedipine application. Other CaV1.3 channel-mediated properties such as metabotropic receptor-induced oscillations and plateau potentials failed to be induced in culture. This study shows that changes specifically affecting CaV1.3 channel contribution to intrinsic oscillatory property expression may occur in culture. The results contribute also to understanding further the potential for plasticity of mature vertebrate neurons.