Ordinary differential equation linearization

Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems. 

No assumption has been made as to continuity, in general, but it will now be assumed that all functions have continuous derivatives of order n + 1. Then the t satisfy a linear ordinary differential equation of order n + 1, which can be written in the form... 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

Hie quasi steady state approximation can be conveniently applied to equations 19 to 21, without any significant loss of accuracy, due to tlie high reactivity of tlie reacting species in aqueous solution. Hms, the system of ordinary differential equations is readily reduced to a system of algebraic non linear equations. 

The resulting model of raulticonponent enulsion pjolymerization systems is consituted by the Pffil 17, an integro-differential equation, a set of ordinary differential equations (equation 18 and 25 and the equations for pjoiymer conposltlon) and the system of the remaining non linear algebraic equations. As expected the conputatlonal effor t is concentrated on the solution of the PBE therefore, let us examine this aspect with some detail. 

Equation (A4) is a first order, linear, ordinary differential equation which can be solved analytically for [PJ assuming X, and X, are constant over a small increment in time. Solving for [PJ from some time ti to tj gives Equation (1) (1). When X, is considered a function of time (i.e., initiator concentration is allowed to vary through the small time increment) while maintaining X, constant over the increment. Equation (A4) can again be solved analytically to give Equation (3). 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

In this chapter we concentrate on dynamic, distributed systems described by partial differential equations. Under certain conditions, some of these systems, particularly those described by linear PDEs, have analytical solutions. If such a solution does exist and the unknown parameters appear in the solution expression, the estimation problem can often be reduced to that for systems described by algebraic equations. However, most of the time, an analytical solution cannot be found and the PDEs have to be solved numerically. This case is of interest here. Our general approach is to convert the partial differential equations (PDEs) to a set of ordinary differential equations (ODEs) and then employ the techniques presented in Chapter 6 taking into consideration the high dimensionality of the problem. 

Next let us turn our attention to models described by a set of ordinary differential equations. We are interested in establishing confidence intervals for each of the response variables y, j=l,...,/w at any time t=to. The linear approximation of the output vector at time to,... 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

Classical process control builds on linear ordinary differential equations and the technique of Laplace transform. This is a topic that we no doubt have come across in an introductory course on differential equations—like two years ago Yes, we easily have forgotten the details. We will try to refresh the material necessary to solve control problems. Other details and steps will be skipped. We can always refer back to our old textbook if we want to answer long forgotten but not urgent questions. 

The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. 

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. 

In the following section, we only consider the integration of the equation of linear motion Eq. (20) the procedure for the equation of rotational motion, Eq. (21), will be completely analogous. Mathematically, Eq. (20) represents an initial-value ordinary differential equation. The evolution of particle positions and velocities can be traced by using any kind of method for ordinary differential equations. The simplest method is the first-order integrating scheme, which calculates the values at a time t + 5t from the initial values at time t (which are indicated by the superscript 0 ) via ... 

MADONNA provides an effective means of solving very large and complicated sets of simultaneous non-linear ordinary differential equations. The above complex reaction problem is solved with considerable ease by means of the following MADONNA program, which is used here to illustrate some of the main features of solution. 

In the computer simulation studies of the two preceding chapters, the systems and their describing equations could be quite complex and nonlinear. In the remaining parts of this book only systems described by linear ordinary differential equations will be considered (linearity is defined in Chap. 6). The reason we are limited to linear systems is that practically all the analytical mathematical techniques currently available are applicable only to linear equations. 

The classical analytical techniques discussed in this chapter are limited to linear ordinary differential equations. But they yield general analytical solutions that apply for any values of parameters, initial conditions, and forcing functions. 

We will start by briefly classifying and defining types of systems and types of disturbances. Then we will learn how to linearize nonlinear equations. It is assumed that you have had a course in differential equations, but we will review some of the most useful solution techniques for simple ordinary differential equations. Finally we will show how useful dynamic insights can sometimes be obtained from steadystate equations alone. 

Therefore the last term in Eq. (6.37) is equal to zero. We end up with a linear ordinary differential equation with constant coefficients in terms of perturbation variables. 

J.1 First-Order Linear Ordinary Differential Equation... 

Derive one linear ordinary differential equation that gives the dynamic dependence of process temperature on controller output signal CO. 

Laplace transformation can only be applied to linear ordinary differential equations. So for most of the rest of the book, we will be dealing with linear systems. 

The use of Laplace transfonnations yields some very useful simplifications in notation and computation. Laplace-transforming the linear ordinary differential equations describing our processes in terms of the independent variable t converts them into algebraic equations in the Laplace transform variable s. This provides a very convenient representation of system dynamics. 

Consider the system described by the linear, homogeneous ordinary differential equations... 

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. 

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