Coupled ordinary differential equations

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

Population Density Response Surface. The algorithm method of characteristic is used to reduce the partial differential Equation (1) into a set pf coupled ordinary differential equations. Since T(n,t) is an exact differential, then... 

For both the finite difference and weighted residual methods a set of coupled ordinary differential equations results which are integrated forward in time using the method of lines. Various software packages implementing Gear s method are popular. 

I presented a group of subroutines—CORE, CHECKSTEP, STEPPER, SLOPER, GAUSS, and SWAPPER—that can be used to solve diverse theoretical problems in Earth system science. Together these subroutines can solve systems of coupled ordinary differential equations, systems that arise in the mathematical description of the history of environmental properties. The systems to be solved are described by subroutines EQUATIONS and SPECS. The systems need not be linear, as linearization is handled automatically by subroutine SLOPER. Subroutine CHECKSTEP ensures that the time steps are small enough to permit the linear approximation. Subroutine PRINTER simply preserves during the calculation whatever values will be needed for subsequent study. 

If simplifying conditions such as adiabatic behavior and temperature-independence of (—AHra) and CP are not valid, the material and energy balances may have to be integrated numerically as a system of two coupled ordinary differential equations. 

The transport rates fj will be determined by the turbulent flow field inside the reactor. When setting up a zone model, various methods have been proposed to extract the transport rates from experimental data (Mann et al. 1981 Mann et al. 1997), or from CFD simulations. Once the transport rates are known, (1.15) represents a (large) system of coupled ordinary differential equations (ODEs) that can be solved numerically to find the species concentrations in each zone and at the reactor outlet. 

Here I describe a simple numerical method for solving Maxwell s equation in the frequency domain. As the structure to be analysed is onedimensional, Maxwell s equations turn into a system of two coupled ordinary differential equations that can be solved with standard numerical routines. 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Gray, B. F. and Roberts, M. J. (1988). A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters. Proc. R. Soc., 416, 361-89. 

Further, both f(f) and V (r ) are expanded in the spherical harmonics and thus the problem is reduced to the solution of a finite set of coupled ordinary differential equations. 

The above equations allow us to solve for Tc, 7), Tan, Tca, Nh2, Nh2o- No2 (bars have been dropped) as a function of time. As in the model of the prior section, an additional assumption is needed namely, that the Tumped internal conditions for the anode and cathode gases will be the average of the inlet and outlet values. The given parameters for this analysis are all the cell design parameters (geometry, materials, properties, etc.), the input temperatures, pressures, mass flow, and compositions of the anode and cathode gases, and the load current on the cell. Such a simple set of coupled ordinary differential equations is readily solved via Matlab-Simulink, and a sample case is presented in Section 9.5. 

Similarly, any two coupled ordinary differential equations can be written for the operator equations of ADM... 

The two-coupled ordinary differential equations have been decomposed by parameterized ADM, and the Mathematica code is listed below. 

The above discussion completes our description of the single-nephron model. In total we have six coupled ordinary differential equations, each representing an essential physiological relation. Because of the need to numerically evaluate Ce in each integration step, the model cannot be brought onto an explicit form. The applied parameters in the single-nephron model are specified in Table 12.1. They have all been adopted from the experimental literature, and their specific origin is discussed in Jensen et al. [13]. 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

Formulating appropriate rate laws for CO adsorption, OH adsorption and the reaction between these two surface species, a set of four coupled ordinary differential equations is obtained, whereby the dependent variables are the average coverages of CO and OH, the concentration of CO in the reaction plane and the electrode potential. In accordance with the experiments, the model describes the S-shaped I/U curve and thus also bistability under potentiostatic control. However, neither oscillatory behavior is found for realistic parameter values (see the discussion above) nor can the nearly current-independent, fluctuating potential be reproduced, which is observed for slow galvanodynamic sweeps (c.f. Fig. 30b). As we shall discuss in Section 4.2.2, this feature might again be the result of a spatial instability. 

Larter, R., Speelman, B., and Worth, R., A coupled ordinary differential equation lattice model for the simulation of epileptic seizures, Chaos, Vol. 9, No. 3, 1999, pp. 795-804. 

These are two coupled ordinary differential equations. (With three energy levels instead of two, we would have three coupled equations.) The system can be solved by differentiating Eq. (3.34.11) again with respect to time, using Eq. (3.34.12), and rearranging... 

The balance equations (7.12 and 7.13) form a set of coupled ordinary differential equations, which has to be solved numerically. Analytical integration is possible for special cases only. 

The techniques just described have been extensively used in modeling reactive flow problems at NRL. Efficient solution of the coupled ordinary differential equations associated with these problems has enabled us to perform a wide variety of calculations on H2 °2 anC Ha/Oo mixtures which have greatly extended our understanding of tne combustion and detonation behavior of these systems. In addition numerous atmospheric problems have been studied. Details on these investigations are provided in references (7) and (9). 

Let us consider the following coupled ordinary differential equations ... 

Pl-lSx (a) There are initially 500 rabbits (x) and 200 foxes (y) on Farmer Oat s property, Use POLYMATH or MATLAB to plot the concentration of foxes and rabbits as a function of time for a period of up to 500 days. The predator-prey relationships are given by the following set of coupled ordinary differential equations ... 

We see that we have j coupled ordinary differential equations that imist be solved simultaneously with either a numerical package or by writing an ODE solver. In fact, this procedure has been developed to take advantage of the vast number of computation techniques now available on mainframe (e.g,... 

The numerical solution of the reactor model, consisting of a set of partial differential equations, is most commonly achieved by application of orthogonal collocation in the space coordinates [13,35,38]. The resulting coupled ordinary differential equations may be integrated in time by using routines from the NAG Fortran library. 

Liquid Phase. For liquid-phase reactions in which there is no volume change, concentration is the preferred variable. The mole balances are shown in Table 4-5 in terms of concentration for the four reactor types we have been discussing. We see from Table 4-5 that we have only to specify the parameter values for the system (C O o- tc.) and for the rate law (i.e., fe, a,P) to solve the coupled ordinary differential equations for either PFR, PBR, or batch reactors or to solve the coupled algebraic equations for a CSTR. 

Data evaluation The evaluation of model parameters by non-linear fitting of experimental net diffusion flux densities to theory requires solution of a set of coupled ordinary differential equations which describe diffusion in porous solids according to MTPM (integration of differential equations with splitted boundary conditions). 

Solution The method for solving partial differential equations generally involves finding a method to express them as coupled ordinary differential equations. A similarity transformation is possible if c,- can be expressed as a function of only a new variable. This requirement implies that equation (2.48) can be expressed as a function of only the new variable, and that the three conditions (2.49) in time and position can collapse into two conditions in the new variable. 

J. S. Newman, "Numerical Solution of Coupled, Ordinary Differential Equations," Industrial and Engineering Chemistry Fundamentals, 7 (1968) 514-517. 

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