Technique For Differential Equations Research Articles

Application of ADE-FDTD Method in Lossy Lorentz Media

A finite-difference time-domain method based on the auxiliary differential equation (ADE) technique is used to obtain the formulation of 2-D TM wave propagation in lossy Lorentz media. In the paper, the reflected coefficients calculated by ADE-FDTD method and the exact theoretical result are better agreement.

Parameter identification of aggregated thermostatically controlled loads for smart grids using PDE techniques

This paper develops methods for model identification of aggregated thermostatically controlled loads (TCLs) in smart grids, via partial differential equation (PDE) techniques. Control of aggregated TCLs provides a promising opportunity to mitigate the mismatch between power generation and demand, thus enhancing grid reliability and enabling renewable energy penetration. To this end, this paper focuses on developing parameter identification algorithms for a PDE-based model of aggregated TCLs. First, a two-state boundary-coupled hyperbolic PDE model for homogenous TCL populations is derived. This model is extended to heterogeneous populations by including a diffusive term, which provides an elegant control-oriented model. Next, a passive parameter identification scheme and a swapping-based identification scheme are derived for the PDE model structure. Simulation results demonstrate the efficacy of each method under various autonomous and non-autonomous scenarios. The proposed models can subsequently be employed to provide system critical information for power system monitoring and control.

Comparison Between Roe's Scheme and Cell-centered scheme For Transonic Flow Pass Through a Turbine Blades Section

The finite volume method (FVM) is a discretization technique for partial differential equations, especially those that arise from physical conservation laws. FVM uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations [1]. The present work employs the Cell- centred and Roe's scheme to investigate the inviscid compressible flow pass through a two dimensional blade on two types of blade shapes: Blazek and SE1050 blade models. The governing equation of fluid motion of the flow problem in hand is the Euler Equation. The behavior of this equation will depend on the local Mach number if the governing equation stated in the form as the governing equation of steady flow problem. If the local Mach number is less than one, the governing equation will behave as elliptic type of differential equation while if the Mach number is greater than one it will behave as hyperbolic type of differential equation. Elimination of the presence of those two types governing equation for the case of transonic flow problem can be achieved by representing the Euler equation in unsteady form. So the equation becomes hyperbolic with respect to time. There are various Finite Volume Methods that can used for solving hyperbolic type of equation, such as Cell-centered scheme [2], Roe Upwind Scheme [3] and TVD Scheme [1]. Those computer codes apply in the case of inviscid two dimensional compressible flow pass through blades of turbine. The present work focuses on two computer codes, first based on Cell Centre scheme and the second one based on Roe's scheme.

The OpenMOC method of characteristics neutral particle transport code

The method of characteristics (MOC) is a numerical integration technique for partial differential equations, and has seen widespread use for reactor physics lattice calculations. The exponential growth in computing power has finally brought the possibility for high-fidelity full core MOC calculations within reach. The OpenMOC code is being developed at the Massachusetts Institute of Technology to investigate algorithmic acceleration techniques and parallel algorithms for MOC. OpenMOC is a free, open source code written using modern software languages such as C/C++ and CUDA with an emphasis on extensible design principles for code developers and an easy to use Python interface for code users. The present work describes the OpenMOC code and illustrates its ability to model large problems accurately and efficiently.

Approximate Hedging in a Local Volatility Model with Proportional Transaction Costs

Local volatility models are popular as they can be calibrated to the market of European options by the simple Dupire formula. For such a model, we propose a modified Leland method which allows to approximately replicate a European contingent claim when the market is under proportional transaction costs. The convergence of the scheme is shown by means of a new strategy of proof based on partial differential equations (PDEs) techniques allowing us to obtain appropriate Greek estimations.

Linearized Oscillations in Autonomous Delay Impulsive Differential Equations

In recent years, there have been intensive efforts to establish linearised oscillation results for onedimensional delay, neutral delay and advanced impulsive differential equations. An impressive number of these efforts have yielded fruitful results in many analytical and applied areas. This is particularly obvious in the areas of applied disciplines such as the linear delay impulsive differential equations. However, there still remains a lot more to be explored in this direction, especially, in the area of non-linear autonomous differential equations. In this paper, we are proposing the development of linearised oscillation techniques for some general non-linear autonomous impulsive differential equations with several delays.

Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion

Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm.

Dynamical Behavior of the Stochastic Delay Mutualism System

We discuss the dynamical behavior of the stochastic delay three-specie mutualism system. We develop the technique for stochastic differential equations to deal with the asymptotic property. Using it we obtain the existence of the unique positive solution, the asymptotic properties, and the nonpersistence. Finally, we give the numerical examinations to illustrate our results.

Successive Vaccination and Difference in Immunity of a Delay SIR Model with a General Incidence Rate

A delay SIR epidemic model with difference in immunity and successive vaccination is proposed to understand their effects on the disease spread. From theorems, it is obtained that the basic reproduction number governs the dynamic behavior of the system. The existence and stability of the possible equilibria are examined in terms of a certain threshold condition about the basic reproduction number. By use of new computational techniques for delay differential equations, we prove that the system is permanent. Our results indicate that the recovery rate and the vaccination rate are two factors for the dynamic behavior of the system. Numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease, to support the analytical conclusion, and to illustrate possible behavioral scenarios of the model.

Modified Riccati technique for half-linear differential equations with delay

Modified Riccati technique for half-linear differential equations with delay

Symplectic FDTD algorithm for the simulations of double dispersive materials

Combined with the Lossy Drude-Lorentz dispersive model, a symplectic finite-difference time-domain (SFDTD) algorithm is proposed to deal with the double dispersive model. Based on matrix splitting, symplectic integrator propagator and the auxiliary differential equation (ADE) technique, with the rigorous and artful formula derivation, the algorithm is constructed, and detailed formulations are provided. Excellent agreement is achieved between the SFDTD-calculated and exact theoretical results when transmittance coefficient in simulation of double dispersive film in one dimension is calculated. As to numerical results for a more realistic structure in three dimensions, the simulation of periodic arrays of silver split-ring resonators using the Drude dispersion model are also included. The transmittance, reflectance, and absorptance of the structure are presented to test the efficiency of the proposed method. Our method can be used as an efficiency simulation tool for checking the experimental data.

A Direct Differential Approach to Photometric Stereo with Perspective Viewing

Shape from shading and photometric stereo are two fundamental problems in computer vision aimed at reconstructing surface depth given either a single image taken under a known light source or multiple images taken under different illuminations from the same viewing angle. Whereas the former uses partial differential equation techniques to solve the image irradiance equation, the latter can be expressed as a linear system of equations in surface derivatives when three or more images are given. Therefore, it seems that current photometric stereo techniques do not extract all possible depth information from each image by itself. Extending our previous results on this problem, we consider the more realistic perspective projection of surfaces during the photographic process. Under this assumption, there is a unique weak solution (Lipschitz continuous) to the problem at hand, solving the well-known convex/concave ambiguity of the shape from shading problem. The main contribution of this paper is based on a new differential approach for multi-image photometric stereo. Most of the existing works on this topic do not directly address this problem. The common approach is to estimate the gradient field of the surface by minimizing some functional and integrate it afterwards to find the depth and hence the geometry of the object. Our new differential approach allows us to solve the problem directly, while dealing with images having missing parts. The mathematical well-posedness of the new formulation allows a fast numerical algorithm based on a combination of fast marching and fast sweeping methods.

Free Convection Heat and Mass Transfer MHD Flow in a Vertical Porous Channel in the Presence of Chemical Reaction

The objective of the present study is to examine the fully developed free convective MHD flow of an electrically conducting viscous incompressible fluid in a vertical porous channel under influence of asymmetric wall temperature and concentration in the presence of chemical reaction. The heat and mass transfer coupled with diffusion-thermo effect renders the present analysis interesting and curious. The analytical solution by Laplace transform technique of partial differential equations is used to obtain the expressions for the velocity, temperature, and concentration. It is observed that under the influence of dominating mass diffusivity over thermal diffusivity with stronger Lorentz force the velocity is reduced at all points Further, low rate of thermal diffusion delays the attainment of free stream state. Flow of aqueous solution in the presence of heavier species is prone to back flow.

A symplectic FDTD algorithm for the simulations of lossy dispersive materials

A high-order symplectic finite-difference time-domain (SFDTD) algorithm, based on the matrix splitting, the symplectic integrator propagator and the auxiliary differential equation (ADE) technique, is presented. The algorithm is applied to the lossy Lorentz–Drude dispersive model. With the rigorous and artful formula derivation, the detailed formulations are provided. Moreover, the present algorithm can also be applicable to the lossy Drude and Lorentz dispersive model in a straightforward manner. An excellent agreement is achieved between the SFDTD-calculated and exact theoretical results when calculating the transmission coefficient in simulation of metal films. Focusing action of the matched left-handed materials (LHMs) slab is also achieved as the second example in the two-dimensional space. Numerical results for a more realistic structure, the simulation of periodic arrays of silver split-ring resonators (SRRs) using the Drude dispersion model, are also included, and the results agree well with those obtained by the finite element method (FEM).

Hybrid approach for pest control with impulsive releasing of natural enemies and chemical pesticides: A plant–pest–natural enemy model

The agricultural pests can be controlled effectively by simultaneous use (i.e., hybrid approach) of biological and chemical control methods. Also, many insect natural enemies have two major life stages, immature and mature. According to this biological background, in this paper, we propose a three tropic level plant–pest–natural enemy food chain model with stage structure in natural enemy. Moreover, impulsive releasing of natural enemies and harvesting of pests are also considered. We obtain that the system has two types of periodic solutions: plant–pest-extinction and pest-extinction using stroboscopic maps. The local stability for both periodic solutions is studied using the Floquet theory of the impulsive equation and small amplitude perturbation techniques. The sufficient conditions for the global attractivity of a pest-extinction periodic solution are determined by the comparison technique of impulsive differential equations. We analyze that the global attractivity of a pest-extinction periodic solution and permanence of the system are evidenced by a threshold limit of an impulsive period depending on pulse releasing and harvesting amounts. Finally, numerical simulations are given in support of validation of the theoretical findings.

Delay-Dependent Stability Criteria for Reaction–Diffusion Neural Networks With Time-Varying Delays

This paper studies the global asymptotic stability problem of a class of reaction–diffusion neural networks with time-varying delays. To overcome the difficulty caused by the partial differential term, a novel Lyapunov–Krasovskii functional is proposed, and a partial differential equation technique together with a linear operator approach are also applied to obtain the delay-dependent stability criteria, which are less conservative than the existing results. Finally, simulation examples are given to verify and illustrate the theoretical analysis.

A STAGE-STRUCTURED PREDATOR-PREY SI MODEL WITH DISEASE IN THE PREY AND IMPULSIVE EFFECTS

This paper aims to develop a high-dimensional SI model with stage structure for both the prey (pest) and the predator, and then to investigate the dynamics of it. The model can be used for the study of Integrated Pest Management (IPM) which is a combination of constant pulse releasing of animal enemies and diseased pests at two different fixed moments. Firstly, we use analytical techniques for impulsive delay differential equations to obtain the conditions for global attractivity of the ‘pest-free’ periodic solution and permanence of the population model. It shows that the conditions strongly depend on time delay, impulsive release of animal enemies and infective pests. Secondly, we present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is permanent. Finally, numerical analysis is presented to illustrate our main conclusion.

Haar wavelet–quasilinearization technique for fractional nonlinear differential equations

In this article, numerical solutions of nonlinear ordinary differential equations of fractional order by the Haar wavelet and quasilinearization are discussed. Quasilinearization technique is used to linearize the nonlinear fractional ordinary differential equation and then the Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Haar wavelet method. The results are compared with the results obtained by the other technique and with exact solution. Several initial and boundary value problems are solved to show the applicability of the Haar wavelet method with quasilinearization technique.

Distributed stochastic consensus of multi‐agent systems with noisy and delayed measurements

Networked systems are often subject to environmental uncertainties and communication delays, which make timely and accurate information exchange among neighbours difficult or impossible. This study investigates the distributed consensus problem of dynamical networks of multi-agents in which each agent can only obtain noisy and delayed measurements of the states of its neighbours. The authors consider consensus protocols that take into account both the noisy measurements and the communication time delays, and introduce the notions of almost sure average-consensus and p th moment average-consensus. Using a convergence theorem for continuous-time semimartingales and moment inequality techniques for stochastic delay differential equations, the authors establish sufficient conditions for both almost sure and moment average-consensus. These results naturally generalise to networks with arbitrary and Markovian switching topologies. The consensus protocol considered here can be applied to networks with arbitrary bounded communication delays, which appears to the first consensus algorithm that is both average preserving and robust to arbitrarily sized delays. Numerical simulations are also provided to demonstrate the theoretical results.

The effect of media parameters on wave propagation in a chiral metamaterials slab using FDTD

SUMMARY Because the permittivity, permeability, and chirality parameters of chiral metamaterials (CMMs) are frequency dependent, the wave equations that describe the characters of electromagnetic wave propagation in CMMs are presented and discretized on the basis of auxiliary differential equation technique in finite-difference time-domain method. The total-field/scattered-field, Mur's first-order absorbing and dielectric boundary conditions for CMMs slab are discussed in the paper. Numerical results show that the cross-polarized reflected coefficient of the CMMs slab is zero. Negative index of refraction phenomenon and optical property of giant optical activity in CMMs slabs are illustrated with 1D auxiliary differential equation–finite-difference time-domain method. The effects to positive or negative phase velocity caused by media parameters of CMMs are studied. Copyright © 2013 John Wiley & Sons, Ltd.