Continuous First-order Derivatives Research Articles

Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

In this paper, we consider the traveling wave solutions of the one-dimensional Serre-Green-Naghdi (SGN) equations which are proposed to model dispersive nonlinear long water waves in a one-layer flow over flat bottom. We decouple the traveling wave system of SGN equations into two ordinary differential equations. By studying the bifurcations and phase portraits of each bifurcation set of one equation, we obtain the exact traveling wave solutions of SGN equations for the variable $ u(x, t) $ which represents average horizontal velocity of water wave. For the compacted orbits intersecting with the singular line in phase plane, we obtained two families of solutions: a family of smooth traveling wave solutions including periodic wave solutions and solitary wave solutions, and a family of compacted singular solutions which have continuous first-order derivative but discontinuous second-order derivative.

Robust sampling-sourced numerical retrieval algorithm for optical energy loss function based on log–log mesh optimization and local monotonicity preserving Steffen spline

We introduce a new technique of interpolation of the energy-loss function (ELF) in solids sampled by empirical optical spectra. Finding appropriate interpolation methods for ELFs poses several challenges. The sampled ELFs are usually very heterogeneous, can originate from various sources thus so called “data gaps” can appear, and significant discontinuities and multiple high outliers can be present. As a result an interpolation based on those data may not perform well at predicting reasonable physical results. Reliable interpolation tools, suitable for ELF applications, should therefore satisfy several important demands: accuracy and predictive power, robustness and computational efficiency, and ease of use. We examined the effect on the fitting quality due to different interpolation schemes with emphasis on ELF mesh optimization procedures and we argue that the optimal fitting should be based on preliminary log–log scaling data transforms by which the non-uniformity of sampled data distribution may be considerably reduced. The transformed data are then interpolated by local monotonicity preserving Steffen spline. The result is a piece-wise smooth fitting curve with continuous first-order derivatives that passes through all data points without spurious oscillations. Local extrema can occur only at grid points where they are given by the data, but not in between two adjacent grid points. It is found that proposed technique gives the most accurate results and also that its computational time is short. Thus, it is feasible using this simple method to address practical problems associated with interaction between a bulk material and a moving electron. A compact C++ implementation of our algorithm is also presented.

A Further Research on the Convergence of Wu-Schaback’s Multi-quadric Quasi-Interpolation

The paper discusses the error estimate of Wu-Schaback's quasi-interpolant for a wider class of approximated functions (the functions with lower smoothness order). Three cases are considered: a function with a Lipschitz continuous first-order derivative, a continuous function and a Lipschitz continuous function, respectively.

Asymptotic rejection of unknown sinusoidal disturbances in non-linear dynamic systems with delay

The authors deal with asymptotic rejection of unknown disturbances in non-linear dynamic systems with delay. The disturbances are generated from an unknown linear exosystem whose order is known. The non-linear dynamic system contains non-linear terms in the system state variables with delay. There is no restriction on the non-linear functions, as long as they have continuous first-order derivatives. Here, a state feedback stabilisation control strategy is proposed for the system without the disturbances. An adaptive internal model is then designed and incorporated with the state feedback control design to asymptotically reject the unknown disturbances. A simulation example is included to demonstrate the proposed control design.

Reconstruction of Sensory Stimuli Encoded with Integrate-and-Fire Neurons with Random Thresholds

We present a general approach to the reconstruction of sensory stimuli encoded with leaky integrate-and-fire neurons with random thresholds. The stimuli are modeled as elements of a Reproducing Kernel Hilbert Space. The reconstruction is based on finding a stimulus that minimizes a regularized quadratic optimality criterion. We discuss in detail the reconstruction of sensory stimuli modeled as absolutely continuous functions as well as stimuli with absolutely continuous first-order derivatives. Reconstruction results are presented for stimuli encoded with single as well as a population of neurons. Examples are given that demonstrate the performance of the reconstruction algorithms as a function of threshold variability.

NUMERICAL MODELING OF VOID MIGRATION IN SOLIDS DUE TO TEMPERATURE GRADIENT USING THE BOUNDARY ELEMENT METHOD

Voids (a land of flaw) are not desired in the products of many industrial and manufacturing processes. In this article, we seek effective ways to remove the void by modeling the void migration and predicting the intermediate and the final shape of the cavity. The boundary element method (BEM) is applied to the quasi-steady state void migration process governed by Laplace's equation. The conduction solution depends on the void shape, and the void shape depends on the conduction solution. Hence this is a conjugate problem. The analytical formulation and the numerical approach are outlined. The Overhauser spline elements are used in the BEM to ensure continuous first-order derivatives on the void boundary. Given the material properties, geometry of the physical model, and boundary conditions, this computer model can predict detailed information such as flux, velocity and direction of void motion, and temperature at any stage of the void migration. Different strategies for void removal are investigated.

Distance approximations for rasterizing implicit curves

In this article we present new algorithms for rasterizing implicit curves, i.e., curves represented as level sets of functions of two variables. Considering the pixels as square regions of the plane, a “correct” algorithm should paint those pixels whose centers lie at less than half the desired line width from the curve. A straightforward implementation, scanning the display array evaluating the Euclidean distance from the center of each pixel to the curve, is impractical, and a standard quad-tree-like recursive subdivision scheme is used instead. Then we attack the problem of testing whether or not the Euclidean distance from a point to an implicit curve is less than a given threshold. For the most general case, when the implicit function is only required to have continuous first-order derivatives, we show how to reformulate the test as an unconstrained global root-finding problem in a circular domain. For implicit functions with continuous derivatives up to orderkwe introduce an approximate distance of orderk. The approximate distance of orderkfrom a point to an implicit curve is asymptotically equivalent to the Euclidean distance and provides a sufficient test for a polynomial of degreeknot to have roots inside a circle. This is the main contribution of the article. By replacing the Euclidean distance test with one of these approximate distance tests, we obtain a practical rendering algorithm, proven to be correct for algebraic curves. To speed up the computation we also introduce heuristics, which used in conjunction with low-order approximate distances almost always produce equivalent results. The behavior of the algorithms is analyzed, both near regular and singular points, and several possible extensions and applications are discussed.