Variational iteration method for solving the multi-pantograph delay equation

In this Letter, the variational iteration method is applied to solve the multi-pantograph delay equation. Sufficient conditions are given to assure the convergence of the method. Examples show that the method is effective.     

Differential transform method for solving linear and non-linear systems of partial differential equations

In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

Generalized differential transform method to differential-difference equation

In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Padé technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.     

Solution of singular two-point boundary value problems using differential transformation method

In this Letter, we implemented relatively new, exact series method of solution known as the differential transform method for solving singular two-point boundary value problems. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

Efficient algorithms for discrete lattice calculations

We discuss algorithms for lattice-based computations, in particular lattice reduction, the detection of nearest neighbors, and the computation of clusters of nearest neighbors. We focus on algorithms that are most efficient for low spatial dimensions (typically d=2,3$d=2\text{,}3$) and input data within a reasonably limited range. This makes them most useful for physically oriented numerical simulations, for example of crystalline solids. Different solution strategies are discussed, formulated as algorithms, and numerically evaluated.     

Delay-range-dependent stability for fuzzy BAM neural networks with time-varying delays

This Letter considers the problem of delay-range-dependent stability for fuzzy bi-directional associative memory (BAM) neural networks with time-varying interval delays. Based on Lyapunov-Krasovskii theory, the delay-range-dependent stability criteria are derived in terms of linear matrix inequalities (LMIs). By constructing new Lyapunov-Krasovskii functional, stability conditions are dependent on the upper and lower bounds of the delays, which is made possible by using some advanced techniques for achieving delay dependence. A numerical example is given to illustrate the effectiveness of the proposed method.     

Self-consistent solution of cosmological radiation-hydrodynamics and chemical ionization

We consider a PDE system comprising compressible hydrodynamics, flux-limited diffusion radiation transport and chemical ionization kinetics in a cosmologically-expanding universe. Under an operator-split framework, the cosmological hydrodynamics equations are solved through the piecewise parabolic method, as implemented in the Enzo community hydrodynamics code. The remainder of the model, including radiation transport, chemical ionization kinetics, and gas energy feedback, form a stiff coupled PDE system, which we solve using a fully-implicit inexact Newton approach, and which forms the crux of this paper. The inner linear Newton systems are solved using a Schur complement formulation, and employ a multigrid-preconditioned conjugate gradient solver for the inner Schur systems. We describe this approach and provide results on a suite of test problems, demonstrating its accuracy, robustness, and scalability to very large problems.     

Passivity analysis for uncertain neural networks with discrete and distributed time-varying delays

In this Letter, the passivity problem of uncertain neural networks with discrete and distributed time-varying delays is investigated. New delay-dependent conditions for this problem are obtained by using a novel Lyapunov functional together with the linear matrix inequality (LMI) approach. Numerical examples are given to show the effectiveness of our theoretical results.     

Coupled modes of the torsion pendulum

The coupling of the swing modes of a torsion pendulum to the torsional mode has been solved analytically. Our solution provides a clear explanation of why a magnetic damper is effective in suppressing unwanted modes in most gravitational experiments. The analytic solution also shows that the amplitude of the mode coupling is greatest at the lowest frequencies. This explains why mode coupling at the lowest frequency is all that is observed in the experiments reported here.     

Mechanisms of small clusters production by short and ultra-short laser ablation

The mechanisms involved into the formation of clusters by pulsed laser ablation are studied both numerically and experimentally. To facilitate the model validation by comparison with experimental results, the time and length scales of the simulation are considerably increased. This increase is achieved by using a combination of molecular dynamics (MD) and the direct simulation Monte Carlo (DSMC) methods. The combined MD-DSMC model is then used to compare the relative contribution of the two channels of the cluster production by laser ablation: (i) direct cluster ejection upon the laser-material interaction, and (ii) collisional sticking and aggregation in the ablated gas flow. Calculation results demonstrate that both of these mechanisms play a role. The initial cluster ejection provides cluster precursors thus eliminating the three-body collision bottleneck in the cluster growth process. The presence of clusters thus facilitates the following collisional condensation and evaporation processes. The rates of these processes become considerable, leading to the modification of not only the plume cluster composition, but also the dynamics of the plume expansion. Calculation results explain several recent experimental findings.     

Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences

In this research, two powerful analytical methods are introduced to handle nonlinear good Boussinesq, heat transfer and coupled Burgers' equations. One is the homotopy-perturbation method (HPM) and the other is the variational iteration method (VIM). VIM is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. HPM converts a difficult problem into a simple one, which can be easily handled. The results attained in this paper confirm the idea that HPM and VIM are powerful mathematical tools and that they can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.     

Compact-2D FDTD for waveguides including materials with negative dielectric permittivity, magnetic permeability and refractive index

An efficient compact-2D finite-difference time-domain method is presented for the numerical analysis of guided modes in waveguides that may include negative dielectric permittivity, negative magnetic permeability and negative refractive index materials. Both complex variable and real variable methods are given. The method is demonstrated for the analysis of channel-plasmon-polariton guided modes in triangular groves on a metal surface. The presented method can be used for a range of waveguide problems that were previously unsolvable analytically, due to complex geometries, or numerically, due to computational requirements of conventional three-dimensional finite-difference time-domain methods. A three-dimensional finite-difference time-domain algorithm that also allows analysis in the presence of bound or free electric and equivalent magnetic charges is presented and an example negative refraction demonstrates the method.     

Synthesis of nanoclusters by nanosecond laser ablation: Direct simulation Monte Carlo modelling

Collisional processes leading to the formation of nanoparticles in a laser-ablated plume are numerically simulated with the aid of an atomistic-level model based on direct simulation Monte Carlo (DSMC) method. The formation of nanoparticles in nanosecond laser ablation of a mono-atomic target is investigated in the presence of an inert background gas. The DSMC procedure is modified in order to account for numerous plume species and to describe several reactions (i.e., recombination/dissociation, sticking, evaporation) taking place in the plume and affecting the size and spatial distribution of the produced nanoclusters. Calculation results allow us to visualize the nanoparticles and to correlate their space distributions with plume dynamics. In addition, cluster size distributions are investigated at different pressures. The effects of the background gas on cluster formation within the plume are furthermore shown.     

He's homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients

In this Letter, He's homotopy perturbation method is applied to heat-like and wave-like equations with variable coefficients. The solutions are introduced in this Letter are in recursive sequence forms which can be used to obtain the closed form of the solutions if they are required. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method.     

Calculations of Photo-Ionization Cross Sections for Lithium Atoms

Photo-ionization cross sections for the ground and the ≤5 excited states of lithium atoms are calculated in the photoelectron energy ranging from threshold to 0.5 Rydberg. The wavefunctions for both the bound and continuum states are obtained by solving the SchrSdinger equation numerically in a symplectic scheme. Our results are in excellent agreement with the recent experimental measurements and in harmony with other theoretical calculations.     

Two Kinds of Square-Conservative Integrators for Nonlinear Evolution Equations

Based on the Lie-group and Gauss-Legendre methods, two kinds of square-conservative integrators for square- conservative nonlinear evolution equations are presented. Lie-group based square-conservative integrators are linearly implicit, while Gauss-Legendre based square-conservative integrators are nonlinearly implicit and iterative schemes are needed to solve the corresponding integrators. These two kinds of integrators provide natural candidates for simulating square-conservative nonlinear evolution equations in the sense that these integrators not only preserve the square-conservative properties of the continuous equations but also are nonlinearly stable. Numerical experiments are performed to test the presented integrators.     

Solution of the MHD Falkner-Skan flow by Hankel-Padé method

In this Letter, an analysis is performed to find the unknown skin friction coefficient of the boundary layer Falkner-Skan equation for wedge. The governing equation takes into account the effect of a non-uniform magnetic field. The results are obtained by using Hankel-Padé method.     

Resonant-Like Activation in a Bistable System with Noise and Time Delay

A bistable system with noise and time delay is investigated. Theoretical analysis and stochastic simulation show that： （i） In the ease of a system driven only by multiplicative Gaussian white noise, the mean first-passage time for a particle to reach the other stable state from one stable state exhibits a minimum with respect to delay time, i.e., a resonant-like activation （RA） phenomenon. （ii） In the action of additive and multiplicative noise, as the additive noise intensity increases, no matter whether a correlation between the two types of noise exists or not, the RA gradually disappears. （iii） The correlation strength between the two types of noise does not influence the existence of the RA.     

Transient and Stationary Simulations for a Quantum Hydrodynamic Model

The transient and stationary characteristics of a one-dimensional quantum hydrodynamic model are comparatively studied for semiconductor charge transport in a resonant tunnelling diode. When the bias is not small, our numerical results show a deviation of the asymptotic transient solutions from the stationary ones. A dynamic instability accounts for such deviation. The stationary quantum hydrodynamic model is therefore unsuitable in general for simulating quantum devices.     

On uniqueness of solution of a reverse problem of magnetic location

An analysis of solution uniqueness for a problem of magnetic dipole localization based on known values of magnetic field strength vectors at two points is presented. It is shown that a usage of two triples (pairs) of sensors is sufficient to solve a 3D (2D) problem of arbitrary magnetic dipole localization with a satisfactory number of crude errors.