Ordinary differential equation linear

The analysis of many physicochemical systems yields mathematical models that are sets of linear ordinary differential equations with constant coefficients and can be reduced to the form 

Such examples abound in chemical engineering, The unsteady-state material and energy balances of multiunit processes, without chemical reaction, often yield linear differential equations. 

In order to develop this solution, let us first consider a single linear differential equation of the type 

37) is essentially the scalar form of the matrix set of Eq. (5.34). The solution of the scalar equation can be obtained by separating the variables and integrating both sides of the equation 

First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs). Single first order equations can be integrated by classical methods (Rice and Do, 1995). However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra. In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations. 

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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. 

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). 

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. 

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. 

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. 

A solution to equation (E2.1.2) may be achieved by (1) separating variables and integrating or (2) solving the equation as a second-order, linear ordinary differential equation. We will use the latter because the solution technique is more general. 

Equation (E6.4.6) is a linear ordinary differential equation and has the solution... 

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 ... 

Linear ordinary differential equations are relatively easily manipulated and yield solutions which are simply the sum of exponential terms ... 

Many of the chemistry references appropriate to this chapter have been given in chapter 2. Local stability analysis is covered in most advanced mathematical texts on non-linear ordinary differential equations, for example ... 

All of the terms in these equations have the factor cos fax). Cancelling this leaves a pair of linear ordinary differential equations for the amplitudes of the perturbations ... 

This is a linear ordinary-differential-equation boundary-value problem that can be solved analytically (see Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, 1960). Here, however, proceed directly to numerical finite-difference solution, which can be implemented easily in a spreadsheet. Assuming a cone angle of a = 2° and a rotation rate of 2 = 30 rpm, determine f(0) — v /r. 

With K being a constant, an exact solution to the linear ordinary differential equation gives the velocity profile as... 

The periodic response of a linear viscoelastic cooling tower to a prescribed recurring sequence of pressure fluctuations and earth accelerations are analyzed. An approximate analysis, based on the bending theory of shells, is presented. The problem is reduced to a double sequence of boundary-value problems of linear ordinary differential equations. 19 refs, cited. 

Each complex function may be written as the sum of a real function (index 1) and an imaginary function (index 2). The set of the three coupled Eqs. (2-7), (2-8), (2-9) becomes then a set of six coupled linear ordinary differential equations. 

For the case of the linear spring given in equation (7.45), equation (7.52) is a linear, ordinary differential equation that can be solved exactly. If the initial condition is a qui-... 

The nonlinearity in Eqs. (3.9)—(3.11) occurs in the product of variables and in the exponential temperature term. Expanding these nonlinear terms in a Taylor series and truncating after the first term give three linear ordinary differential equations ... 

Equation E2.5-9 further indicates that, in the absence of a pressure drop, the net flow rate equals the drag flow rate. Note that qp is positive if Pq > PL and pressure flow is in the positive z direction and negative when Pp > Po- The net flow rate is the sum or linear superposition of the flow induced by the drag exerted by the moving plate and that caused by the pressure gradient. This is the direct result of the linear Newtonian nature of the fluid, which yields a linear ordinary differential equation. For a non- Newtonian fluid, as we will see in Chapter 3, this will not be the case, because viscosity depends on shear rate and varies from point to point in the flow field. 

If there are n species, we have n simultaneous linear ordinary differential equations, which can be solved by well-known techniques. 

This is a two-order linear ordinary differential equation. The general solution of Equation B. 13 can be written as... 

This is a system of two linear ordinary differential equations. The eigenvalues of the system are... 

Therefore, the two boundary conditions can be specified at the same boundary, and it is not necessary to specify them at different locations. In fact, the fundamental theorem of linear ordinary differential equations guarantees that a unique solution exists when both conditions are specified at the same location. 

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