First-order ordinary differential application

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,... 

For the application of exponentialy fitted methods an accurate estimation of the frequency is required. Recently Vanden Berghe et al. ° have introduced a new method for the determination of the frequency of the problem. Vanden Berghe et alf have also constructed multistep exponentialy fitted methods for first order Ordinary Differential Equations. 

Here we will review a few methods for solving the first-order ordinary differential equations. Following each method are examples demonstrating the application of that method. Also, the notion of translating prose into mathematical symbolism is introduced as Problem Setup in Section 2.4. 

The traditional approach of outlining the theory and presenting some supporting examples has been followed up to now. However, a needed deviation from traditimi is a how to or a problem setup section. This section is included to demraistrate one approach to formulating a physically applicable first-order ordinary differential equation. 

Chapter 2 deals with select first order ordinary differential equations and provides chemical engineering examples that demonstrate the use of solution techniques. A section addressing the formulation of some physically applicable first-order ordinary differential equations (problem setup) is included. 

Many drugs undergo complex in vitro drug degradations and biotransformations in the body (i.e., pharmacokinetics). The approaches to solve the rate equations described so far (i.e., analytical method) cannot handle complex rate processes without some difficulty. The Laplace transform method is a simple method for solving ordinary linear differential equations. Although the Laplace transform method has been used for more complex applications in physics, engineering, and other research areas, here it will be applied to ordinary differential equations of first-order rate processes. 

There are several reasons for going first to this level of generality for the n-compartment model. First/ it points out clearly that the theories of noncompartmental and compartmental models are very different. While the theory underlying noncompartmental models relies more on statistical theory/ especially in developing residence time concepts [see/ e.g./ Weiss (11)]/ the theory underlying compartmental models is really the theory of ordinary/ first-order differential equations in which/ because of the nature of the compartmental model applied to biological applications/ there are special features in the theory. These are reviewed in detail in Jacquez and Simon (5)/ who also refer to the many texts and research articles on the subject. 

The above are three examples of regular perturbation. The method is applicable to both partial and ordinary differential equations. The next example is one in which the meaning of the term regular is emphasized. Consider now the first-order problem... 

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