System Of Ordinary Dierential Equations Research Articles

Continuous Stirred Tank Reactor: A Process Design for Interesterification of Macauba (Acrocomia aculeata) palm oil

Other than the edible oils extracted from the Acrocomia aculeata fruit, there is a growing interest in the palm to generate other high value-added products. Relatively high amounts of carotenoids (up to 378 mg kg-1) have been found in the esculent oils mechanically obtained from the fruit mesocarp. From industrial application perspectives, several processes have been proposed to modify native vegetable oils to yield high functional properties of structured lipids. For interesterified products, the thermal effects of industrial reactors are crucial in reaction mechanisms. The present study has taken into account previously estimated kinetic parameters for the overall disappearances of all-trans β-carotene in the Acrocomia aculeata oil (ko= 2.6 x 10-4 min-1; Ea = 105.0003 kJ mol-1; ΔH = 9.8 x 104 J kg-1) to develop a continuous stirred tank reactor (CSTR) kinetic treatment that obeys first-order kinetics. A system of ordinary dierential equations - mass and energy balances - was solved by the 4th order Runge-Kutta method (GNU Octave software). Under research conditions related to interesterification processing (2 h; 393.15 K), the initial concentration of carotenoids (around 11%) showed no significant decrease. Overall, realistic processing effects and conditions have been assessed, integrating results and knowledge, improving prospects of Acrocomia aculeata as a promising source of high-quality raw material, for producing functional ingredients and food with nutraceutical properties.

Mathematical Modelling of the Dynamics of COVID-19 Disease Transmission

We construct a simple mathematical model that describes the dynamics of the transmission of COVID-19 disease in a human population. It accounts for the various phases of the disease and its mode of contact through infectious humans and surfaces. The contribution of asymptomatic humans in the dynamics of the disease is well represented. The model is a system of ordinary dierential equations that describes the evolution of humans in a range of COVID-19 states due to emergence of an index case. The analysis includes establishment of the basic reproduction number, R0, where, R0 < 1 signifies a disease free state that is locally asymptotically stable. A key result in this study shows some long term damped oscillatory behaviour that do not seem to end soon.

How to identify the physiological parameters and run the optimal race

This paper shows how a system of ordinary dierential equations describing the evolution of the anaerobic energy, the oxygen uptake, the propulsive force and the velocity of a runner accurately describes pacing strategy. We find a protocol to identify the physiological parameters needed in the model using numerical simulations and time splits measurements for an 80m and a 1600m race. The velocity curve of the simulations is very close to the experimental one. This model could allow to study the influence of training and improving some specific parameters for the pacing strategy.

Model for reliable location of information in storage systems

In this work, the model of reliable location of information in systems of related nodes containing the physical disks is proposed. The hypotheses on the fault-tolerance of an individual node of system are formulated. On the basis of hypotheses, there are oered dependencies of speed and fault-tolerance of node on the density of its lling up. A system of ordinary dierential equations for the failure probability of the entire system is obtained. Examples of the dynamic information distribution in the system of two and three nodes are considered. Mathematics Subject Classication: 34B15, 37N40

Mathematical analysis of a multistable switch model of cell differentiation

Non-binary simultaneous decision network of gene regulation represents a cell dierentiation process that involves more than two possible cell lineages. The simultaneous decision network is an alternative to the hierarchical models of gene regulation and it exhibits possible presence of multistable master switches. To investigate the qualitative behavior of the dynamics of the simultaneous decision network, we employ geometric techniques in the analysis of the network’s corresponding system of ordinary dierential equations (ODE). We determine the location and the maximum number of equilibrium points given a set of parameter values. Our analysis shows that the solution to the ODE model always converge to a stable equilibrium point. Varying the values of some parameters, such as the degradation rate and the amount of exogenous stimulus, can decrease the size of the basin of attraction of an

Peculiarities of elastic wave refraction from the layer with fractal distribution of density

In this study, we underline the peculiarities of the refraction problem of elastic waves from a layer with a fractal density distribution. The refraction problem is reduced to the system of ordinary dierential equations with linear coecients. Analytic solutions for each of its equations are found. The case for the layer with fractal density distribution is investigated numerically. Characteristic maxima of the reected wave

Modelling Endemic Malaria and the Dangers of Partial Immunity

In this talk we present human-mosquito interaction model that describes the development of malaria in a human population. The model accounts for the various phases of the disease in humans and mosquitoes, together with treatment of both sick and partially immune humans. The partially immune humans (termed asymptomatic) have recovered from the worst of the symptoms, but can still transmit the disease. We present a mathematical model consisting of a system of ordinary dierential equations that describes the evolution of humans and mosquitoes in a range of malarial states. A new feature, in what turns out to be a key class, is the consideration of reinfected asymptomatic humans. The analysis to be presented includes establishment of the basic reproduction number, R 0 , and asymptotic analysis to draw out the major timescale of events in the process of malaria becoming non-endemic to endemic in a region following introduction of a few infected mosquitoes. The results suggests that intervention programmes may yield better results if implemented during the time scale when the feedback from infectious humans osets the linear growth eect of the initial small amount of infected mosquitoes. The reinfected asymptomatic class is signicant to the continuation of malaria in endemic areas and in our analysis of a model describing treatment we show evidence of effective control and possible eradication of the disease, with a ''moderate level of treatment, if partially immune carriers of malaria parasites are treated alongside with sick humans.

Proper Orthogonal Decomposition for Reduced-Order Thermal Solution in Hypersonic Aerothermoelastic Simulations

The ability to perform full-order aerothermoelastic simulations of hypersonic vehicles is hindered by the strong coupling exhibited between the aerodynamics, heat transfer, and structural dynamic response in the hypersonic ight regime. As a result of these interactions, alternative techniques are necessary to obtain computationally tractable systems of governing equations and their solutions. This work addresses the use of proper orthogonal decomposition for reduced-order solution of the heat transfer problem within a hypersonic modeling framework. The specic challenge of handling time-dependent boundary conditions due to transient aerodynamic heating is discussed. An overview of the proper orthogonal decomposition is given and two methods for solution of the reduced system of ordinary dierential equations are outlined. The methodology is applied to a representative hypersonic vehicle control surface model for two cases in which the time-history of the thermal load vector is known a priori: one in which the boundary conditions are time-independent and another in which they are time-varying. Results demonstrate the ability of the reduced-order solution to approximate the full-order solution with reasonable accuracy. Finally, a time-marching hypersonic aerothermoelastic framework is described in which proper orthogonal decomposition is used for the transient thermal solution.

Novel algorithm for simulation of3Dquantum reactive atom-diatom scattering

A new approach is described to the evaluation of the quantum scattering S-matrix in 3D atom-diatom reactive collision. The theory is developed in terms of natural collision coordinates where the coordinate reaction fulfills the same role as a time in a time-dependent scattering formulation. Having written the full wavefunction of the particles system in the coupled-channel representation we have proved that the 3D multi-channel scattering problem can be reduced to the inelastic single-arrangement problem which is described by system of ordinary dierential equations (ODE) of second order. The system of coupled-channel second order ODEs exactly is reduced to the system of integro-dierential equations (IDE) of first order which is solved with the initial conditions. The problem of Koshi for the system of IDEs is proposed to be solved by the method of Runge-Kutta of fourth order. The detailed algorithm for parallel simulation of initial 3D scattering problem is proposed. In result of simulation of IDEs the full wavefunction and all S-matrix elements of reactive transitions and state-to-state cross section are obtained simultaneously without other extra calculations.

Magneto-optic Study on a Transverse Gyrotropic Grating

A two-dimensional scalar Helmholtz equation is derived for a transverse electric mode transverse gyrotropic grating. In the frame of a rigorous coupled-wave analysis, a Bloch-type wave solution and a Fourier expansion of the dielectric tensors are substituted into the Helmholtz equation to obtain a system of ordinary dierential equations, and an Airy-like internal-reflection series is applied for better convergence. In the derivation, no approximation or estimate has been employed, and it is easy to implement. Finally, the numerical method is tested with various parameters.

Parallelization in time for thermo‐viscoplastic problems in extrusion of aluminium

AbstractThe ParaReal algorithm (C.R. Acad. Sci. Paris 2001; 332:1–6) is a parallel approach for solving numerically systems of ordinary differential equations by exploiting parallelism across the steps of the numerical integrator. The method performs well for dissipative problems and problems of fluid–structure interaction (Int. J. Numer. Methods Engng 2003; 58:1397–1434). We consider here a convergence analysis for the method and we report the performance achieved from the parallelization of a Stokes/Navier–Stokes code via the ParaReal algorithm. Copyright © 2009 John Wiley & Sons, Ltd.

Topological properties of attainability sets of linear systems

Linear systems are best studied in control theory. They are both of applied interest in themselves and of theoretical importance. Indeed, the local investigation of a nonlinear control system by linearization results in a linear system whose behavior largely characterizes the local behavior of the original nonlinear system. The structure and properties of attainability sets play an important role in the analysis of control systems. In the present paper, we consider topological properties of attainability sets of linear systems. Consider an object whose motion is governed by the linear system of ordinary dierential equations _ x = A(t)x +B(t)u(t) ;t 2 [0;T]; (1) with the initial condition x(0) = 0: (2)

Spatial homogenization and internal layers in a reaction-diffusion system

For a system of reaction-diusion equations of activator-inhibitor type, we show that solutions undergo at least three stages of dynamical behaviour when the activator diuses slowly and reacts fast, and the inhibitor diuses fast. In the first stage, the inhibitor quickly decays to its spatial average (spatial homogenization of the inhibitor). In the second stage, the activator develops internal layers (formation of internal layers). In the third stage, the layers move according to a certain motion law (motion of interfaces) which is described by a system of ordinary dierential equations on finite time intervals. Asymptotic behaviour of the solutions of the interface equation is also analyzed. To describe the behaviour of the solutions of the reaction-diusion equations after the last interface equation becomes powerless, another type of interface equation is proposed.

On non-Lie ansatzes and new exact solutions of the classical Yang-Mills equations

We ssuggest an eective method for reducing Yang-Mills equations to systems of ordinary dierential equations. With the use of this method, we construct wide families of new exact solutions of the Yang-Mills equations. Analysis of the solutions obtained shows that they correspond to conditional symmetry of the equations under study.

On the stability of solutions of linear boundary value problems for a system of ordinary differential equations

Linear boundary value problems for a system of ordinary dierential equations are considered. The stability of the solution with respect to small perturbations of coecients and boundary val- ues is investigated.