Ordinary differential equation order

The differential equations are solved using the ordinary differential equation order ode as presented earlier. 

Gear C W 1966 The numerical integration of ordinary differential equations of various orders ANL 7126... 

The remaining terms in equation set (4.125) are identical to their counterparts derived for the steady-state case (given as Equations (4.55) to (4.60)). By application of the 9 time-stepping method, described in Chapter 2, Section 2.5, to the set of first-order ordinary differential equations (4.125) the working equations of the solution scheme are obtained. The general form of tliese equations will be identical to Equation (2.111) in Chapter 2,... 

In Chapter 3, the analytieal method of solving kinetie sehemes in a bateh system was eonsidered. Generally, industrial realistie sehemes are eomplex and obtaining analytieal solutions ean be very diffieult. Beeause this is often the ease for sueh systems as isothermal, eonstant volume bateh reaetors and semibateh systems, the designer must review an alternative to the analytieal teehnique, namely a numerieal method, to obtain a solution. Eor systems sueh as the bateh, semibateh, and plug flow reaetors, sets of simultaneous, first order ordinary differential equations are often neeessary to obtain die required solutions. Transient situations often arise in die ease of eontinuous flow stirred tank reaetors, and die use of numerieal teehniques is die most eonvenient and appropriate mediod. 

The ordinary differential equations for f and C now form a fifth-order system which can be solved using a standard NAG library routine. The results are shown in Fig. 10.73. This figure also shows the numerical results for concentration obtained using a full numerical approach, and there is reasonable agreement between the two. 

The steady state TMB model equations are obtained from the transient TMB model equations by setting the time derivatives equal to zero in Equations (25) and (26). The steady state TMB model was solved numerically by using the COLNEW software [29]. This package solves a general class of mixed-order systems of boundary value ordinary differential equations and is a modification of the COLSYS package developed by Ascher et al. [30, 31]. 

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... 

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. 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

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). 

This definition is a formal analog of an mth order ordinary differential equation... 

Bagmut, G. (1969) Difference schemes of higher-order accuracy for an ordinary differential equations with singularity. Zh. Vychisl. Mat. i Mat. Fiz., 9, 221-226 (in Russian) English transl. in USSR Comput. Mathem. and Mathem. Physics. 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

The scope of this book deals primarily with the parameter estimation problem. Our focus will be on the estimation of adjustable parameters in nonlinear models described by algebraic or ordinary differential equations. The models describe processes and thus explain the behavior of the observed data. It is assumed that the structure of the model is known. The best parameters are estimated in order to be used in the model for predictive purposes at other conditions where the model is called to describe process behavior. 

The order of an ordinary differential equation is the order of its highest derivative. Thus, an ordinary differential equation of order n is an equation of the form... 

A phase space is established for a typical particle, whose coordinates specify the location of the particle as well as its quality. Then, ordinary differential equations describe how these phase coordinates evolve in time. In other words, the state of a particle in a processing system is specified by the values of a number of phase coordinates z. The only requirement on z is that they describe the state of the particle fully enough to permit one to write a set of first order ode s of the form ... 

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