The fully implicit stochastic-alpha method for stiff stochastic differential equations

A fully implicit integration method for stochastic differential equations with significant multiplicative noise and stiffness in both the drift and diffusion coefficients has been constructed, analyzed and illustrated with numerical examples in this work. The method has strong order 1.0 consistency and has user-selectable parameters that allow the user to expand the stability region of the method to cover almost the entire drift-diffusion stability plane. The large stability region enables the method to take computationally efficient time steps. A system of chemical Langevin equations simulated with the method illustrates its computational efficiency.

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier-Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e(vertical bar k vertical bar/kd) at high wavenumbers vertical bar k vertical bar. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e-(C(k/kd) ln(vertical bar k vertical bar/kd)) for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C-* = 1/ ln 2. The same behavior with a universal constant C-* is conjectured for the Navier-Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier-Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.

Spiral-Wave Turbulence and Its Control in the Presence of Inhomogeneities in Four Mathematical Models of Cardiac Tissue

Regular electrical activation waves in cardiac tissue lead to the rhythmic contraction and expansion of the heart that ensures blood supply to the whole body. Irregularities in the propagation of these activation waves can result in cardiac arrhythmias, like ventricular tachycardia (VT) and ventricular fibrillation (VF), which are major causes of death in the industrialised world. Indeed there is growing consensus that spiral or scroll waves of electrical activation in cardiac tissue are associated with VT, whereas, when these waves break to yield spiral- or scroll-wave turbulence, VT develops into life-threatening VF: in the absence of medical intervention, this makes the heart incapable of pumping blood and a patient dies in roughly two-and-a-half minutes after the initiation of VF. Thus studies of spiral- and scroll-wave dynamics in cardiac tissue pose important challenges for in vivo and in vitro experimental studies and for in silico numerical studies of mathematical models for cardiac tissue. A major goal here is to develop low-amplitude defibrillation schemes for the elimination of VT and VF, especially in the presence of inhomogeneities that occur commonly in cardiac tissue. We present a detailed and systematic study of spiral- and scroll-wave turbulence and spatiotemporal chaos in four mathematical models for cardiac tissue, namely, the Panfilov, Luo-Rudy phase 1 (LRI), reduced Priebe-Beuckelmann (RPB) models, and the model of ten Tusscher, Noble, Noble, and Panfilov (TNNP). In particular, we use extensive numerical simulations to elucidate the interaction of spiral and scroll waves in these models with conduction and ionic inhomogeneities; we also examine the suppression of spiral- and scroll-wave turbulence by low-amplitude control pulses. Our central qualitative result is that, in all these models, the dynamics of such spiral waves depends very sensitively on such inhomogeneities. We also study two types of control chemes that have been suggested for the control of spiral turbulence, via low amplitude current pulses, in such mathematical models for cardiac tissue; our investigations here are designed to examine the efficacy of such control schemes in the presence of inhomogeneities. We find that a local pulsing scheme does not suppress spiral turbulence in the presence of inhomogeneities; but a scheme that uses control pulses on a spatially extended mesh is more successful in the elimination of spiral turbulence. We discuss the theoretical and experimental implications of our study that have a direct bearing on defibrillation, the control of life-threatening cardiac arrhythmias such as ventricular fibrillation.

Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes

This paper represents the effect of nonlocal scale parameter on the wave propagation in multi-walled carbon nanotubes (MWCNTs). Each wall of the MWCNT is modeled as first order shear deformation beams and the van der Waals interactions between the walls are modeled as distributed springs. The studies shows that the scale parameter introduces certain band gap region in both flexural and shear wave mode where no wave propagation occurs. This is manifested in the wavenumber plots as the region where the wavenumber tends to infinite (or group speed tends to zero). The frequency at which this phenomenon occurs is called the ``Escape frequency''. The analysis shows that, for a given N-walled carbon nanotube (CNT). the nonlocal scaling parameter has a significant effect on the shear wave modes of the N - 1 walls. The escape frequencies of the flexural and shear wave modes of the N-walls are inversely proportionl to the nonlocal scaling parameter. It is also shown that the cut-off frequencies are independent of the nonlocal scale parameter. (C) 2009 Elsevier B.V. All rights reserved.

A new framework for solution of multidimensional population balance equations

A new framework is proposed in this work to solve multidimensional population balance equations (PBEs) using the method of discretization. A continuous PBE is considered as a statement of evolution of one evolving property of particles and conservation of their n internal attributes. Discretization must therefore preserve n + I properties of particles. Continuously distributed population is represented on discrete fixed pivots as in the fixed pivot technique of Kumar and Ramkrishna [1996a. On the solution of population balance equation by discretization-I A fixed pivot technique. Chemical Engineering Science 51(8), 1311-1332] for 1-d PBEs, but instead of the earlier extensions of this technique proposed in the literature which preserve 2(n) properties of non-pivot particles, the new framework requires n + I properties to be preserved. This opens up the use of triangular and tetrahedral elements to solve 2-d and 3-d PBEs, instead of the rectangles and cuboids that are suggested in the literature. Capabilities of computational fluid dynamics and other packages available for generating complex meshes can also be harnessed. The numerical results obtained indeed show the effectiveness of the new framework. It also brings out the hitherto unknown role of directionality of the grid in controlling the accuracy of the numerical solution of multidimensional PBEs. The numerical results obtained show that the quality of the numerical solution can be improved significantly just by altering the directionality of the grid, which does not require any increase in the number of points, or any refinement of the grid, or even redistribution of pivots in space. Directionality of a grid can be altered simply by regrouping of pivots.

Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements

This paper presents a formulation of an approximate spectral element for uniform and tapered rotating Euler-Bernoulli beams. The formulation takes into account the varying centrifugal force, mass and bending stiffness. The dynamic stiffness matrix is constructed using the weak form of the governing differential equation in the frequency domain, where two different interpolating functions for the transverse displacement are used for the element formulation. Both free vibration and wave propagation analysis is performed using the formulated elements. The studies show that the formulated element predicts results, that compare well with the solution available in the literature, at a fraction of the computational effort. In addition, for wave propagation analysis, the element shows superior convergence. (C) 2007 Elsevier Ltd. All rights reserved.

Inverted slot-mode slow-wave structures for traveling-wave tubes

The dispersion and impedance characteristics of an inverted slot-mode (ISM) slow-wave structure computed by three different techniques, i.e., an analytical model based on a periodic quasi-TEM approach, an equivalent-circuit model, and 3-D electromagnetic simulation are obtained and compared. The comparison was carried out for three different slot-mode structures at S-, C-, and X-bands. The approach was also validated with experimental measurements on a practical X-band ISM traveling-wave tube. The design of ferruleless ISM slow-wave structures, both in circular and rectangular formats, has also been proposed and the predicted dispersion characteristics for these two geometries are compared with 3-D simulation and cold-test measurements. The impedance characteristics for all three designs are also compared.

Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives

A modified form of Green's integral theorem is employed to derive the energy identity in any water wave diffraction problem in a single-layer fluid for free-surface boundary condition with higher-order derivatives. For a two-layer fluid with free-surface boundary condition involving higher-order derivatives, two forms of energy identities involving transmission and reflection coefficients for any wave diffraction problem are also derived here by the same method. Based on this modified Green's theorem, hydrodynamic relations such as the energy-conservation principle and modified Haskind–Hanaoka relation are derived for radiation and diffraction problems in a single as well as two-layer fluid.

Bond-order wave phase, spin solitons, and thermodynamics of a frustrated linear spin-1/2 Heisenberg antiferromagnet

The linear spin-1/2 Heisenberg antiferromagnet with exchanges J(1) and J(2) between first and second neighbors has a bond-order wave (BOW) phase that starts at the fluid-dimer transition at J(2)/J(1)=0.2411 and is particularly simple at J(2)/J(1)=1/2. The BOW phase has a doubly degenerate singlet ground state, broken inversion symmetry, and a finite-energy gap E-m to the lowest-triplet state. The interval 0.4 < J(2)/J(1) < 1.0 has large E-m and small finite-size corrections. Exact solutions are presented up to N = 28 spins with either periodic or open boundary conditions and for thermodynamics up to N = 18. The elementary excitations of the BOW phase with large E-m are topological spin-1/2 solitons that separate BOWs with opposite phase in a regular array of spins. The molar spin susceptibility chi(M)(T) is exponentially small for T << E-m and increases nearly linearly with T to a broad maximum. J(1) and J(2) spin chains approximate the magnetic properties of the BOW phase of Hubbard-type models and provide a starting point for modeling alkali-tetracyanoquinodimethane salts.

Modeling and analysis of a tunable piezoelectric structure for transverse shear wave generation

An analytical investigation of the transverse shear wave mode tuning with a resonator mass (packing mass) on a Lead Zirconium Titanate (PZT) crystal bonded together with a host plate and its equivalent electric circuit parameters are presented. The energy transfer into the structure for this type of wave modes are much higher in this new design. The novelty of the approach here is the tuning of a single wave mode in the thickness direction using a resonator mass. First, a one-dimensional constitutive model assuming the strain induced only in the thickness direction is considered. As the input voltage is applied to the PZT crystal in the thickness direction, the transverse normal stress distribution induced into the plate is assumed to have parabolic distribution, which is presumed as a function of the geometries of the PZT crystal, packing mass, substrate and the wave penetration depth of the generated wave. For the PZT crystal, the harmonic wave guide solution is assumed for the mechanical displacement and electric fields, while for the packing mass, the former is solved using the boundary conditions. The electromechanical characteristics in terms of the stress transfer, mechanical impedance, electrical displacement, velocity and electric field are analyzed. The analytical solutions for the aforementioned entities are presented on the basis of varying the thickness of the PZT crystal and the packing mass. The results show that for a 25% increase in the thickness of the PZT crystal, there is ~38% decrease in the first resonant frequency, while for the same change in the thickness of the packing mass, the decrease in the resonant frequency is observed as ~35%. Most importantly the tuning of the generated wave can be accomplished with the packing mass at lower frequencies easily. To the end, an equivalent electric circuit, for tuning the transverse shear wave mode is analyzed.

The third post-Newtonian gravitational wave polarizations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits

The gravitational waveform (GWF) generated by inspiralling compact binaries moving in quasi-circular orbits is computed at the third post-Newtonian (3PN) approximation to general relativity. Our motivation is two-fold: (i) to provide accurate templates for the data analysis of gravitational wave inspiral signals in laser interferometric detectors; (ii) to provide the associated spin-weighted spherical harmonic decomposition to facilitate comparison and match of the high post-Newtonian prediction for the inspiral waveform to the numerically-generated waveforms for the merger and ringdown. This extension of the GWF by half a PN order (with respect to previous work at 2.5PN order) is based on the algorithm of the multipolar post-Minkowskian formalism, and mandates the computation of the relations between the radiative, canonical and source multipole moments for general sources at 3PN order. We also obtain the 3PN extension of the source multipole moments in the case of compact binaries, and compute the contributions of hereditary terms (tails, tails-of-tails and memory integrals) up to 3PN order. The end results are given for both the complete plus and cross polarizations and the separate spin-weighted spherical harmonic modes.

Semigroups on Frechet spaces and equations with infinite delays

In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions.

The partition of unity finite element method for elastic wave propagation in Reissner-Mindlin plates

This paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four-noded element for the Reissner-indlin plate is derived in this paper, which is free of shear locking. Such a locking-free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency-dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright (C)006 John Wiley & Sons, Ltd.

Formal Verification of Full-Wave Rectifier: A Case Study

We present a case study of formal verification of full-wave rectifier for analog and mixed signal designs. We have used the Checkmate tool from CMU [1], which is a public domain formal verification tool for hybrid systems. Due to the restriction imposed by Checkmate it necessitates to make the changes in the Checkmate implementation to implement the complex and non-linear system. Full-wave rectifier has been implemented by using the Checkmate custom blocks and the Simulink blocks from MATLAB from Math works. After establishing the required changes in the Checkmate implementation we are able to efficiently verify, the safety properties of the full-wave rectifier.

Boundary value problems for third-order nonlinear ordinary differential equations

In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here.