Second Time Derivative Research Articles

Numerical integrator for highly oscillatory differential equations based on the Neumann series

AbstractWe propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

The solution of the nonlinear Poisson-Boltzmann equation using lattice-Boltzmann

In this paper we present the solution of the Poisson-Boltzmann equation using the Lattice-Boltzmann method. In order to obtain the solution, we use a redefinition of tensor n°, which is declared as a symmetric tensor whose diagonal components are chosen as the second derivative in time of the first moment of the distribution function, and the components outside of the diagonal give account of the nonlinear terms. The results are presented in two dimensions employing the D2Q9 lattice velocity scheme. We obtain results for the scalar field and its gradient for several kinds of initial conditions.

A new variational discretization technique for initial value problems bypassing governing equations

Motivated by the fact that both the classical and quantum description of nature rest on causality and a variational principle, we develop a novel and highly versatile discretization prescription for classical initial value problems (IVPs). It is based on an optimization (action) functional with doubled degrees of freedom, which is discretized using a single regularized summation-by-parts (SBP) operator. Formulated as optimization task it allows us to obtain classical trajectories without the need to derive an equation of motion. The novel regularization we develop in this context is inspired by the weak imposition of initial data, often deployed in the modern treatment of IVPs and is implemented using affine coordinates. We demonstrate numerically the stability, accuracy and convergence properties of our approach in systems with classical equations of motion featuring both first and second order derivatives in time.

Limits of solutions to the semilinear plate equation with small parameter

We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.

Fermat’s principle and the effect of jerk on light motion near the Sun

In classical mechanics, in the case of gravitational and electromagnetic interactions, the force on a particle is usually proportional to its acceleration: The force acts locally on the particle. However, there are situations possible-if the particle moves through a suitable medium, for example, in which the force depends also on the first-time derivative of its acceleration, the jerk, and on its second-time derivative, the snap, and possibly also on higher-time derivatives. Such forces are called nonlocal, and this work investigates such nonlocal forces, mainly those depending on the jerk. In particular, we implement jerk and acceleration in geodesics by means of the nonlocal-in-time kinetic energy approach to spacetime physics. We describe a framework that can be used to estimate the quantum nonlocal time parameter by studying the deflection of light around the Sun. Comparing our results with long baseline interferometry (VLBI) observations, we concluded that the nonlocal time parameter [Formula: see text] s.

Leg Dominance Effects on Postural Control When Performing Challenging Balance Exercises.

Leg dominance reflects the preferential use of one leg over another and is typically attributed to asymmetries in the neural circuitry. Detecting leg dominance effects on motor behavior, particularly during balancing exercises, has proven difficult. The current study applied a principal component analysis (PCA) on kinematic data, to assess bilateral asymmetry on the coordinative structure (hypothesis H1) or on the control characteristics of specific movement components (hypothesis H2). Marker-based motion tracking was performed on 26 healthy adults (aged 25.3 ± 4.1 years), who stood unipedally on a multiaxial unstable board, in a randomized order, on their dominant and non-dominant leg. Leg dominance was defined as the kicking leg. PCA was performed to determine patterns of correlated segment movements (“principal movements” PMks). The control of each PMk was characterized by assessing its acceleration (second-time derivative). Results were inconclusive regarding a leg-dominance effect on the coordinative structure of balancing movements (H1 inconclusive); however, different control (p = 0.005) was observed in PM3, representing a diagonal plane movement component (H2 was supported). These findings supported that leg dominance effects should be considered when assessing or training lower-limb neuromuscular control and suggest that specific attention should be given to diagonal plane movements.

Numerical analysis of quasilinear parabolic equations under low regularity assumptions

In this paper, we carry out the numerical analysis of a class of quasilinear parabolic equations, where the diffusion coefficient depends on the solution of the partial differential equation. The goal is to prove error estimates for the fully discrete equation using discontinuous Galerkin discretization in time DG(0) combined with piecewise linear finite elements in space. This analysis is performed under minimal regularity assumptions on the data. In particular, we omit any assumption regarding existence of a second derivative in time of the solution.

On the convergence rates of energy-stable finite-difference schemes

We consider constant-coefficient initial-boundary value problems, with a first or second derivative in time and a highest spatial derivative of order q, and their semi-discrete finite difference approximations. With an internal truncation error of order p≥1, and a boundary error of order r≥0, we prove that the convergence rate is: min⁡(p,r+q).The assumptions needed for these results to hold are: i) The continuous problem is linear and well-posed (with a smooth solution). ii) The numerical scheme is consistent, nullspace consistent, nullspace invariant, and energy stable. These assumptions are often satisfied for Summation-By-Parts schemes.

A PDE approach to fractional diffusion: a space-fractional wave equation

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $$s \in (0,1)$$ , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $$\varOmega $$ . We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $$\mathcal {C}= \varOmega \times (0,\infty )$$ . We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space–time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: the trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $$\varOmega $$ with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates. We consider a first-degree FEM in $$\varOmega $$ with mesh refinement near corners and the aforementioned hp-FEM in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open, bounded, polytopal but not necessarily convex domains $$\varOmega \subset {\mathbb {R}}^2$$ . We discuss implementation details and report several numerical examples.

High accuracy solution of bi-directional wave propagation in continuum mechanics

Solution of partial differential equations by numerical method is strongly affected due to numerical errors, which are caused mainly by deviation of numerical dispersion relation from the physical dispersion relation. To quantify and control such errors and obtain high accuracy solutions, we consider a class of problems which involve second derivative of unknowns with respect to time. Here, we analyse numerical metrics such as the numerical group velocity, numerical phase speed and the numerical amplification factor for different methods in solving the model bi-directional wave equation (BDWE). Such equations can be solved directly, for example, by Runge–Kutta–Nyström (RKN) method. Alternatively, the governing equation can be converted to a set of first order in time equations and then using four-stage fourth order Runge–Kutta (RK4) method for time integration. Spatial discretisation considered are the classical second and fourth order central difference schemes, along with Lele's central compact scheme for evaluating second derivatives. In another version, we have used Lele's scheme for evaluating first derivatives twice to obtain the second derivative. As BDWE represents non-dissipative, non-dispersive dynamics, we also consider the canonical problem of linearised rotating shallow water equation (LRSWE) in a new formulation involving second order derivative in time, which represents dispersive waves along with a stationary mode. The computations of LRSWE with RK4 and RKN methods for temporal discretisation and Lele's compact schemes for spatial discretisation are compared with computations performed with RK4 method for time discretisation and staggered compact scheme (SCS) for spatial discretisation by treating it as a set of three equations as reported in Rajpoot et al. (2012) [1].

Modelling and simulation of a low speed passenger car collision

The development of a model for predicting passenger car response to low speed collisions with a test pendulum or a barrier is described. The collision scenarios are set forth in Federal Motor Vehicle Safety Standard (FMVSS) 215, Part 581. It was developed to provide an efficient tool for use in bumper design evaluation for compliance with FMVSS 215, Part 581, which requires that vehicle bumpers prevent damage 10 surrounding vehicle safety components in low speed (4.0 kph; 2.5 mph) collisions. The six degrees of freedom model utilises vehicle and bumper data including properties of energy absorbing devices common to modern passenger car bumper systems. The simulation code (BUMPER) is written in FORTRAN. It produces time histories of each of the generalised coordinates and of their first and second time derivatives, as well as of the magnitudes of the bumper absorber and impact forces. These simulation results can then be used in a subsequent bumper survivability analysis.

Summation-By-Parts in Time: The Second Derivative

We analyze the extension of summation-by-parts operators and weak boundary conditions for solving initial boundary value problems involving second derivatives in time. A wide formulation is obtained by first rewriting the problem on first order form. This formulation leads to optimally sharp fully discrete energy estimates that are unconditionally stable and high order accurate. Furthermore, it provides a natural way to impose mixed boundary conditions of Robin type, including time and space derivatives. We apply the new formulation to the wave equation and derive optimal fully discrete energy estimates for general Robin boundary conditions, including nonreflecting ones. The scheme utilizes wide stencil operators in time, whereas the spatial operators can have both wide and compact stencils. Numerical calculations verify the stability and accuracy of the method. We also include a detailed discussion on the added complications when using compact operators in time and give an example showing that an energy estimate cannot be obtained using a standard second order accurate compact stencil.

Cough Sound Analysis

Abstract Cough is the most common symptom of many respiratory diseases. Currently, no standardized methods exist for objective monitoring of cough, which could be commercially available and clinically acceptable. Our aim is to develop an algorithm which will be capable, according to the sound events analysis, to perform objective ambulatory and automated monitoring of frequency of cough. Because speech is the most common sound in 24-hour recordings, the first step for developing this algorithm is to distinguish between cough sound and speech. For this purpose we obtained recordings from 20 healthy volunteers. All subjects performed continuous reading of the text from the book with voluntary coughs at the indicated instants. The obtained sounds were analyzed using by linear and non-linear analysis in the time and frequency domain. We used the classification tree for the distinction between cough sound and speech. The median sensitivity was 100% and the median specificity was 95%. In the next step we enlarged the analyzed sound events. Apart from cough sounds and speech the analyzed sounds were induced sneezing, voluntary throat and nasopharynx clearing, voluntary forced ventilation, laughing, voluntary snoring, eructation, nasal blowing and loud swallowing. The sound events were obtained from 32 healthy volunteers and for their analysis and classification we used the same algorithm as in previous study. The median sensitivity was 86% and median specificity was 91%. In the final step, we tested the effectiveness of our developed algorithm for distinction between cough and non-cough sounds produced during normal daily activities in patients suffering from respiratory diseases. Our study group consisted from 9 patients suffering from respiratory diseases. The recording time was 5 hours. The number of coughs counted by our algorithm was compared with manual cough counts done by two skilled co-workers. We have found that the number of cough analyzed by our algorithm and manual counting, as well, were disproportionately different. For that reason we have used another methods for the distinction of cough sound from non-cough sounds. We have compared the classification tree and artificial neural networks. Median sensitivity was increasing from 28% (classification tree) to 82% (artificial neural network), while the median specificity was not changed significantly. We have enlarged our characteristic parameters of the Mel frequency cepstral coefficients, the weighted Euclidean distance and the first and second derivative in time. Likewise the modification of classification algorithm is under our interest

Simultaneous Formulation of the Theoretical Equations for Conductor Currents and Displacement Currents and Method of Three Dimensional Calculations

We propose theoretical formulation to connect the integral equations of conductor currents and differential equations of displacement currents in order to develop analytical techniques related to high-frequency electrical characteristics for noise and surge protection designs. Discretization of the obtained equations by the finite element method leads to only one matrix equation of 3D space and second-time derivative with the unknowns of current vector potentials because the displacement currents can be eliminated. In the frequency domain, the frequency characteristics of displacement currents in insulators and currents in conductors as well as impedances can be obtained with this method. To verify this method, we developed a 3D high-frequency currents calculation program. We calculated the impedances of test objects, and compared them with our measured impedances. The results were as follows. (1) The calculated resonance and anti-resonance frequencies of a simple 1D transmission line were within a 4.5% difference off measured ones below 1GHz. (2) The first resonance frequency of an LCR series circuit formed by the parasitic capacitance of a commercial power module was calculated by taking into account the wiring shape, and this frequency matched the measured value within 4.2%. These results validate our theoretical formulation and method.

Assessment of added mass effects on flutter boundaries using the Leishman–Beddoes dynamic stall model

We consider the dynamics of a typical airfoil section both in forced and free oscillations and investigate the importance of the added mass terms, i.e. the second derivatives in time of the pitch angle and plunge displacement. The structural behaviour is modelled by linear springs in pitch and plunge and the aerodynamic loading represented by our interpretation of the state-space version of the Leishman–Beddoes semi-empirical model. The added mass terms are often neglected since this leads to an explicit system of ODEs amenable for solution using standard ODE solvers. We analyse the effect of neglecting the added mass terms in forced oscillations about a set of mean angles of incidence by comparing the solutions obtained with the explicit and implicit systems of ODEs and conclude that their differences amount to a time lag that increases at a constant rate with increases of the reduced frequency. To determine the effect of the added mass terms in free oscillations, we introduce a spring offset angle to obtain static equilibrium positions at various degrees of incidence. We analyse the stability of the explicit and implicit aeroelastic systems about those positions and compare the locations of the respective flutter points calculated as Hopf bifurcation points. For low values of the spring offset angle, added mass effects are significant for low values of the mass ratio, or the ratio of natural frequencies, of the aeroelastic system. For high values of the spring offset angle, corresponding to stall flutter, we observe that their effect is greater for large values of the mass ratio.

Inhomogeneous tachyon condensation

We investigate the spacetime-dependent condensation of the tachyon in effective field theories. Previous work identified singularities in the field which appear in finite time: infinite gradients at the kinks, and (in the eikonal approximation) caustics near local minima. By performing a perturbation analysis, and with numerical simulations, we demonstrate and explain key features of the condensation process: perturbations generically freeze, and minima develop singular second derivatives in finite time (caustics). This last has previously been understood in terms of the eikonal approximation to the dynamics. We show explicitly from the field equations how this approximation emerges, and how the caustics develop, both in the DBI and BSFT effective actions. We also investigate the equation of state parameter of tachyon matter showing that it is small, but generically non-zero. The energy density tends to infinity near field minima with a charateristic profile. A proposal to regulate infinities by modifying the effective action is also studied. We find that although the infinities at the kinks are successfully regularised in the time-dependent case, caustics still present.

Damped-Wave Conduction and Relaxation in a Finite Sphere and Cylinder

Damped-Wave Conduction and Relaxation in a Finite Sphere and Cylinder

Hyperbolicity of scalar-tensor theories of gravity

Two first order strongly hyperbolic formulations of scalar-tensor theories of gravity allowing non- minimal couplings (Jordan frame) are presented along the lines of the 3 � 1 decomposition of spacetime. One is based on the Bona-Massoformulation, while the other one employs a conformal decomposition similar to that of Baumgarte-Shapiro-Shibata-Nakamura. A modified Bona-Massoslicing condition adapted to the scalar-tensor theory is proposed for the analysis. This study confirms that the scalar-tensor theory has a well-posed Cauchy problem even when formulated in the Jordan frame. Scalar-tensor theories of gravity (STT) are alternative theories of gravitation where a scalar field is coupled non- minimally to the curvature associated with the physical metric (this is the so-called Jordan frame representation). The term ''physical metric'' refers to a situation where test particles follow the geodesics of that metric. The variation of the action of the STT with respect to the physical metric gives rise to field equations which contain an effective energy-momentum tensor (EMT) involving second order derivatives in time and space of the scalar field. Such EMT has the property that ''ordinary matter,'' i.e., matter asso- ciated with fields other that the scalar field, obeys the (weak) equivalence principle which mathematically trans- lates into a conserved EMT for ordinary matter alone. Since a priori, it was not clear how such second order derivatives could be eliminated (in terms of lower order derivatives), or managed so as to obtain a quasilinear system of hyperbolic equations for which the Cauchy problem was well-posed (in the Hadamard sense), many people decided to abandon this approach in favor of the so- called Einstein frame representation where the nonminimal coupling is absorbed into the curvature by means of a conformal transformation of the metric. The new confor- mal metric is unphysical in the sense that (non-null) test particles do not follow the geodesics of that metric. However, the mathematical advantage is that the field equations for the nonphysical metric resemble the standard Einstein field equations with an unphysical effective EMT which involves at most first order derivatives of a suitable transformed scalar field (this EMT is unphysical because the ''ordinary matter'' part is not separately conserved). In the Einstein frame one can show that by using standard gauges (e.g., harmonic gauge) the field equations acquire the form required in the application of theorems that establish the well-posedness of the Cauchy problem. In view of the apparent mathematical advantages and disadvantages of the Jordan and Einstein frames, several

Green's Function Method for Perturbed sine-Gordon Equation**The project supported by National Natural Science Foundation of China under Grant Nos. 10375041 and 10174054

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.

Translational and rotational effects upon the Rayleigh spectra of ethane

The time-correlation functions (tcfs) corresponding to the spectral intensities and band shapes of the Rayleigh spectra for ethane at several densities and a fixed temperature are examined. The usual tcfs are supplemented by second time derivative (STD) tcfs. It is argued that these functions give a much improved description of the relative importance of translation and rotation in determining the spectra. Evaluations of the STD tcfs for a simple two-site model of the ethane molecule plus the dipole-induced dipole (DID) model for the polarizability induction show rather clearly that it is the translational degrees of freedom that are most important in determining the interaction-induced spectra and tcfs. The density dependence of the separate 2-, 3-, and 4-body terms in the STD tcfs is also shown. The most important factors in determining the Rayleigh spectra of ethane under the chosen conditions are deduced from these tcfs.