Set Of Linear Differential Equations Research Articles

MHD equations in the atmosphere of the stars

In this paper, we find the perturbations depending on the magnetohydrodynamics time in a static and homogeneous plasma, with the help of the set of nonlinear equations. However, we only focus on low-amplitude perturbations, and therefore we can find a set of linear differential equations with the corresponding wave form solutions. We should also mention that if the non-perturbed velocity is non-zero but uniform, one can always translate the plasma to a framework where it is stationary there. Hence, we suppose that the plasma is in equilibrium state and its initial velocity is zero. This static equilibrium is changed to the small perturbations in the magnetic field, in the pressure fluid, and in the mass density.

Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field

The effect of a uniform external magnetic field on the onset of convection in an electrically conducting nanofluid layer is examined numerically based on non-homogeneous two-phase model (i.e., classical Buongiorno’s mathematical model) which incorporates the effects of Brownian motion and thermophoresis of nanoparticles in the thermal transport mechanism of nanofluids. In this investigation, we consider that the nanofluid is Newtonian, heated from below and confined horizontally in a Darcy-Brinkman porous medium between two infinite rigid boundaries, with different nanoparticle configurations at the horizontal boundaries (i.e., top heavy and bottom heavy nanoparticle distributions). The linear stability theory has been wisely used to obtain a set of linear differential equations which are transformed to an eigenvalue problem, so that the thermal Rayleigh number Ra is the corresponding eigenvalue. The thermal Rayleigh number Ra and its corresponding wave number a are found numerically using the Chebyshev-Gauss-Lobatto collocation method for each set of fixed nanofluid parameters. The marginal instability threshold (Rac, ac) characterizing the onset of stationary convection is computed accurately for wide ranges of the modified magnetic Chandrasekhar number Q, the modified specific heat increment NB, the nanoparticle Rayleigh number RN, the modified Lewis number Le, the modified diffusivity ratio NA and the Darcy number Da. Based on these control parameters and the notions of streamlines, isotherms and iso-nanoconcentrations, the stability characteristics of the system and the development of complex dynamics at the critical state are discussed in detail for both nanoparticle distributions.

Rate Constant Change of Photo Reaction of Bacteriorhodopsin Observed in Trimeric Molecular System.

To elucidate the time evolution of photo reaction of bacteriorhodopsin in glycerol mixed purple membrane at around 196 K under irradiation by red light, a kinetic model was constructed. The change of absorption with irradiation at times of 560 nm and 412 nm was analyzed for the purpose of determining reaction rates of photo reaction of bacteriorhodopsin and its product M intermediate. In this study it is shown that reaction rates of conversion from bacteriorhodopsin to the M intermediate can be explained by a set of linear differential equations. This model analysis concludes that bacteriorhodopsin in which constitutes a trimer unit with other two bacteriorhodopsin molecules changes into M intermediates in the 1.73 of reaction rate, in the initial step, and according to the number of M intermediate in a trimer unit, from three to one, the reaction rate of bacteriorhodopsin into M intermediates smaller as 1.73, 0.80, 0.19 which caused by influence of inter-molecular interaction between bacteriorhodopsin.

Explicit analytic solution for heat and mass transfer in a desiccant wheel using a simplified model

Heat and mass transfer in a desiccant wheel was modeled into a set of linear differential equations under the linearization assumptions on the temperature and humidity profiles and the psychrometric relation. The explicit analytic solutions to the temperature and humidity ratio of the outlet air were obtained using the simplified model and assessed for the validity by comparison with a numerical simulation at the range of low regeneration temperatures. The RMS (Root Mean Square) errors were evaluated less than 10% for most operation range of the air velocity, the dehumidification period and the psychrometric conditions of the regeneration air. The analysis revealed that the behavior of a desiccant wheel depends on four major dimensionless parameters: κ, σ, N, and ψ, each of which represents the sorption capacity, the thermal capacity, the transfer capacity of the wheel and the psychrometric relation of the air, respectively. The psychrometric progress of the outlet air was observed to be divided into two distinct phases, each dominated by an exponential function identified by the four parameters. Asymptotic analysis on the solutions showed that the early phase is governed by the thermal characteristics while the later phase is by the sorption characteristics of the desiccant wheel.

Accurate calculation of diffraction-limited encircled and ensquared energy.

Mathematical properties of the encircled and ensquared energy functions for the diffraction-limited point-spread function (PSF) are presented. These include power series and a set of linear differential equations that facilitate the accurate calculation of these functions. Asymptotic expressions are derived that provide very accurate estimates for the relative amount of energy in the diffraction PSF that fall outside a square or rectangular large detector. Tables with accurate values of the encircled and ensquared energy functions are also presented.

Checking Time-Dependent Disorders in Homogeneous and Stationary Plasma Magneto Resistance

Our aim in this article is the finding time-dependent disorders in homogeneous and stationary plasma magnetoresistance that it will be described by using the equations described in the text these equations are non-linear. However, we restrict our attention to Low-amplitude disturbances and so we expect that finding a set of linear differential equations that describe the Low-amplitude differences then the wave form of the equations will be searched, if the velocity is non-zero non-impaired, but uniform. Always, we can move plasma to a framework in which it is at stasis. Therefore, we assume that the initial velocity is zero and also assume that the plasma is in equilibrium. If the static balance by entering a slight disturbance in the plasma velocity, this disturbance leads this small magnetic field disturbance disorders, fluid pressure and mass density. We infer that if the system does not establish that the magnetic field or magnetic field intensity is low. So that it becomes possible to ignore its effect and when the gas temperature is uniform, wave properties of the system can be studied using analytical methods.Keywords: Approximation Conductivity, Magneto Resistance Hydrodynamics, Plasma, Plasmoid Divergence, Sunspot, The Sheet Flow

On non-autonomous dynamical systems

In usual realistic classical dynamical systems, the Hamiltonian depends explicitly on time. In this work, a class of classical systems with time dependent nonlinear Hamiltonians is analyzed. This type of problems allows to find invariants by a family of Veronese maps. The motivation to develop this method results from the observation that the Poisson-Lie algebra of monomials in the coordinates and momenta is clearly defined in terms of its brackets and leads naturally to an infinite linear set of differential equations, under certain circumstances. To perform explicit analytic and numerical calculations, two examples are presented to estimate the trajectories, the first given by a nonlinear problem and the second by a quadratic Hamiltonian with three time dependent parameters. In the nonlinear problem, the Veronese approach using jets is shown to be equivalent to a direct procedure using elliptic functions identities, and linear invariants are constructed. For the second example, linear and quadratic invariants as well as stability conditions are given. Explicit solutions are also obtained for stepwise constant forces. For the quadratic Hamiltonian, an appropriated set of coordinates relates the geometric setting to that of the three dimensional manifold of central conic sections. It is shown further that the quantum mechanical problem of scattering in a superlattice leads to mathematically equivalent equations for the wave function, if the classical time is replaced by the space coordinate along a superlattice. The mathematical method used to compute the trajectories for stepwise constant parameters can be applied to both problems. It is the standard method in quantum scattering calculations, as known for locally periodic systems including a space dependent effective mass.

Lie Group Method for Studying the Heat Generation Effect on Freeconvection Laminar Boundary-layer Flow Over a Vertical Flat Plate

The nonlinear equations of heat and mass transfer in two-dimensional free-convection, laminar, boundary layer flow of a viscous incompressible fluid over a vertical plate with thermophoresis and heat generation effect have been considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Liegroup method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the heat generation parameter He, the Prandtl number Pr, the Schimted number Sc, the thermophoretic parameter τ, the solutal Grashof number Gc and the thermal Grashof number Gr on velocity, concentration and temperature have been studied and the results are plotted.

From microscopic to macroscopic description of Josephson dynamics in one-dimensional arrays of weakly-coupled superconducting islands

By starting from a microscopic quantum mechanical description of Josephson dynamics of a one-dimensional array of N coupled superconductors, we obtain a set of linear differential equations for the system order parameter and for additional macroscopic physical quantities. With opportune considerations, we adapt this description to two coupled superconductors, obtaining the celebrated Feynman model for Josephson junctions. These results confirm the correspondence between the microscopic picture and the semi-classical Ohta’s model adopted in describing the superconducting phase dynamics in multi-barrier Josephson junctions.

Robust Control for PWM-Based DC–DC Buck Power Converters With Uncertainty Via Sampled-Data Output Feedback

This paper investigates the sampled-data output feedback control problem for dc-dc buck power converters taking consideration of components uncertainties. A reduced-order observer and a robust output feedback controller, both in the sampled-data form, have been explicitly constructed with strong robustness in the presence of uncertain parameters. A delicate stability analysis process is presented to show that, by carefully selecting the design gains and the tunable sampling period, the output voltage of the hybrid closed-loop dc-dc buck converter system will globally asymptotically tend to the desired value even though the separation principle is out of reach and the controller is only switched at the sampling points. The proposed controller consists of a set of linear difference equations which will lead to direct and easier digital implementation. Numerical simulations and experimental results are shown to illustrate the performance of the proposed control scheme.

A reduced simulation applied to the viscoelastic fatigue of polymers

The paper extends the use of the PGD method to viscoelastic evolution problems described by a large number of internal variables and with a large spectrum of relaxation times. The internal variables evolution is described by a set of linear differential equations that involve many time scales. The feasibility and the robustness of the method are discussed in the case of a polymer in a non-equilibrium state under creep and cyclic loading. The relationships between different time scales (loading and internal variables) are also discussed.

Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments

In this paper, the probabilistic characterization of a nonlinear system enforced by Poissonian white noise in terms of complex fractional moments (CFMs) is presented. The main advantage in using such quantities, instead of the integer moments, relies on the fact that, through the CFMs the probability density function (PDF) is restituted in the whole domain. In fact, the inverse Mellin transform returns the PDF by performing integration along the imaginary axis of the Mellin transform, while the real part remains fixed. This ensures that the PDF is restituted in the whole range with exception of the value in zero, in which singularities appear. It is shown that using Mellin transform theorem and related concepts, the solution of the Kolmogorov–Feller equation is obtained in a very easy way by solving a set of linear differential equations. Results are compared with those of Monte Carlo simulation showing the robustness of the solution pursued in terms of CFMs. Further, a very elegant strategy, to give a relationship between the CFMs evaluated for different value of real part, is introduced as well.

Nonlinear electro-dynamic analysis of micro-actuators: Effect of material nonlinearity

This paper presents a nonlinear dynamic analysis of a micro-actuator made of nonlinear elasticity materials. The theoretical formulations are based on Bernoulli–Euler beam theory and include the effects of mid-plane stretching due to large deformation and material nonlinearity. By employing Linstedt–Poincaré perturbation method, the nonlinear governing equation is transformed into a set of linear differential equations which are then solved using Galerkin’s method. Numerical results show that the linear constitutive relationship used in previous studies is valid for small deformation only whereas for large deformation, the nonlinear elasticity constitutive relationship must be used for accurate analysis. The effects of initial gap and beam length on the nonlinear electro-dynamic behavior of the micro-actuator are also discussed.

New methods for characterizing phases of 2D supersymmetric gauge theories

We study the physics of two-dimensional N=(2,2) gauged linear sigma models (GLSMs) via the two-sphere partition function. We show that the classical phase boundaries separating distinct GLSM phases, which are described by the secondary fan construction for abelian GLSMs, are completely encoded in the analytic structure of the partition function. The partition function of a non-abelian GLSM can be obtained as a limit from an abelian theory; we utilize this fact to show that the phases of non-abelian GLSMs can be obtained from the secondary fan of the associated abelian GLSM. We prove that the partition function of any abelian GLSM satisfies a set of linear differential equations; these reduce to the familiar A-hypergeometric system of Gel'fand, Kapranov, and Zelevinski for GLSMs describing complete intersections in toric varieties. We develop a set of conditions that are necessary for a GLSM phase to admit an interpretation as the low-energy limit of a non-linear sigma model with a Calabi-Yau threefold target space. Through the application of these criteria we discover a class of GLSMs with novel geometric phases corresponding to Calabi-Yau manifolds that are branched double-covers of Fano threefolds. These criteria provide a promising approach for constructing new Calabi-Yau geometries.

New solutions for solving problem of particle trajectories in linear deep-water waves via Lie-group method

The nonlinear equations of the two-dimensional inviscid incompressible fluid in a constant gravitational field describing the wave propagation on the water surface are considered. The Lie-group method has been applied for determining symmetry reductions of the system of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. Effects of the wavelength λ and time t on the particle path have been studied and the results are plotted.

Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors

Nonlinear dynamical systems have been used in many disciplines to model complex behaviors, including biological motor control, robotics, perception, economics, traffic prediction, and neuroscience. While often the unexpected emergent behavior of nonlinear systems is the focus of investigations, it is of equal importance to create goal-directed behavior (e.g., stable locomotion from a system of coupled oscillators under perceptual guidance). Modeling goal-directed behavior with nonlinear systems is, however, rather difficult due to the parameter sensitivity of these systems, their complex phase transitions in response to subtle parameter changes, and the difficulty of analyzing and predicting their long-term behavior; intuition and time-consuming parameter tuning play a major role. This letter presents and reviews dynamical movement primitives, a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques. The essence of our approach is to start with a simple dynamical system, such as a set of linear differential equations, and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term. Both point attractors and limit cycle attractors of almost arbitrary complexity can be generated. We explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics.

Coupled free vibration of composite beams with asymmetric cross-sections

A set of linear differential equations of motion for coupled free vibrations of composite beams with asymmetric cross-sections is obtained in this paper based on Euler–Bernoulli beam theory. More coupling terms are included in the governing equations; and hence, both even and odd order spatial derivatives are included. All these features lead to much difficulty in solving the resultant equations. The axially unloaded condition of F1=0, instead of e=0, is adopted to simplify the equations of motion for the case of inextensional beams. An algorithm is developed for solving the resultant equations to obtain natural frequencies and modes of the beams. Numerical examples for validation show that the equations of motion obtained in this paper are correct and the corresponding algorithm is effective and easy to use.

Semi-Global Stabilization via Linear Sampled-Data Output Feedback for a Class of Uncertain Nonlinear Systems

Abstract This paper studies the semi-global sampled-data stabilization problem via output feedback for a class of uncertain nonlinear systems under mismatched uncertainties and a higher-order, rather than linear growth lower-triangular condition. A systematic design scheme is developed to construct a linear output feedback control law in sampled-data form that semi-globally stabilizes the uncertain nonlinear system. By extending the output domination approach and restraining the state growth under a zero-order-hold input, explicit stability analysis procedures are presented for the hybrid closed-loop system. With less control ambition, the proposed semi-global sampled-data controller consists of a set of linear difference equations which will lead to easier digital implementations.

Comparison of a double-Λ atomic scheme with single- and two-fold coupled transitions

A four-level double-Λ atomic scheme, i.e. two Λ systems sharing the same two ground levels, that interacts with four laser light fields is studied theoretically. The peculiarity is that each of the two ground states can be coupled to each excited state by two laser light fields. A certain energy difference exists between excited-state levels. We consider this energy small enough so that a laser resonant to either transition can also couple the other transition. We test whether coupling of the more detuned laser is not non-negligible. Multiply connected states were also recently analyzed (and a comparison with experiment was presented), but for the simpler, two- and three-level atomic schemes (Stacey et al 2008 J. Phys. B: At. Mol. Opt. Phys. 41 085502). Theoretical treatment of the double-Λ atomic scheme is commonly done by solving optical Bloch equations (OBEs). When the rotating-wave approximation (RWA) is applied, OBEs become, by their form, a set of linear differential equations with constant coefficients. Theoretical treatment of the interaction scheme treated here leads to OBEs with coefficients that are not constant, but oscillate with time, even after RWA is applied. Under certain assumptions, the approximation can be used where the time-dependent coefficients are averaged over their periods. The method yields a new system of equations (similar to standard OBEs), but with more independent variables, and can also be solved in the usual way. The results presented here analyze the validity of this approximation by comparing the results for the double-Λ atomic scheme with single- and two-fold coupled transitions. We test whether in the limit of large-energy splitting between excited-state levels both approaches lead to similar results.

Modeling of Passenger Delays at a Multimodal Transport Access Point of a University Campus

This study models the egress behaviour of passengers at a transportation access point with multimodal service with the objective of evaluating the delays associated with their departure from the access area. Passengers arriving at this access area have the taxi mode, minibus mode and the bus mode to choose from. A discrete choice model with the utility function based on mode fare and transit time was used to describe the egress behaviour of passengers. The resulting queues at the access area for the three modes were transient in character with time-varying arrival λ(t), time varying service rates µ(t), a truncated bulk service single channel with first-in-first out queue discipline. The queuing systems were described by M/M * /K/FIFO with random clearing of all queued taxi and minibus passengers. A computer simulation package using Runge-Kutta numerical methods was used to solve the set of linear differential equations developed for the three transport modes serving the area. The simulation models were tested with field data collected from the access area on all trips made during the study period and were analysed at fifteen minute time intervals. The probabilities of delay predicted by the simulation models agreed closely with field observations. The chi-squared test indicated that there were no significant differences between the models and the field values at the 95% significant level. Passengers at the access area experienced an average delay of 16.4minutes, 12.4minutes and 23.5minutes for the taxi, minibus and bus modes respectively. The methodology would need to be tested with data from other areas with similar demand characteristics to build confidence into it. Research Article