Nonlinear ordinary differential

P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991)... 

In a moving co-ordinate system, the traveling wave equations typically reduce to a system of parameterized nonlinear ordinary differential equations. The solutions of this system corresponding to pulses and fronts for the original reaction-diffusion equation are called homoclinic and heteroclinic orbits, correspondingly, or just connecting orbits. 

These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncoupled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case. 

The three first-order nonlinear ordinary differential equations given in Eqs. (3.3) are the mathematical model of the system. The parameters that must be known are Fj, 2, 3, ife, fcj, and k. The variables that must be specified before these equations can be solved are F and C o Specified does not mean that they must be constant. They can be time-varying, but they must be known or given functions of time. They are the forcing functions. 

Our mathematical model now contains six first-order nonlinear ordinary differential equations. Parameters that must be known are k kj, 3, n. 

Example 6.4. onsider the nonlinear ordinary differential equation for the gravity-flow tank of Example 2.9. 

Here, a class of biochemical processes whose mass balances can be described by the following nonlinear ordinary differential equations are considered. 

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. 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

Clearly, efficiency of the symmetry reduction procedure is subject to our ability to integrate the reduced systems of ordinary differential equations. Since the reduced equations are nonlinear, it is not at all clear that it will be possible to construct their particular or general solutions. That it why we devote the first part of this subsection to describing our technique for integrating the reduced systems of nonlinear ordinary differential equations (further details can be found in Ref. 33). 

Inserting ansatz (53), where ( ) are given by formulas (54) and (91), into the Yang-Mills equations yields systems of nonlinear ordinary differential equations of the form (87), where... 

Theory of bifurcations of dynamic systems on a plane. Wiley, New York. Jordan, D. W. and Smith, P. (1977). Nonlinear ordinary differential equations. Clarendon Press, Oxford. 

Overall the system of equations (continuity and momentum) is third order, nonlinear, ordinary-differential equation, boundary-value problem. The boundary conditions require no-slip at the plates and specified wall-injection velocities,... 

For a fixed value of x, this is a third-order, nonlinear, ordinary differential equation. Recall that the metric coefficient h x,y) depends on both coordinates, but that the cone angle 9 is a fixed constant. 

One of the simplest practical examples is the homogeneous nonisothermal and adiabatic continuous stirred tank reactor (CSTR), whose steady state is described by nonlinear transcendental equations and whose unsteady state is described by nonlinear ordinary differential equations. 

The unsteady thermal behavior of the dense phase is described by the nonlinear ordinary differential equation... 

Hence, the time independent Navier-Stokes equations are put in form of three coupled, nonlinear, ordinary differential equations, functions of rj only ... 

The dynamic model of the reactor and jacket consists of four nonlinear ordinary differential equations ... 

These three nonlinear ordinary differential equations will be used to simulate the dynamic performance of the CSTR. The openloop behavior applies when no controllers are used. In this case the flowrate of the cooling water is held constant. With closedloop behavior, a temperature controller is installed that manipulates cooling water flow to maintain reactor temperature. 

Nonlinear Dynamic Simulation The nonlinear ordinary differential equations are numerically integrated in the Matlab program given in Figure 4.2. A simple Euler integration algorithm is used with a step size of 2 s. The effects of several equipment and operating parameters are explored below. 

After the series of metabolic pathways had been elucidated for the three model compounds 1-3, these data were implemented into the mathematical model PharmBiosim. The nonlinear system s response to varying ketone exposure was studied. The predicted vanishing of oscillatory behavior for increasing ketone concentration can be used to experimentally test the model assumptions in the reduction of the xenobiotic ketone. To generate such predictions, we employed as a convenient tool the continuation of the nonlinear system s behavior in the control parameters. This strategy is applicable to large systems of coupled, nonlinear, ordinary differential equations and shall together with direct numerical simulations be used to further extend PharmBiosim than was sketched here. This model already allows more detailed predictions of stereoisomer distribution in the products. 

The coefficients in the above series in if/p(r) alternate between the hyperbolic sine and cosine value of the uniform contribution, i/r0(z). In contrast to a full linear treatment, which is the usual procedure followed, the 0(if/p) term here does not vanish. As if/0(z) is large we must regard it as satisfying the nonlinear, ordinary differential form of the PB equation,... 

Irvine, D. and Savageau, M., Efficient solution of nonlinear ordinary differential equations expressed in S-system canonical form, SIAM Journal of Numerical Analysis, Vol. 27, 1990, pp. 704-735. 

Perfect reactor level control is assumed. The reactor effluent flowrate F is fixed in control structure CS2. The two state variables of the system are the two reactor compositions zA and zB. The two nonlinear ordinary differential equations describing the system are... 

The two nonlinear ordinary differential equations can be linearized around the steady-state values of the reactor compositions zA and zs. Laplace transforming gives the characteristic equation of the system. It is important to remember that we are looking at the closed-loop system with control structure CS2 in place. Therefore Eq. (2.13) is the closed-loop characteristic equation of the process ... 

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