Series Of Ordinary Differential Equations Research Articles

Melnikov's method for chaos of a two-dimensional thin panel in subsonic flow with external excitation

In this brief communication, Melnikov's method is adopted to study the chaotic behaviors of a two-dimensional thin panel subjected to subsonic flow and external excitation. The nonlinear governing equations of the subsonic panel system are reduced to a series of ordinary differential equations by using Galerkin method. The critical parameters for chaos are obtained. It is found that the critical parameters obtained by the theoretical analysis are in agreement with the numerical simulations. The method suggested in this paper can also be extended for other fluid-structure dynamic systems, such as the fluid-conveying system.

Exact solutions for one‐dimensional transient response of fluid‐saturated porous media

AbstractBased on the Biot theory, the exact solutions for one‐dimensional transient response of single layer of fluid‐saturated porous media and semi‐infinite media are developed, in which the fluid and solid particles are assumed to be compressible and the inertial, viscous and mechanical couplings are taken into account. First, the control equations in terms of the solid displacement u and a relative displacement w are expressed in matrix form. For problems of single layer under homogeneous boundary conditions, the eigen‐values and the eigen‐functions are obtained by means of the variable separation method, and the displacement vector u is put forward using the searching method. In the case of nonhomogeneous boundary conditions, the boundary conditions are first homogenized, and the displacement field is constructed basing upon the eigen‐functions. Making use of the orthogonality of eigen‐functions, a series of ordinary differential equations with respect to dimensionless time and their corresponding initial conditions are obtained. Those differential equations are solved by the state‐space method, and the series solutions for three typical nonhomogeneous boundary conditions are developed. For semi‐infinite media, the exact solutions in integral form for two kinds of nonhomogeneous boundary conditions are presented by applying the cosine and sine transforms to the basic equations. Finally, three examples are studied to illustrate the validity of the solutions, and to assess the influence of the dynamic permeability coefficient and the fluid inertia to the transient response of porous media. Copyright © 2010 John Wiley & Sons, Ltd.

Correspondence: Cyclic neutropenia associated with T cell immunity to granulocyte proteases and a double de novo mutation in GFI1, a transcriptional regulator of ELANE

correspondence: Cyclic neutropenia associated with T cell immunity to granulocyte proteases and a double <i>de novo</i> mutation in <i>GFI1</i>, a transcriptional regulator of <i>ELANE</i>

Kundt spacetimes as solutions of topologically massive gravity

We obtain new solutions of topologically massive gravity. We find the general Kundt solutions, which in three dimensions are spacetimes admitting an expansion-free null geodesic congruence. The solutions are generically of algebraic type II, but special cases are types III, N or D. Those of type D are the known spacelike-squashed AdS3 solutions and of type N are the known AdS pp-waves or new solutions. Those of types II and III are the first known solutions of these algebraic types. We present explicitly the Kundt solutions that are constant scalar invariant (CSI) spacetimes, for which all scalar polynomial curvature invariants are constant, whereas for the general case, we reduce the field equations to a series of ordinary differential equations. The CSI solutions of types II and III are deformations of spacelike-squashed AdS3 and the round AdS3, respectively.

Quasi-One-Dimensional Model of Hydrogen-Fueled Scramjet Combustors

A computationally efficient, quasi-one-dimensional, supersonic combustion ramjet (scramjet) propulsion model has been produced for use in hypersonic system design studies. The model solves a series of ordinary differential equations using a fourth-order Runge–Kutta method to describe the gas dynamics within the scramjet duct. Additional models for skin friction and wall heat transfer are also included. The equations are derived assuming an open thermodynamic system with equilibrium or simplified-chemistry combustion models. The combustion is also assumed to be mixing-limited rather than kinetically limited. This assumption allows simplification of the modeling and is justified when the model is compared against experimental results. Three test cases are used to validate the performance of the scramjet propulsion model: 1) a reflected-shock-tunnel hydrogen-fueled scramjet experiment, 2) a continuous-flow hydrogen-fueled scramjet ground test, and 3) a segment of the HyShot II flight test. The results show that the model simulates scramjet propulsion with a reasonable degree of accuracy.

Development of Nailed Wood Joint Element in ABAQUS

Testing showed that the hysteretic behavior of nailed wood joints governs the response of many wood systems when subjected to lateral loadings. Unfortunately, commercially available software does not have the appropriate hysteretic element for a nailed wood joint, and the accuracy and versatility of the previously developed nail joint elements are not satisfactory. A general hysteretic model, Bouc-Wen-Barer-Wen, was modified to represent the hysteretic behavior of a nailed joint, in which the hysteretic constitutive law is characterized by a series of ordinary differential equations. It can produce versatile and smoothly varying hysteresis curves, is nonlinear, history dependent, and includes stiffness and strength degradation, and pinching. Based on the test data, the suitable parameters for different joint configurations can be estimated through genetic algorithms. This model was embedded in commercially available software, ABAQUS/Standard (Version 6.5), as a user-defined element which accounted for the coupling property of the nail joint action. A detailed shear wall was modeled and analyzed, and the results agreed well with the test data.

Solving differential equations with constructed neural networks

A novel hybrid method for the solution of ordinary and partial differential equations is presented here. The method creates trial solutions in neural network form using a scheme based on grammatical evolution. The trial solutions are enhanced periodically using a local optimization procedure. The proposed method is tested on a series of ordinary differential equations, systems of ordinary differential equations as well as on partial differential equations with Dirichlet boundary conditions and the results are reported.

Modeling track–terrain interaction for transient robotic vehicle maneuvers

This article describes integration of a realistic and efficient track–terrain interaction model with a multibody dynamics model of a robotic tracked vehicle. The track–terrain interface continuum is approximated by discretized and parameterizable force elements. Of particular note is a kinematic model used to estimate dynamic shear displacement, taking the form of a partial differential equation. This equation is approximated by a series of ordinary differential equations, making it compatible with multibody dynamics model formulations. Comparisons between simulated results and those obtained from field testing with a remotely-operated unmanned tracked vehicle are made to evaluate the effectiveness of this approach and to validate the use of nominal parameter data from the literature. The test vehicle was subjected to four different types of maneuvers (go-and-stop, j-turn, double lane change, and zero radius turn) on asphalt and dry sand. Simulated results using both the dynamic and steady-state track–terrain interaction models match very well with those obtained from the tests, except for the zero radius turning maneuver in sand. In this case, bulldozing effects must be incorporated to improve prediction of lateral forces.

An Explicit, Multi-Factor Credit Default Swap Pricing Model with Correlated Factors

AbstractWith the recent significant growth in the single-name credit default swap (CDS) market has come the need for accurate and computationally efficient models to value these instruments. While the model developed by Duffie, Pan, and Singleton (2000) can be used, the solution is numerical (solving a series of ordinary differential equations) rather than explicit. In this paper, we provide an explicit solution to the valuation of a credit default swap when the interest rate and the hazard rate are correlated by using the “change of measure” approach and solving a bivariate Riccati equation. CDS transaction data for the period 2/15/2000 through 4/8/2003 for 60 firms are used to test both the goodness of fit of the model and provide estimates of the influence of economic variables in the market for credit-risky bonds.

Mathematical Modeling of Closed-End Pulsating Heat Pipes Operating with a Bottom Heat Mode

The mathematical model of a closed-end pulsating heat pipe (CEPHP) with a bottom heat mode at different inclination angles was constructed. The closed-end pulsating heat pipe was modeled with specified assumptions that were observed visually (i.e., the scaling factor for geometrical size and the frequency of bubble generation inside the liquid slugs). The solution for all of the basic governing equations of liquid film, liquid slugs, and vapor plugs, in which the effects of surface tension, viscous friction of the working fluid, and perfect gas were included, has been numerically obtained by solving a series of ordinary differential equations by means of the explicit method. However, the solution for the momentum equation of liquid slugs was numerically obtained by solving a series of partial differential equations by using the implicit method. Results from the model clearly simulated the dynamics of the internal working fluid in the CEPHP. Moreover, the results were compared with existing experimental data, and good agreement was found with an error range of ± 13%. It was also noted that the maximum heat transfer rate of the CEPHP with bottom heat mode occurred at the highest evaporator temperature (150°C for this study) and inclination angles of 70–80 degrees from horizontal axis. The boiling frequencies in this range of inclination angles were observed by visual experiment and seen to be at their highest values. This has been justified by the higher amount of liquid in the evaporator section as well as the change in flow pattern to a stratified flow (inclination tube).

Modeling the effects of a Staphylococcal Enterotoxin B (SEB) on the apoptosis pathway

BackgroundThe lack of detailed understanding of the mechanism of action of many biowarfare agents poses an immediate challenge to biodefense efforts. Many potential bioweapons have been shown to affect the cellular pathways controlling apoptosis [1-4]. For example, pathogen-produced exotoxins such as Staphylococcal Enterotoxin B (SEB) and Anthrax Lethal Factor (LF) have been shown to disrupt the Fas-mediated apoptotic pathway [2,4]. To evaluate how these agents affect these pathways it is first necessary to understand the dynamics of a normally functioning apoptosis network. This can then serve as a baseline against which a pathogen perturbed system can be compared. Such comparisons can expose both the proteins most susceptible to alteration by the agent as well as the most critical reaction rates to better instill control on a biological network.ResultsWe explore this through the modeling and simulation of the Fas-mediated apoptotic pathway under normal and SEB influenced conditions. We stimulated human Jurkat cells with an anti-Fas antibody in the presence and absence of SEB and determined the relative levels of seven proteins involved in the core pathway at five time points following exposure. These levels were used to impute relative rate constants and build a quantitative model consisting of a series of ordinary differential equations (ODEs) that simulate the network under both normal and pathogen-influenced conditions. Experimental results show that cells exposed to SEB exhibit an increase in the rate of executioner caspase expression (and subsequently apoptosis) of 1 hour 43 minutes (± 14 minutes), as compared to cells undergoing normal cell death.ConclusionOur model accurately reflects these results and reveals intervention points that can be altered to restore SEB-influenced system dynamics back to levels within the range of normal conditions.

Dissolution kinetic model for metal powders — a case of varying surface area

A kinetic model for leaching particulate metals in a batch system was developed. The leaching mechanism was analyzed by solving a series of ordinary differential equations relating heterogeneous reaction kinetics at the solid/liquid interface to mass transfer of reactant and products. Special emphasis was given to the effects of substrate surface area changes and initial particle distribution on the overall leaching rate. The dissolution of gold powders in acidic solution with bromine as an oxidant was used as a model system. The effects of particle size. particle distribution, chemical reaction order, rate constant and mass transfer coefficient on the overall metal powder dissolution rate are analyzed and discussed.

Electrical and mechanical fully coupled theory and experimental verification of Rosen-type piezoelectric transformers

A piezoelectric transformer is a power transfer device that converts its input and output voltage as well as current by effectively using electrical and mechanical coupling effects of piezoelectric materials. Equivalent-circuit models, which are traditionally used to analyze piezoelectric transformers, merge each mechanical resonance effect into a series of ordinary differential equations. Because of using ordinary differential equations, equivalent circuit models are insufficient to reflect the mechanical behavior of piezoelectric plates. Electromechanically, fully coupled governing equations of Rosen-type piezoelectric transformers, which are partial differential equations in nature, can be derived to address the deficiencies of the equivalent circuit models. It can be shown that the modal actuator concept can be adopted to optimize the electromechanical coupling effect of the driving section once the added spatial domain design parameters are taken into account, which are three-dimensional spatial dependencies of electromechanical properties. The maximum power transfer condition for a Rosen-type piezoelectric transformer is detailed. Experimental results, which lead us to a series of new design rules, also are presented to prove the validity and effectiveness of the theoretical predictions.

Long‐term prediction of the water level and salinity in the Dead Sea

AbstractThe long‐term water level variations of the Dead Sea (DS) were assessed using a previously developed simulation model. The model establishes the condition of the DS by evaluating a series of ordinary differential equations describing mass balances on the water and major chemical species. The DS was modelled as a two‐layer system. The model was modified using up‐to‐date inflow data and recent hypsometric graphs to derive the volume–area–level relationships. Three scenarios were studied: continuation of current conditions; a cessation in industrial activity when the DS water level drops to a certain level; and a simplified weather change scenario. The model predicted that the DS will not dry up, but its level will continue to drop with a decelerating rate with no equilibrium level in 500 years. Changing climate would accelerate the level drop. In the 500 year period, after an initial increase, the DS salinity drops. The opposite behaviour is noted in the evaporation rate, which increases after an initial decrease. Ceasing industrial pumping would eventually restore the DS to its normal level, but with changed conditions. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Dynamic simulation of the Dead Sea

The Dead Sea is a hypersaline terminal lake located in the Rift Valley between Jordan and Israel. In this work a generalised mathematical model describing the behaviour of the Dead Sea has been developed. The model established the condition of the Sea by evaluating a series of ordinary differential equations describing mass balances on the water and major chemical species in the Sea. The Sea was modelled as a two-layer system. The model was validated by comparing its predictions to measured level records. The results obtained highlighted the importance of detailed evaporation modelling, showed the necessity to model the Sea as a two-layer system, validated the usage of average distribution data to estimate the flowrates of rivers, and justified ignoring diffusion effects in further modelling. The model predicted that in the case of continuing current conditions, the level will continue to decline, at a decelerating rate, because the area and evaporation rate are both decreasing. Under these conditions, the model shows that the salinity of both layers will continue to increase, and that seasonal stratification and seasonal crystallisation of gypsum and aragonite will continue.

Weakly nonlinear oscillatory flow over an isolated seamount

Abstract Numerical models can show large errors in the modelling of flow around tall topography. We present a weakly nonlinear analytical solution of barotropic oscillatory flow over a tall axisymmetric seamount and compare the results to a numerical tidal model. The comparison is formulated for a flat topped seamount with parabolic sides and for a standing wave boundary condition in order to make the problem analytically tractable and to reduce it to a series of ordinary differential equations. The comparison shows reasonable agreement within the confines of the analytical formulation confirming that the numerical model can exploit tall topography. Comparisons to a Gaussian seamount and Kelvin wave forcing illustrate the generality of the results. The analytical formulation produces an equation relating relative errors in the comparison to discontinuities in topographical slope; the numerical model should have smaller errors over smoother topography. Residual velocities are found to be proportional to th...

Crystallization and precipitation engineering-VI. Solving population balance in the case of the precipitation of silver bromide crystals with high primary nucleation rates by using the first order upwind differentiation

This paper deals with the numerical simulation of the nucleation and crystal growth processes involved in the batch precipitation of photographic emulsions by the simultaneous addition of solutions of silver nitrate and sodium bromide to an aqueous solution of sodium bromide and gelatin. These processes have been successfully modelled in a previous paper (Muhr et al., 1995, Chem. Engng Sci. 50,345–355). Two methods used to solve the population balance equation in order to follow the crystal size distribution are described. The partial differential equation which represents the dynamics of the particle size density function ψ is transformed into a series of ordinary differential equations by means of the method of classes or by the direct discretization of ψ. After a comparison of both methods, the second option was selected for the calculations. To this series of equations are added balances for the total volume, moles of silver and moles of bromide. The aim of the present paper is to point out several details of the simulation. Results from example simulations are reported; the total number of crystals, the average crystal size, the standard deviation of the crystal size, and the crystal size density function are calculated. Special emphasis is set on the problems of grid spacing and representation of the primary nucleation process.

Magnetohydrodynamics equilibrium of a self-confined elliptical plasma ball

A variational principle is applied to the problem of magnetohydrodynamics (MHD) equilibrium of a self-contained elliptical plasma ball, such as elliptical ball lightning. The principle is appropriate for an approximate solution of partial differential equations with arbitrary boundary shape. The method reduces the partial differential equation to a series of ordinary differential equations and is especially valuable for treating boundaries with nonlinear deformations. The calculations conclude that the pressure distribution and the poloidal current are more uniform in an oblate self-confined plasma ball than that of an elongated plasma ball. The ellipticity of the plasma ball is obviously restricted by its internal pressure, magnetic field, and ambient pressure. Qualitative evidence is presented for the absence of sighting of elongated ball lightning.

Numerical Solution for Response of Beams with Moving Mass

An analytical‐numerical method is presented that can be used to determine the dynamic behavior of beams, with different boundary conditions, carrying a moving mass. This article demonstrates the transformation of a familiar governing equation into a new, solvable series of ordinary differential equations. The correctness of the results has been ascertained by a comparison, using finite element models, and very good agreement has been obtained. Furthermore, the article shows that the response of structures due to moving mass, which has often been neglected in the past, must be properly taken into account because it often differs significantly from the moving force model.

The origin and subsequent development in space of Tollmien-Schlichting waves in a boundary layer

Two kinds of small-parameter expansions are used to derive a series of ordinary differential equations governing spatially growing or decaying disturbances of the travelling-wave type in non-parallel laminar boundary layers. Eigensolutions of the lowest-order equation are shown to tend to resemble eigensolutions of the unsteady boundary-layer equation as the Reynolds number Re decreases, and to the Tollmien-Schlichting wave solutions of the Orr-Sommerfeld equation as Re increases. Examination of the next-order approximation reveals that the unsteady boundary-layer equation has eigensolutions only if Re exceeds a critical value depending on the frequency of disturbances, while it remains a partial differential equation for lower Reynolds numbers. Thus the Tollmien-Schlichting wave may be considered to originate at the above critical value of Re. The subsequent development of the wave with fixed frequency can be followed by integrating the spatial growth rate of the eigensolutions with respect to the downstream distance. The amplitude is at first damped but grows even higher than its original value farther downstream, provided the frequency is less than a certain value.