Second-order Time Derivatives Research Articles

Collocation methods for the solution of von-Kármán dynamic non-linear plate systems

The von-Kármán nonlinear, dynamic, partial differential system over rectangular domains is considered, and numerically solved using both the Chebyshev-collocation and Legendre-collocation methods for the spatial discretization and the implicit Newmark- β scheme combined with a non-linear fixed point algorithm for the temporal discretization. As the system is non-linear, involving operators of different orders, different timescales, and may contain initial/boundary incompatible conditions at the domain's corners, it is our aim to highlight some of the difficulties inherent in its numerical treatment using collocation methods. We begin by examining the Chebyshev-collocation scheme considered in [Comput. Meth. Appl. Mech. Engrg. 193 (6–8) (2004) 575]. In that work, filtering was used to stabilize the full von-Kármán system, however the source of the instability was not discussed. We first show that IC/BC incompatibilities are not the culprits and demonstrate empirically that the source of the instability was the numerical treatment of the linear cross-derivative terms found in the in-plane equations. We prove mathematically that although the continuous linear system is well-posed, the Chebyshev-collocation solution of problems involving cross-derivatives and second-order time derivatives is unstable. Instead of adding sufficient filtering to counter the energy growth due to the linear terms, we employ the Legendre-collocation method. We show empirically and prove that the Legendre-collocation scheme for the linear system has the necessary characteristics to remain stable. Nevertheless, the collocation treatment of the non-linear terms still causes the solution to be unstable at long time. We thus return to advocating a filtering solution, but in this case only filtering the non-linear terms to compensate for the inherent aliasing error due to our collocation treatment of these terms. We conclude by comparing the results of filtered Chebyshev-collocation solution, non-linear terms filtered Legendre-collocation solution, and a simplified von-Kármán (solution omitting the inertial terms in the in-plane equations) as presented in (loc. cit).

New experimental proposals for testing Dirac equation

The advent of phenomenological quantum gravity has ushered us in the search for experimental tests of the deviations from general relativity predicted by quantum gravity or by string theories, and as a by-product of this quest the possible modifications that some field equations, for instance, the motion equation of spin-1/2-particles, have already been considered. In the present Letter a modified Dirac equation, whose extra term embraces a second-order time derivative, is taken as mainstay, and three different experimental proposals to detect it are put forward. The novelty in these ideas is that two of them do not fall within the extant approaches in this context, to wit, red-shift, atomic interferometry, or Hughes–Drever type-like experiments.

Observers for a class of 3D continuous chaotic systems

In this paper the design of observers of a class of 3D continuous chaotic systems is discussed. Only using a state variable and its (first-and second-order) time derivatives, one can construct a control law to implement the synchronization between the investigated chaotic systems and its observers.

Fractional Diffusive Waves

By fractional diffusive waves we mean the solutions of the so-called time-fractional diffusion-wave equation. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with a fractional derivative of order β∈(0,2) and is expected to govern evolution processes intermediate between diffusion and wave propagation when β∈(1,2). Here it is shown to govern the propagation of stress waves in viscoelastic media which, by exhibiting a power law creep, are of relevance in acoustics and seismology since their quality factor turns out to be independent of frequency. The fundamental solutions for the Cauchy and signaling problems are expressed in terms of entire functions (of Wright type) in the similarity variable. Their behaviors turn out to be intermediate between those found in the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. Furthermore, their scaling properties and the relations with some stable probability distributions are outlined.

FRACTIONAL DIFFUSIVE WAVES

By fractional diffusive waves we mean the solutions of the so-called time-fractional diffusion-wave equation. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with a fractional derivative of order β∈(0,2) and is expected to govern evolution processes intermediate between diffusion and wave propagation when β∈(1,2). Here it is shown to govern the propagation of stress waves in viscoelastic media which, by exhibiting a power law creep, are of relevance in acoustics and seismology since their quality factor turns out to be independent of frequency. The fundamental solutions for the Cauchy and signaling problems are expressed in terms of entire functions (of Wright type) in the similarity variable. Their behaviors turn out to be intermediate between those found in the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. Furthermore, their scaling properties and the relations with some stable probability distributions are outlined.

A Differential Equation of the Second Order with Respect to Time in a Domain with Moving Boundary

0 <k< m, with second-order time derivatives. This equation was considered in the work [1, Chap. 5, Sec. 3; 2] in the case of cylindrical domains (t 2 [0; 1], x 2 [0; 2]). Earlier, problems in noncylindrical domains (domains with moving boundary) were considered for the heat equation [3, Chap. 6, Sec. 3], a parabolic equation of the rst order with respect to time [4], and the wave equation [5] [which can be obtained from (1) by setting k =0a ndm = 1 and by omitting the second term on the left-hand side]. In the present paper, we analyze Eq. (1) of the second order with respect to t for 0 <k<m (the parabolic type) in a domain with moving boundary and prove the existence and uniqueness theorem for an initial-boundary value problem for this equation. Let us describe the domain. By G1 we denote the domain in the (t;x)-plane bounded by the lines t =0 ,t =1 , andx =0a nd the curvex = (t), and by G2 we denote the domain bounded by the lines t =0a ndx =2a nd the curvex = (t). Here (t) is a given monotone increasing function on the interval [0; 1] such that (0) = x02 (0; 2] and (1) = 2. Consider the following problem. Find a solution U = U(t;x) of Eq. (1) in the rectangle G1[G2[f(t;x ): x = (t) ;t 2 (0; 1)g such that U vanishes on G2 and satises the initial conditions U(0;x )= U0(x) ;U 0 t (0;x )= U1(x) ;x 2 [0;x0]; (2) U0(x )=0 ,U1(x )=0 ,x2 (x0; 2], and the boundary conditions U(t; 0) = U(t; (t)) = @U(t; 0)=@x = @U(t; (t))=@x = = @ m 1 U(t; 0)=@x m 1 = @ m 1 U(t; (t))=@x m 1 =0 ; @ 2m U(t; 0)=@x 2m = @ 2m U(t; (t))=@x 2m = = @ 2m+k 1 U(t; 0)=@x 2m+k 1 = @ 2m+k 1 U(t; (t))=@x 2m+k 1 =0

Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time

This paper is concerned with numerical analysis of the transient elastodynamic problems by the dual reciprocity BEM (DRBEM) in space combined with the differential quadrature method (DQM) in time. Emphasis is placed on a comparative study of various time-marching schemes. Three numerical examples considered are for the longitudinal forced vibration of plates and the transverse free vibration of membranes. To authors’ best knowledge, this is the first attempt to apply the DQM to the second-order time derivative in the DRBEM elastodynamic formulations. A recent approach using boundary conditions in the DQM was here extended to handle the initial conditions of elastodynamic problems. The resulting algebraic formulation is a Lyapunov matrix equation, which can be very efficiently solved by the Bartels–Stewart algorithm. It is revealed that the DQM is an unconditionally stable algorithm and gives much better accuracy than the standard finite difference schemes such as the Wilson θ, Newmark and Houbolt methods, for the same time step size for the cases considered.

Diffusion Waves and their Uses

The concept of waves is an integral part of our scientific culture and has nourished physicists, pure and applied alike, for centuries. Many important discoveries in physics, including quantum mechanics, have involved wave phenomena. The wave concept owes some of its scientific success to its mathematical tractability. Linear wave equations—the sort that describe the transmission of sound and radio waves through air—contain a nonzero second-order time derivative, which gives rise mathematically to the rich and familiar array of properties we associate with waves, such as wavefront propagation, reflection, and refraction.

Wright functions as scale-invariant solutions of the diffusion-wave equation

The time-fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α (0< α⩽2). Using the similarity method and the method of the Laplace transform, it is shown that the scale-invariant solutions of the mixed problem of signalling type for the time-fractional diffusion-wave equation are given in terms of the Wright function in the case 0< α<1 and in terms of the generalized Wright function in the case 1< α<2. The reduced equation for the scale-invariant solutions is given in terms of the Caputo-type modification of the Erdélyi–Kober fractional differential operator.

Numerical coupled Liouville approach: Application to second hyperpolarizability of molecular aggregate

In our previous study Int. J. Quant. Chem., to appear , we have developed a novel numerical calculation scheme for a dynamics of quantum network for linear molecular aggregates under intense time-dependent electric fields. In this approach, each molecule is assumed to be an electric dipole arranged linearly with an angle from the longitudinal axis, and the molecular interactions are taken into account by adding the radiations from these dipoles to the external electric fields. The effects of the radiations from all the dipoles involve the intermolecular distance, the speed of light, retarded polarization, and its first- and second-order time derivatives at the position of each dipole. The quantum dynamics is performed by solving coupled Liouville equations composed of the Liouville equation for each dipole. In the present study, we develop a calculation approach of nonperturbative second hyperpolarizability g in our novel approach and examine the g of dimer models composed of two-state molecules under the one-photon near resonant intense laser fields. Similar phase transition-like behavior in the field-intensity dependence of the g is observed. We also investigate the second hyperpolarizability spectra in the three-photon resonant region for dimers composed of three-state molecules, which mimic the electronic states of allyl cation. Contrary to the one-photon resonant case, phase transition-like behavior is not observed in the intensity dependence of g in the three-photon resonant region. Q 1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 295)306, 1999

On the viscous and resistive dissipation of magnetohydrodynamic waves

The dissipative magnetohydrodynamic (MHD) equations, in the presence of Ohmic electrical resistance and bulk and shear viscosities, are linearized relative to a mean state of rest, and eliminated for the velocity perturbation. The result is a vector dissipative MHD equation consisting of (i) nondissipative terms, involving second-order space and time derivatives; (ii) dissipative terms, linear or quadratic in the diffusivities, involving fourth- and fifth-order space–time derivatives. Considering plane wave solutions, the dispersion matrix shows that Alfvén waves decouple, and are damped by Ohmic electrical resistance and shear viscosity. The latter two, and the bulk viscosity, damp the coupled slow and fast modes, which satisfy a dispersion relation of degree five in the frequency; it reduces to degree three in the case of magnetosonic waves, i.e., wave vector transverse to the direction of propagation. The roots of these dispersion relations of degree three to five, are obtained by a novel method, which applies first to the case of weak damping, and then can be extended to strong damping, e.g., taking into account products of diffusivities. For most applications, e.g., MHD waves in the solar atmosphere, the second-order effects of products of diffusivities are small.

A Maximum Principle for Optimal Control Using Spatially Distributed Pointwise Controllers

The problem of characterizing optimal controls for a class of damped distributed parameter systems is considered. The system is governed by a variable-coefficient, linear-partial differential equation involving up to second-order time derivatives and up to fourth-order space derivatives of the state variable. Pointwise controllers extending over the spatial region occupied by the system are considered. A class of performance indexes is introduced that includes convex functions of the state velocity, spatial state derivatives, and the control force. The state variable and its derivatives are evaluated at a preassigned terminal time. A maximum principle is given that facilitates the determination of the optimal control, which is shown to be unique. The use of the maximum principle is demonstrated by determining the optimal pointwise control of the vibrations for a uniform undamped Euler-Bernoulli beam.

Electrodynamic response of disordered Josephson-junction arrays

The dynamic impedance $Z$ of a diluted Josephson-junction array (with missing bonds or missing sites) is calculated by the coherent potential approximation and by Mori's technique for evaluating dynamic correlation functions. The linearized equations of motion for the superconducting phases take into account the Josephson coupling between neighboring sites, resistive losses (either between nearest-neighbor sites or between each site and the substrate), and a possible ``capacitive'' term, given by a second-order time derivative. The results exhibit a crossover between a low-frequency ($Z$ constant) and a ``fractonic'' regime in which $Z$ varies as a power of frequency. Good overall agreement is found with experimental data, but some finer details of the latter need further investigation.

What happens with an initially kicked soliton?

Abstract A simple method to study the response of solitons in overdamped dissipative systems to external impulsive perturbations is developed. Thanks to the nontrivial choice of a small parameter, the perturbative scheme captures genuine nonlinear phenomena. The method is developed and tested by numerical simulations for kinks in 1 + 1 dimensions and for skyrmions in 2 + 1 dimensions. An extension to models including second-order time derivatives is also discussed.

The fundamental solutions for the fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0 < β ≤ 1 2 or 1 2 < β ≤ 1 , respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and Signalling problems can be expressed in terms of an auxiliary function M( z; β), where z = |x| t β is the similarity variable. Such function is proved to be an entire function of Wright type.

The time fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0<β≤1/2 or 1/2<β≤1, respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M (z; β), where z is the similarity variable. Such function, which reduces to the well-known Gaussian function for β=1/2 (ordinary diffusion), is proved to be an entire function of Wright type.

Multilayer perceptrons as labelers for hidden Markov models

A novel combination of multilayer perceptrons (MLPs) and hidden Markov models (HMMs) is presented. Instead of using MLPs as probability generators for HMMs, the authors propose to use MLPs as labelers for discrete parameter HMM's. Compared with the probabilistic interpretation of MLPs, this gives them the advantage of flexibility in system design (e.g., the use of word models instead of phonetic models while using the same MLPs). Moreover, since they do not need to reach a global minimum, they can do with MLPs with fewer hidden nodes, which can be trained faster. In addition, they do not need to retrain the MLPs with segmentations generated by a Viterbi alignment. Compared with Euclidean labeling, their method has the advantages of needing fewer HMM parameters per state and obtaining a higher recognition accuracy. Several improvements of the baseline MLP labeling are investigated. When using one MLP, the best results are obtained when giving the labels a fuzzy interpretation. It is also possible to use parallel MLPs where each is based on a different parameter set (e.g., basic parameters, their time derivatives, and their second-order time derivatives). This strategy increases the recognition results considerably. A final improvement is the training of MLPs for subphoneme classification.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The inertia of a crack near a dislocation

Abstract The dynamics of a mode-III crack in interaction with a nearby screw dislocation is studied within the framework of elasticity theory by thinking of the crack as a continuous superposition of infinitesimal dislocations and using an expansion valid for short distances and small accelerations. The resulting equation of motion involves (at least) second-order time derivatives of the crack tip position due to the presence of the dislocation. Thus, contrary to what happens for an isolated crack, the crack considered here has a dislocation-induced inertia. The method employed should be generalizable to situations other than an antiplane.

Infrared behavior of quasilocal systems at finite temperature.

We adopt the method developed recently for treating the infrared behavior of finite-size and quasilocal systems to study the symmetry behavior of dynamical quantum fields at finite temperature in cosmological spacetimes. We find that for slowly varying fields at high temperature the critical temperature is increased by an amount proportional to the second-order time derivative (``deceleration'') of the background field. This work can be viewed as a preliminary step towards a fuller investigation of critical dynamics in the early Universe involving nonequilibrium quantum fields.

Linear growth of a liquid droplet divided from its vapour by a “soap bubble”-like fluid interface

The theory proposed in [1,3] is particularized to describe the spherically symmetric growth, in the neighbourhood of an equilibrium state far from the critical temperature, of a liquid incompressible droplet surrounded by its vapour when the interface between the phases behaves as a fluid membrane which resembles a soap bubble. A free moving boundary problem for an integro-differential equation of parabolic type is deduced in which the second-order time derivative of the radius R(t) of the droplet appears, together with a volume source term which depends on the history of R.