Systems of ordinary differential equations with initial values

5 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS, WITH INITIAL VALUES 

As has been seen in Sect. 3, the equations of mass, energy and momentum balances for batch and plug flow reactors generally constitute a system of ordinary differential equations, with initial values. It is convenient to write such a system in a compact vector form, viz. 

Sometimes, instead of an initial value problem, the mathematical model of a chemical process is a boundary value problem in which values of the dependent variables are specified at different values of the independent variable t. The shooting technique consists of solving an initial value problem, but with an initial value vector a considered as a parameter to estimate (by optimization techniques) so that boundary conditions are satisfied. In this way, a boundary value problem is transformed into an initial value problem. 

Numerical algorithms for integrating ordinary differential equations with initial conditions have been reviewed in general books [127—134] and specialized books [19, 139—145], as well as in an avalanche of papers. But, before describing the main methods for solving such systems, 

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With the exception of a few cases (simple reaction orders, isothermal condition), the solution of the above system of ordinary differential equations can be obtained only numerically. For this, there are several numerical software tools available, (e.g., Matlab, Mathematica, GNU Octave, etc.) that provide advanced solvers for such initial value problems. A good overview can be found, e.g., in Press et al., 2002. 

Abstract. A class (m,k)-methods is discussed for the numerical solution of the initial value problems for impHcit systems of ordinary differential equations. The order conditions and convergence of the numerical solution in the case of implementation of the scheme with the time-lagging of matrices derivatives for systems of index 1 are obtained. At A < 4 the order conditions are studied and schemes optimal computing costs are obtained. 

For one special case, isothermal reaction at constant density, the set of differential equations comprising the time derivatives of all species concentrations is sufficient to determine the evolution of a system described by a given reaction mechanism for any assumed starting concentrations. While this special case does apply for some experiments of interest in combustion research, it does not pertain to the conditions under which most combustion processes occur. Usually we must expand our set of differential equations so as to describe the effects of chemical reaction on the physical conditions and the effects of changes in the physical conditions on the chemistry. In either case, the evolution of the system is found by numerical integration of the appropriate set of ordinary differential equations with a computer. This procedure is known in the language of numerical analysis as the solution of an initial value problem. 

Armed with techniques for solving linear and nonlinear algebraic systems (Chapters 1 and 2) and the tools of eigenvalue analysis (Chapter 3), we are now ready to treat more complex problems of greater relevance to chemical engineering practice. We begin with the study of initial value problems (IVPs) of ordinary differential equations (ODEs), in which we compute the trajectory in time of a set of N variables Xj(t) governed by the set of first-order ODEs... 

The solution of boundary value problems depends to a great degree on the ability to solve initial value problems.) Any n -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition. In general... 

Figure 8 depicts our view of an ideal structure for an applications program. The boxes with the heavy borders represent those functions that are problem specific, while the light-border boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically either systems of nonlinear algebraic equations, ordinary differential equation initial or boundary value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the user must write the subroutine that defines his particular system of equations. However, that subroutine should be able to make calls to problem-independent software to return many of the components that are needed to assemble the governing equations. Specifically, such software could be called to return in-... 

CONP Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA. A Fortran program (conp.f) that solves the time-dependent kinetics of a homogeneous, constant pressure, adiabatic system. The program runs in conjunction with CHEMKIN and a stiff ordinary differential equation solver such as LSODE (lsode.f, Hindmarsh, A. C. LSODE and LSODI, Two Initial Value Differential Equation Solvers, ACM SIGNUM Newsletter, 15, 4, (1980)). The simplicity of the code is particularly valuable for those not familiar with CHEMKIN. 

This is a partial differential equation, as we should expect from a plug-flow tubular reactor with a single reaction. We note in passing that the solution requires the specification of an initial distribution and a boundary, or feed, value. These are both functions (the first of z because t = 0 the second of t because z = 0) in the distributed system. Of the corresponding quantities, c0 and cin, in the lumped system, the latter is embodied in the ordinary differential equation itself and the former is the initial value. 

In the above definitions, 9 represents a set of parameters of the system, having constant values. These parameters are also called control parameters. The set of the system s variables forms a representation space called the phase space [32]. A point in the phase space represents a unique state of the dynamic system. Thus, the evolution of the system in time is represented by a curve in the phase space called trajectory or orbit for the flow or the map, respectively. The number of variables needed to describe the system s state, which is the number of initial conditions needed to determine a unique trajectory, is the dimension of the system. There are also dynamic systems that have infinite dimension. In these cases, the processes are usually described by differential equations with partial derivatives or time-delay differential equations, which can be considered as a set of infinite in number ordinary differential equations. The fundamental property of the phase space is that trajectories can never intersect themselves or each other. The phase space is a valuable tool in dynamic systems analysis since it is easier to analyze the properties of a dynamic system by determining... 

Ordinary differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractions are nonlinear because the coefficients of Xij change with time. Therefore, numerical methods of integration with respect to time must be employed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear (Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

Initiated by photorecovered FMNH2, the bioluminescent reaction is a short flash of light with pronounced maximum. In this case the kp value is very large" and the enzyme makes, at this, one cycle. It means, that interaction of enzyme with FMNH2 very quickly decreases. So the kinetic behavior of the system can be described as the system of ordinary linear differential equations with constant coefficients ... 

These are systems where the state variables are varying in one or more directions of the space coordinates. The simplest chemical reaction engineering example is the plug flow reactor. These systems are described at steady state either by an ordinary differential equation (where the variation of the state variables is only in one direction of the space coordinates, i.e. one dimensional models, and the independent variable is this space direction), or partial differential equations (when the variation of the state variables is in more than one direction of the space coordinates, i.e. two dimensional models, and the independent variables are these space directions). The ordinary differential equations of the steady state of the one-dimensional distributed model can be either initial value differential equations (e.g. plug flow models) or two-point boundary value differential equations (e.g. models with superimposed axial dispersion). The equations describing the unsteady state of distributed models are invariably partial difierential equations. 

Fiiiite Differences. Many of the equations used in numerical analysis contain ordinary or partial derivatives. One of the most important techniques used in numerical analysis is to replace the derivatives in an equation, or system of equations, with equivalent finite differences of the same order, and then develop an iterative formula from the equation. For example, in the case of a first-order differential equation with an initial value condition, such as/(x) = F[x,/( )],... 

A characteristic of a differential equation is that it involves an unknown function and one or more of the function s derivatives. If the unknown function depends on only one independent variable, it is classified as an ordinary differential equation (ODE). The order of the differential equation is simply the order of the highest derivative that appears in the equations. Consequently, a first-order ODE contains only first derivatives, whilst a second-order ODE may contain both second and first derivatives. The ODEs can also be classified as linear or non-hnear. Linear ODEs are the ones in which all dependent variables and their derivatives appear in a linear form. This implies that they cannot be multiphed or divided by each other, and they must be raised to the power of 1. An ODE has an infinite number of solutions, but with the appropriate conditions that describe systems, i.e. the initial value or the boundary value, the solutions can be determined uniquely. 

In the numerical integrations of the one parameter Equations (77) and (78), numerical difficulties caused by the small value of Z can be avoided by the use of the prompt jump approximation which replaces the differential Equation (77) for N by the ordinary Equation (101). This reduction in the overall order of the system of equations must be accompanied by a changed set of initial conditions. Specifically, the initial conditions to be used are those which apply immediately following the prompt jump. Thus, in a problem where p changes suddenly, the calculation starts with the value of N following the prompt jump. If dpjdt changes suddenly, then the calculation... 

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