Ordinary Differential Equations as Initial Value Problems

Note that all the conditions are known at one time, t = 0. Thus it is possible to calculate the function on the right-hand side at f = 0 to obtain the derivative there. This makes the set of equations initial value problems. The equations are ordinary differential equations because there is only one independent variable. Any higher-order ordinary differential equation can be turned into a set of first-order ordinary differential equations they are initial value problems if all the conditions are known at the same value of the independent variable [Finlayson, 1980, 1997 (p. 3-54), 1990 (Vol. BI, p. 1-55)]. The methods for initial value problems are explained here for a single equation extension to multiple equations is straightforward. These methods are used when solving plug-flow reactors (Chapter 8) as well as time-dependent transport problems (Chapters 9-11). 

In numerical solutions of ODE-IVP, the solution y at the point t is represented by y . The simplest method is Euler s method, which is obtained by writing a difference expression for the derivative  

The first-order approximation of the derivative is not very accurate, so this method is not very accurate either. It is easy to program, though. Rearrange Eq. (F.16) as 

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NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS... 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

When it is necessary to include these effects - slow reaction rates, catalysts, heat transfer, and mass transfer - it can make an engineering problem extremely difficult to solve. Numerical methods are a must, but even numerical methods may stumble at times. This chapter considers only relatively simple chemical reactors, but to work with these you must leam to solve ordinary differential equations as initial value problems. 

Ordinary differential equations as initial value problems ... 

The boundary conditions also need to be included, leading to a set of differential-algebraic equations as an initial value problem. Standard methods for ordinary differential equations as initial value problems can be applied if the boundary condihons are incorporated into the differential equations, which is possible when the boundary conditions are linear. 

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