First-order nonlinear ordinary differential

These equations are a set of nonlinear first-order ordinary differential equations that describe the evolution of the n species as a function of time starting from a set of initial conditions... 

We now have to solve the following system of two nonlinear coupled first-order ordinary differential equations for the given initial conditions ... 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

Combining the mass eonservation (Eq. (1)), momentum eonservation equation (Eq. (2)), mass balanee and energy eonservation, (Eq. (3)) and pressure balance (Eq. (5)) along with Eq. (6) for the eurvature for the miero-region results in a set of three nonlinear first-order ordinary differential Eqs. (7), (8) and (9), as derived in [15],... 

Suppose that we have a nonlinear dynamical system, that is, first-order ordinary differential equations,... 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

AUT097 (http //indy.cs.concordia.ca/auto) A code for tracking by continuation the solution of systems of nonlinear algebraic and/or first-order ordinary differential equations as a function of a bifurcation parameter (available only for UNIX-based computers)... 

State-space models provide a convenient representation of dynamic models that can be expressed as a set of first-order, ordinary differential equations. State-space models can be derived from first principles models (for example, material and energy balances) and used to describe both linear and nonlinear dynamic systems. 

Analytical solution of film thickness. The relation governing film thickness (= involves and is a nonlinear first-order ordinary differential equation. The following series solution can easily be developed ... 

The problem in obtaining a state space model for the dynamics of the CSD from this physical model is that the population balance is a (nonlinear) first-order partial differential equation. Consequently, to obtain a state space model the population balance must be transformed into a set of ordinary differential equations. After this transformation, the state space model is easily obtained by substitution of the algebraic relations and linearization of the ordinary differential equations. 

This equation is a second-order ordinary differential equation. It is nonlinear when is other than zero- or first-order. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

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 4.3 Plagan Poiseuille flow - laminar, steady incompressible flow in a long pipe with a linear pressure gradient (first-order, nonlinear solution to an ordinary differential equation)... 

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. 

These two ordinary differential equations are a nonlinear, first-order coupled system, with the axial coordinate z as the independent variable. The dependent variables are U and p. 

The equations (2.1 - 2.5) are the set of five first-order, nonlinear, eoupled ordinary differential equations with five unknown variables r, ui, Uv, Pi and Pv. This system need to be solved numerically with the following boundary eonditions ... 

The solution of Equations 47, 48, and 49 requires numerical techniques. For such nonlinear equations, it is usually wise to employ a simple numerical integration scheme which is easily understood and pay the price of increased computational time for execution rather than using a complex, efficient, numerical integration scheme where unstable behavior is a distinct possibility. A variety of simple methods are available for integrating a set of ordinary first order differential equations. In particular, the method of Huen, described in Ref. 65, is effective and stable. It is self-starting and consists of a predictor and a corrector step. Let y = f(t,y) be the vector differential equation and let h be the step size. 

Several of these simple mass balances with basic rate expressions were solved analytically. In the case of multiple reactions with nonlinear rate expressions (i.e., not first-order reaction rates), the balances must be solved numerically. A high-quality ordinary differential equation (ODE) solver is indispensable for solving these problems. For a complex equation of state and nonconstant-volume case, a differential-algebraic equation (DAE) solver may be convenient. 

The mass balance given by (18-5), (18-6), and (18-7) corresponds to an ordinary differential eqnation that is second-order dne to diffusion and nonlinear when n 0, 1 due to the rate of depletion of reactant A via chemical reaction. Numerical integration is required to generate basic information for 4 a( A), except when n = 0, 1. Second-order ODEs are solved numerically by reducing them to a set of two coupled first-order ODEs, which require two boundary conditions for a unique solution. The procedure is illustrated for porous wafers. If the dimensionless gradient of molar density is defined by d /dr] = then... 

The most frequently encountered numerical problem in nonlinear chemical dynamics is that of solving a set of ordinary, nonlinear, first-order, coupled, autonomous differential equations, such as those describing the BZ reaction. We hope you understand by now what nonlinear means, but let us comment on the other modifiers. The equations are ordinary because they do not contain partial derivatives (we consider partial differential equations in the next section), first order because the highest derivative is the first derivative, and coupled because the time derivative of one species depends on the concentrations of other species. In the absence of time-dependent external forcing, rate equations are autonomous, meaning that time does not appear explicitly on the right-hand side. 

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