Single Ordinary Differential Equation Research Articles

A discrete-vortex model for the arbitrary motion of a thin airfoil with fluidic control

A low-order model for the arbitrary motion of a thin airfoil with trailing edge fluidic control is derived from basic fluid mechanics principles. The model consist of solving a single ordinary differential equation with a special treatment of a vortex shedding criteria. The model is compared with experimental and high-order numerical simulations and the results give a reasonable means of predicting the lift and moment on a thin airfoil. Furthermore, the model is extended to account for the actuation and control due to the synthetic jet actuation near the trailing edge. The model response is compared to experimental results.

Shear-flow suppression of hotspot growth

We consider the role of fluid shear in maintaining anomalous “hotspot” solutions reported for a chaotically stirred bistable chemical reaction model [S.M. Cox, Persistent localized states for a chaotically mixed bistable reaction, Phys. Rev. E 74 (2006) 056206]. In the well-mixed regime, the chemical concentration is governed by a single autonomous ordinary differential equation with two stable equilibria. Whether the reaction goes to extinction or to an excited state depends on whether the initial concentration lies below or above some unstable threshold. By contrast, when the concentration varies spatially, and the chemical is stirred, the interplay between advection, diffusion and reaction is much more complicated, and the fate of the reaction depends sensitively on the initial conditions. It has previously been shown that, if the stirring is temporally periodic, a localised hotspot may form, and so the system never becomes fully extinct, nor fully excited. In this work, we first demonstrate that hotspots are in some sense generic, in that they are easily found by trial and error, by choosing reaction parameter values between those giving rise to global extinct and excited states. We also show that hotspots may be associated with hyperbolic, elliptic or even parabolic orbits associated with the underlying stirring. We observe that fluid shear is an important mechanism in localising a hotspot, and derive a reduced ordinary differential equation model which can predict the fate of a chemical stripe in a shear flow accurately.

Nonsimilar solutions of the viscous shallow water equations governing weak hydraulic jumps

The steady viscous shallow water equations are often used for the study of hydraulic jumps. We cast these as a single parametric ordinary differential equation with global continuity as a constraint. The solution provides both the local velocity profile and the downstream evolution of the film height. Moreover this is an exact approach, in contrast with existing approaches which encounter a closure problem and need modeling. There is only one solution which is supercritical initially. This shows a jumplike behavior at a Froude number close to unity, in consonance with predictions of inviscid theory. At low Froude number, it is shown that two solutions are possible, one with a separated profile and one without. Flow downstream of a real hydraulic jump must switch to the second solution, calling into question the validity of the shallow water approach in resolving the region of the switch. A series solution of the velocity profile shows that the first correction to a streamwise-varying parabolic profile is a quartic term. Circular and planar solutions are qualitatively similar.

Fractional kinetics in multi-compartmental systems

Fractional calculus, the branch of calculus dealing with derivatives of non-integer order (e.g., the half-derivative) allows the formulation of fractional differential equations (FDEs), which have recently been applied to pharmacokinetics (PK) for one-compartment models. In this work we extend that theory to multi-compartmental models. Unlike systems defined by a single ordinary differential equation (ODE), considering fractional multi-compartmental models is not as simple as changing the order of the ordinary derivatives of the left-hand side of the ODEs to fractional orders. The latter may produce inconsistent systems which violate mass balance. We present a rationale for fractionalization of ODEs, which produces consistent systems and allows processes of different fractional orders in the same system. We also apply a method of solving such systems based on a numerical inverse Laplace transform algorithm, which we demonstrate that is consistent with analytical solutions when these are available. As examples of our approach, we consider two cases of a basic two-compartment PK model with a single IV dose and multiple oral dosing, where the transfer from the peripheral to the central compartment is of fractional order α<1, accounting for anomalous kinetics and deep tissue trapping, while all other processes are of the usual order 1. Simulations with the studied systems are performed using the numerical inverse Laplace transform method. It is shown that the presence of a transfer rate of fractional order produces a non-exponential terminal phase, while multiple dose and constant infusion systems never reach steady state and drug accumulation carries on indefinitely. The IV fractional system is also fitted to PK data and parameter values are estimated. In conclusion, our approach allows the formulation of systems of FDEs, mixing different fractional orders, in a consistent manner and also provides a method for the numerical solution of these systems.

Delay-induced dynamics of an axially moving string with direct time-delayed velocity feedback

The local dynamics of an axially moving string under aerodynamic forces is investigated with a time-delayed velocity feedback controller. The retarded differential difference governing equation is obtained in modal coordinates of a two-degree-of-freedom system through Galerkin’s discretization procedure. The stability of trivial equilibrium is examined with the change of counting multiplicity of eigenvalue with positive real part. The Hopf bifurcation curves are determined in the controlling parameter spaces. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differential equation of one complex variable on the center manifold. The approximate analytical solutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. The curves of excitation–response and frequency–response curves are shown with the effect of time delay. The stability analysis for steady-state periodic solutions of the reduced system indicates the onset of local control parameter for vibration control and response suppression. Moreover, the Poincaré–Bendixson theorem and energy considerations are used to investigate the existences and characteristics of quasi-periodic solutions when stability of the periodic solution is lost. Numerical results demonstrate the validity of the analytical prediction. Two different kinds of quasi-periodic solutions are found.

The effect of “robust” integrator dynamics on decision-making performance

The drift-diffusion model of decision-making provides a framework to explain performance statistics of two-alternative perceptual decision-making tasks. In addition, the cortical activity of rhesus monkeys performing these tasks has been observed to stochastically fluctuate until a threshold level of activity is reached, implying a neuronal implementation of this decision-making paradigm [1]. The observed ramping of activity is hypothesized to represent the accumulation of sensory evidence over timescales much longer than those of the underlying cortical hardware. One proposed mechanism to implement this short-term cortical memory is by a self-excitatory recurrent neural integrator [2]. However, it has been observed that naive excitatory recurrent circuits require extremely fine-tuning to accurately integrate an input signal [3]. We investigated the consequences of implementing a short-term evidence accumulation circuit via a recurrent network designed to be robust to mistuning in the global self-excitation parameter [4], [5]. Following the approach of previous work, we derived a firing rate model for the dynamics of the integrator parameterized by biophysically relevant quantities. In particular, a single piecewise-defined ordinary differential equation models the average firing rate activity of a number of bistable sub-populations exhibiting hysteresis. A major consequence of this robust integration scheme is the exclusion of instantaneous sensory evidence that does not sufficiently favor one alternative over the other. In particular, if stimulus is below a certain evidence-exclusion threshold, average integrator activity does not change. This evidence exclusion mechanism reveals a potential performance tradeoff between gaining robustness to mistuning on one hand, but ignoring potentially relevant information on the other. To our surprise, we find that a surprisingly large fraction of sensory evidence can be ignored before decision-making performance is compromised. To explain this observation, we derived equations that describe a hierarchy of sequential hypothesis tests that vary in their evidence-exclusion threshold. These models yield quantitative predictions of how decision-making performance is diminished as model robustness is increased. We conclude that for a fixed task difficulty and a constant decision threshold, a substantial fraction of the stimulus can be ignored before error rates or reaction times increase, if decision boundaries are modified appropriately. This observation implies that an evidence-accumulating neural integrator circuit robust to feedback mistuning might be able to underlie perceptual decision-making, without a strong compromise in performance.

Non-Newtonian fluid flow induced by peristaltic waves in a curved channel

This paper is devoted to the study of peristaltic flow of a non-Newtonian fluid in a curved channel. The constitutive relationship between stress and shear rate for a non-Newtonian third grade fluid is used. The problem is governed by a set of two nonlinear partial differential equations. These equations are then transformed into a single nonlinear ordinary differential equation in the stream function under long wavelength and low Reynolds number assumptions. This nonlinear ordinary differential equation is solved for stream function by the shooting method using Mathematica. The important phenomenon of pumping and trapping is presented graphically and discussed in detail. It is found that for a non-Newtonian third grade fluid an increase in the curvature of the channel helps in reducing the pressure rise over one wavelength in pumping region. This result is in contrast to the previous result obtained for the pressure rise over one wavelength for a Newtonian fluid. For a Newtonian fluid, the pressure rise over one wavelength increases with an increase in the curvature. The trapping phenomenon is also altered with the presence of curvature and as a result the symmetry observed for a bolus of the trapped fluid in the case of a straight channel is destroyed and splits into two asymmetrical parts for the curved channel. The outer bolus suppresses the inner bolus towards the lower wall. It is also noted that an increase in size and circulation of boluses achieve a maximum for large values of the shear thickening parameter β . Moreover, the size of two boluses in a third grade fluid is larger in comparison with their counterparts in a Newtonian fluid. Further, the lower trapping limit of the flow rate Θ is also changed in the curved channel. In fact the lower trapping limit of the curved channel exceeds that of the straight channel.

On analytic solutions of [formula omitted] relativistic ideal hydrodynamic equations

In this paper, we find various analytic ( 1 + 3 ) D solutions to relativistic ideal hydrodynamic equations based on embedding of known low-dimensional scaling solutions. We first study a class of flows with 2D Hubble embedding, for which a single ordinary differential equation for the remaining velocity field can be derived. Using this equation, all solutions with transverse 2D Hubble embedding and power law ansatz for the remaining longitudinal velocity field will be found. Going beyond the power law ansatz, we further find a few solutions with transverse 2D Hubble embedding and nontrivial longitudinal velocity field. Finally we investigate general scaling flows with each component of the velocity fields scaling independently, for which we also find all possible solutions.

Euler-Maclaurin Closed Form Finite State Space Model for a String Applied to Broadband Plate Vibrations

The Euler-Maclaurin sun formula is applied to the infinite series Green's function solution in the space-time Laplace transform domain for the one dimensional wave equation for a string fixed at each end. The resulting approximate closed form solution is used to derive a single third order input-output ordinary differential equation to model the string dynamics. The average modal density of a plate is shown to be comparable to a string. A finite three state-space model is developed for the string and applied to the vibrations of a plate subjected to broadband random and impulse inputs. The applications include the direct problem of determining the response to a disturbance input and the inverse problem of identifying the disturbance input with a finite state observer based on the finite string model. Numerical simulations using many plate modes are obtained in the time and frequency domains and are used to compare the multimodal plate model to the finite string based model and to demonstrate how the finite string based model can be used to represent the multimodal vibrations of the plate.

The structure of thin accretion discs around magnetised stars

Aims. We determine the steady-state of an axisymmetric thin accretion disc with an internal dynamo around a magnetised star. Methods. Starting from the vertically integrated equations of magnetohydrodynamics we derive a single ordinary differential equation for a thin accretion disc around a massive magnetic dipole and integrate this equation numerically from the outside inwards. Results. Our numerical solution shows that the torque between the star and the accretion disc is dominated by the contribution from the dynamo in the disc. The location of the inner edge of the accretion disc varies between RA and 10 RA depending mainly on the strength and direction of the magnetic field generated by the dynamo in the disc

Black holes in higher dimensional gravity theory with corrections quadratic in curvature

Static spherically symmetric black holes are discussed in the framework of higher dimensional gravity with quadratic in curvature terms. Such terms naturally arise as a result of quantum corrections induced by quantum fields propagating in the gravitational background. We focus our attention on the correction of the form ${\mathcal{C}}^{2}={C}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}{C}^{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}$. The Gauss-Bonnet equation in four-dimensional spacetime enables one to reduce this term in the action to the terms quadratic in the Ricci tensor and scalar curvature. As a result the Schwarzschild solution which is Ricci flat will be also a solution of the theory with the Weyl scalar ${\mathcal{C}}^{2}$ correction. An important new feature of the spaces with dimension $D&gt;4$ is that in the presence of the Weyl curvature-squared term a necessary solution differs from the corresponding ``classical'' vacuum Tangherlini metric. This difference is related to the presence of secondary or induced hair. We explore how the Tangherlini solution is modified by ``quantum corrections,'' assuming that the gravitational radius ${r}_{0}$ is much larger than the scale of the quantum corrections. We also demonstrated that finding a general solution beyond the perturbation method can be reduced to solving a single third order ordinary differential equation (master equation).

Cavitation in elastic and hyperelastic sheets

Plane stress, axially symmetric, cavitation patterns are examined for large sheets with an embedded central circular hole. Both (radially-uniform) remote tension and internal pressure loads are considered. Material behavior is modeled by finite strain Hookean-type elastic and hyperelastic constitutive relations with logarithmic strains. Governing field equations are reduced to a single ordinary differential equation with the principal stress difference as the independent variable. Calculations reveal that under internal pressure the usual definition of cavitation state does not apply as load increases monotonously along the deformation path. Introducing the specific cavitation energy, as a practical measure in cavitation analysis, provides a unified framework for assessment of cavitation phenomena. Comparison with related patterns of spherical and cylindrical cavitation fields supports the main findings of this study.

Modeling the transient behavior of an experimental reverse osmosis tubular membrane

A dynamic model is developed for a tubular reverse osmosis unit. The proposed model considers the unit as a series of single tubes, each tube is described by a single ordinary differential equation (ODE) and the whole module is to be described by sets of differential algebraic equations. The tubes are modeled and solved sequentially where the output of any tube becomes the input for the next tube. The predictions of steady-state and dynamic models are in good agreement with the experimental results of a lab scale RO unit. The model equations are simpler and faster to solve than the currently used distributed parameter models. Furthermore, since it is based on first law principles, the proposed model offers better insight than input–output (black box) models into the effect of different design and operating parameters.

Isospectral Euler–Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler–Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler–Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge–Kutta method.

Effective computation of complex-shaped quantum-dot structures

The possibility of growing complex-shaped nanodot structures of various material composition allows optimization of certain physical parameters. In the present work, we present effective analytical methods for computing conduction-band eigenstates in quantum-dot structures of complex shape. Comparison with detailed finite-element computations is made. The electronic bandstructure model used is a one-band \(\vec{k}\cdot\vec {p}\) model assuming infinite barriers. Results based on two semi-analytical models are presented. The first model employs geometrical perturbation theory to obtain the quantitative effect of quantum-dot surface perturbations on electron energy levels. Furthermore, the method output includes the level of degeneracy and variations with geometry to be assessed. The second model allows both energy levels and eigenstates to be easily determined for three-dimensional axisymmetrical GaAs structures of varying radius embedded in an AlGaAs matrix by extending a method originally due to Stevenson on electromagnetic waveguide structures (Stevenson in J. Appl. Phys. 22:1447, 1951) to account for electron states. The latter model simplifies the description of a three-dimensional partial-differential equation problem into a small set of ordinary differential equations. For structures with a large aspect ratio, the small set reduces to a single ordinary differential equation yet maintaining high accuracy. A case study is presented to exemplify the models shown.

Chaotic Bouncing of a Droplet on a Soap Film

We examine the complex dynamics arising when a water droplet bounces on a horizontal soap film suspended on a vertically oscillating circular frame. A variety of simple and complex periodic bouncing states are observed, in addition to multiperiodicity and period-doubling transitions to chaos. The system is simply and accurately modeled by a single ordinary differential equation, the numerical solution of which captures all the essential features of the observed behavior. Iterative maps and bifurcation diagrams indicate that the system exhibits all the features of a classic low-dimensional chaotic oscillator.

INVARIANT LINEARIZATION CRITERIA FOR SYSTEMS OF CUBICALLY NONLINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalent of the Lie conditions for such systems. We explicitly solve this branch of the linearization problem by point transformations in the case of a square system of two second-order ODEs. Necessary and sufficient conditions for linearization to the simplest system by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a rederivation of Lie's linearization conditions for a single second-order ODE and sheds light on more recent results for them. In particular we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover these results contain the quadratic case as a special case. Examples are given to illustrate our results.

Interfacial and wetting properties of a binary point Yukawa fluid

We investigate the interfacial phase behavior of a binary fluid mixture composed of repulsive point Yukawa particles. Using a simple approximation for the Helmholtz free energy functional, which yields the random phase approximation for the pair direct correlation functions, we calculate the equilibrium fluid density profiles of the two species of particles adsorbed at a planar wall. We show that for a particular choice (repulsive exponential) of the wall potentials and the fluid pair-potential parameters, the Euler-Lagrange equations for the equilibrium fluid density profiles may be transformed into a single ordinary differential equation and the profiles obtained by a simple quadrature. For certain other choices of the fluid pair-potential parameters fluid-fluid phase separation of the bulk fluid is observed. We find that when such a mixture is exposed to a planar hard wall, the fluid exhibits complete wetting on the species 2 poor side of the binodal, i.e., we observe a thick film of fluid rich in species 2 adsorbed at the hard wall. The thickness of the wetting film grows logarithmically with the concentration difference between the fluid state point and the binodal and is proportional to the bulk correlation length of the intruding (wetting) fluid phase. However, for state points on the binodal that are further from the critical point, we find there is no thick wetting film. We determine the accompanying line of first-order (prewetting) surface phase transitions which separate a thin and thick adsorbed film. We show that for some other choices of repulsive wall potentials the prewetting line is still present, but its location and extent in the phase diagram is strongly dependent on the wall-fluid interaction parameters.

Ice accretion simulation on multi-element airfoils using extended Messinger model

In the current article, the problem of in-flight ice accumulation on multi-element airfoils is studied numerically. The analysis starts with flow field computation using the Hess-Smith panel method. The second step is the calculation of droplet trajectories and droplet collection efficiencies. In the next step, convective heat transfer coefficient distributions around the airfoil elements are calculated using the Integral Boundary-Layer Method. The formulation accounts for the surface roughness due to ice accretion. The fourth step consists of establishing the thermodynamic balance and computing ice accretion rates using the Extended Messinger Model. At low temperatures and low liquid water contents, rime ice occurs for which the ice shape is determined by a simple mass balance. At warmer temperatures and high liquid water contents, glaze ice forms for which the energy and mass conservation equations are combined to yield a single first order ordinary differential equation, solved numerically. Predicted ice shapes are compared with experimental shapes reported in the literature and good agreement is observed both for rime and glaze ice. Ice shapes and masses are also computed for realistic flight scenarios. The results indicate that the smaller elements in multielement configurations accumulate comparable and often greater amount of ice compared to larger elements. The results also indicate that the multi-layer approach yields more accurate results compared to the one-layer approach, especially for glaze ice conditions.

A theory of narrow vortex relativistic electron beams with an elliptic section

Narrow vortex relativistic beams with an elliptic section are considered. The beams propagate along an arbitrary smooth spatial curve in the presence of an external nonuniform magnetic field. A model is proposed for description of such beams in a coordinate frame associated with current tubes. It is shown that, contrary to known results, the axial distributions of the potential and magnetic field and the shape of a beam can be specified independently. Solution of the problem is reduced to integration of a single second-order ordinary differential equation.