Exact Differential Equation Research Articles

Weak coupling regime of the Landau-Zener transition for association of an atomic Bose-Einstein condensate

In the framework of a basic semiclassical time-dependent nonlinear two-state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo- and magneto-association of an atomic Bose-Einstein condensate. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom-molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Landau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al. [Phys. Rev. A 69, 043612 (2004); J. Phys. A 38, 3505 (2005)] by order of magnitude.

Holographic dark energy with varying gravitational constant

We investigate the holographic dark energy scenario with a varying gravitational constant, in flat and non-flat background geometry. We extract the exact differential equations determining the evolution of the dark energy density-parameter, which include G-variation correction terms. Performing a low-redshift expansion of the dark energy equation of state, we provide the involved parameters as functions of the current density parameters, of the holographic dark energy constant and of the G-variation.

Inhomogeneous electron liquids generated by Coulomb confinement of electrons with and without interparticle repulsions

The inhomogeneous electron liquid created through Coulomb confinement by a single-centre potential energy −Ze 2/r remains central to the fundamental 1/Z expansion: a pillar of analytic atomic theory. Therefore, some recent progress in this area of non-relativistic quantum mechanics is first discussed, the main focus being the Feynman propagator. The closely related Slater sum is also treated, differential equations being cited in both three and two dimensions. The effect of interparticle interactions on the density of the electron liquids will next be considered in two areas: (i) for models of He-like atomic ions with nuclear charge Ze where analytic ground state electron densities can be obtained, as well as the exact differential equations they satisfy, and (ii) fine-tuning of Hartree–Fock densities for four spherical atoms, Be, Ne, Mg and Ar. In (ii), a semi-empirical framework is now available in which the experimental ionisation potential replaces that of Koopmans. Densities having almost QMC quality are thereby available. Finally, the inhomogeneous electron liquids in short linear chains of H nuclei stabilised by one electron plus an intense magnetic field along the chain axis will be treated, fields considered being as found at the surfaces of some neutron stars.

Approximations for unsteady‐state diffusion and reaction in porous catalyst and their application to packed‐bed reactor

AbstractIn this study, three approximations for unsteady‐state diffusion, a linear adsorption, and a first‐order reaction in a catalyst particle are presented. The time‐domain approximations are based on a first‐, a second‐, and a third‐order approximation of the Laplace‐domain solution of the exact partial differential equation for the catalyst. All the coefficients in the approximate equations are obtained as explicit functions of the Thiele modulus and easy to evaluate. In reactor modeling with the approximations, it is observed that the ratio of the space time in the reactor and the diffusion time in the catalyst is also an important parameter affecting the accuracy of the resulting model. A longer space time or shorter diffusion time improves the prediction of the approximate reactor model so that even the first‐order approximation is found to be sufficiently accurate when the ratio is over 0.9. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Evolution of density perturbations inf(R)theories of gravity

In the context of $f(R)$ theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. Using a completely general procedure, we find the exact fourth-order differential equation for the matter density perturbations in the longitudinal gauge. In the case of sub-Hubble modes, the expression reduces to a second-order equation which is compared with the standard (quasistatic) equation used in the literature. We show that for general $f(R)$ functions the quasistatic approximation is not justified. However, for those functions adequately describing the present phase of accelerated expansion and satisfying local gravity tests, it provides a correct description for the evolution of perturbations.

The diagonal Ising susceptibility

We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions χ(1)d and χ(2)d of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution χ(5)d, but only modulo a given prime. We use these exact linear differential equations to show that not only the Russian-doll structure but also the direct sum structure on the linear differential operators for the n-particle contributions χ(n)d are quite directly inherited from the direct sum structure on the form factors f(n). We show that the nth particle contributions χ(n)d have their singularities at roots of unity. These singularities become dense on the unit circle |sinh 2Ev/kTsinh 2Eh/kT| = 1 as n → ∞.

Theoretical principles of experimental determination of fibre flexural rigidity

An exact differential equation for the equilibrium of an elastic fibre loaded on the free end with a concentrated force and over the entire length of the elastic line of the distributed power load caused by the weight of the fibre was obtained in conditions of large movements. A method was developed for determining the flexural rigidity of the fibre which consists of bending the fibres and numerically solving the equation from the geometric nonlinear theory of bending of elastic fibres.

Exact Travelling Solutions of Discretesine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach,which was used to find new exact travelling wave solutions ofnonlinear partial differential equations or couplednonlinear partial differential equations, to nonlineardifferential-difference equations. As illustration, twoseries of exact travelling wave solutions of the discretesine-Gordon equation are obtained by means of the extendedtanh-function approach.

New Extended Jacobi Elliptic Function RationalExpansion Method and Its Application

In this paper, an extendedJacobi elliptic function rational expansion method is proposed forconstructing new forms of exact Jacobi elliptic function solutionsto nonlinear partial differential equations by means of making amore general transformation. For illustration, we apply the methodto the (2+1)-dimensional dispersive long wave equation andsuccessfully obtain many new doubly periodic solutions, whichdegenerate as soliton solutions when the modulus m approximates1. The method can also be applied to other nonlinear partialdifferential equations.

A differential approach to calculating osmotic equilibrium for multisolvent systems

A differential approach to the calculation of osmotic pressure of multisolvent systems within the Lewis–Randall framework is presented in this paper. Exact differential equations relating the osmotic pressure and the system composition along paths of constant solvent chemical potential are obtained and numerically solved. Although even for the simple case of an ideal solution no analytic expression for the osmotic pressure can be obtained, the system of differential equations does not pose numerical difficulties to be solved. Examples of the use of the proposed methodology are presented using the two-suffix Margules and Flory–Huggins equations, allowing an assessment of the influence of liquid-phase non-ideality on the performance of the method, and showing that it can be applied even for systems wherein liquid–liquid phase equilibrium occurs.

Fault-Tolerant Structured Adaptive Model Inversion Control

An adaptive dynamic inversion control formulation is presented that takes advantage of the inherent dynamic structure of the state-space description of a large class of systems. The formulations impose the exact kinematic differential equations, thereby restricting the adaptation process that compensates for model errors to the acceleration level. The utility of this formulation is demonstrated for the problem of fault tolerance to actuator failures on redundantly actuated systems. The approach incorporates an actuator failure model in the controller formulation, so that actuator failure can be identified as a change in the parameters of the failure model. Tracking of reference trajectories is imposed, and initial error conditions and structured parametric uncertainties are incorporated explicitly in both the plant parameters and the control influence matrix. A numerical example consisting of a nonlinear model of an F-16 type aircraft with thrust vectoring is presented. Simulation results show that the fault-tolerant adaptive controller is capable of simultaneously handling parametric uncertainties, large initial condition errors, and actuator failures while maintaining adequate tracking performance. N recent years, there has been much interest in the development of reconfigurable control systems that can accommodate actuator failures without compromising mission integrity. There has been substantial progress in the development of real-time failure detection and isolation algorithms, system identification after failure, and control reconfiguration techniques in aerospace applications. In Ref. 1, a survey of various reconfigurable flight control methodologies is presented and it is shown that most traditional reconfiguration flight control approaches rely on failure detection and isolation. The complexity of such a system with this feature grows with the increase in the number of failures, and there tends to be a significant possibility of false alarms. 1,2 A different approach to reconfigurable flight control is based on adaptive control theory, in which the adaptive control structure implicitly reconfigures the control law using adaptive estimates of the altered dynamics after failure. 3 In Ref. 3, an adaptive control scheme is presented that uses a linear approximation of the plant model to compute the control, and a neural network based adaptive control law for flight reconfiguration has been developed and successfully flight tested. 4−6 A robust fault-tolerant controller has also been developed to reject state-dependent disturbances. 7 The approach presented in this paper uses a structured nonlinear adaptive dynamic inversion control methodology. Instead of using an explicit failure detection and isolation algorithm, this methodology is based on the adaptive control theory where the controller is constantly updating itself. This methodology is applicable to a general class of nonlinear systems that are affine in the control with uncertain parameters appearing linearly. Fault-tolerance capability is introduced by incorporating a failure model in the controller so that a failure can be identified and compensated for by a change in the parameters of the failure model. First, model reference adaptive control, structured model reference adaptive control, and structured adaptive model inversion

A Unified Mathematical Framework for Strapdown Algorithm Design

A Unified Mathematical Framework for Strapdown Algorithm Design

New developments in differential ebulliometry: Experimental and theoretical

A new ebulliometer has been developed in which superheat is discharged from the vapour–liquid mixture issuing from the Cottrell pump before it impinges on an otherwise vacuum-insulated temperature sensor. The convoluted chamber through which the mixture passes should also assist the attainment of phase equilibrium in binary mixtures. Measurements for several solvents showed unusually broad plateau regions with slopes much smaller than those reported by other researchers. Vapour pressures for several solvents were in close agreement with literature values. For the measurement of limiting activity coefficients, a new experimentally very simple procedure was developed for finding the difficult-to-determine evaporation ratio, ϕ, based on the steady-state macroscopic energy balance. Results are presented for two pure solvents, viz. ethylacetate and cyclohexane. Exact equations, based on material balances and equilibrium relationships involving ϕ then permit calculation of limiting activity coefficients. Measured temperature-composition data are presented for the ethylacetate (1)–toluene (2) and hexane (1)–ethylacetate (2) systems and the ebulliometer characterization procedure is illustrated for three binary systems. Exact differential equations for the temperature and pressure derivatives with respect to composition, and incorporating the full Gibbs-Duhem equation, are derived. These can be used to find the very useful temperature–pressure derivative (∂ T/∂ p) x 1 at any composition and also form the basis for a new thermodynamic consistency test.

Further Extended Jacobi Elliptic Function Rational Expansion Method andNew Families of Jacobi Elliptic Function Solutions to (2+1)-Dimensional DispersiveLong Wave Equation

In this paper, a further extended Jacobi elliptic function rational expansionmethod is proposed for constructing new forms of exact solutionsto nonlinear partial differential equations by making a moregeneral transformation. For illustration, we apply the method to(2+1)-dimensional dispersive long wave equation andsuccessfully obtain many new doubly periodic solutions. When themodulus m→1, these solutions degenerate as solitonsolutions. The method can be also applied to other nonlinearpartial differential equations.

Elastic Analysis of Circular Membrane and Its Application to an Inflatable Paraboloidal Reflector (Effect of Initial In-Plane Tension)

In this paper, we describe the generation of an inflatable reflector. The reflector is generated by applying pressure to a membrane surface. The in-plane force of the radius direction is also applied to the surrounding boundary of the membrane. The exact differential equations to describe the deformation of the membrane are introduced. The deformed shape of the membrane is obtained by numerical calculation. Furthermore, the deformed shape that was calculated is compared with a parabola shape. The standard deviation of shape is introduced to compare the shapes. Finally, the effect of the in-plane force in the radius direction to generating the inflatable reflector is discussed.

Fermion particle density equations in relation to relativistic density functional theory

AbstractRelativistic density functional theory goes back at least to the work of Vallarta and Rosen. This represents the generalization of the Thomas–Fermi method to embody the kinetic energy of an electron of momentum p and rest mass m0 as given by special relativity theory in the chemical potential Euler equation. Here we set up a relativistic density functional theory by heuristic arguments that start from well‐established exact nonrelativistic differential equations for the Fermion density n(r) in model systems. We focus first on the differential equation of Lawes and March for one‐dimensional harmonic confinement. Into their result we introduce the finiteness of the velocity of light c by replacing the Lawes–March differential equation for the Fermion density n(x) by a difference equation for the relativistic Fermion density in which the Compton wavelength h/m0c, with m0 the Fermion rest mass, plays an important role. As we take the nonrelativistic limit c → ∞, we regain the exact nonrelativistic result. Contact is then established with the Vallarta–Rosen treatment in the limit when a large number N of Fermions are harmonically confined. The remaining models investigated are two‐electron in character. Relativistic difference equations to determine the density are again presented, for two such models. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

A Nonstandard Finite Difference Scheme for an Advection-Reaction Equation

We provide the details as to how a nonstandard finite differencescheme should be constructed for an advection-reaction PDE for which the exact difference equation model is known. The computational advantages of the nonstandard scheme are discussed.

Calculation of the vortex pinning energy in a long periodically modulated Josephson contact

An analytical method for evaluating the Josephson and magnetic energies of a vortex, as well as the vortex pinning energy and its components, in a long periodically modulated Josephson contact is suggested. The method allows one to take into account the variation of the vortex shape with the position of the vortex. The results obtained with this method are much closer to those of exact computer analysis compared with results of the conventional techniques. The discrepancy between the exact shape of the vortex and a function obtained by solving a differential equation approximating the exact difference equation is studied.

Exact Finite Difference Scheme for an Advection-Reaction Equation

We construct an exact finite difference scheme for a non-linear PDE having linear advection and an odd-cubic reaction term. This construction is based on the fact that the general initial-value problem for the equation can be completely solved. We also give a detailed discussion of the mathematical structure of the exact finite difference equation.

Quark and Nuclear Matter in the Linear Chiral Meson Model

We present an analytical description of the phase transitions from a nucleon gas to nuclear matter and from nuclear matter to quark matter within the same model. The equation of state for quark and nuclear matter is encoded in the effective potential of a linear sigma model. We exploit an exact differential equation for its dependence upon the chemical potential μ associated to conserved baryon number. An approximate solution for vanishing temperature is used to discuss possible phase transitions as the baryon density increases. For a nucleon gas and nuclear matter we find a substantial density enhancement as compared to quark models which neglect the confinement to baryons. The results point out that the latter models are not suitable to discuss the phase diagram at low temperature.