Properties of the Laplace Transformation

In this section, we will outline only those properties of the Laplace transform that are directly relevant to the solution of systems of linear differential equations with constant coefficients. A more extensive coverage can be found, for example, in the text book by Franklin [6]. 

When we apply these two properties of the Laplace transform to differential equations of our pharmacokinetic model in eq. (39.46), we obtain ... 

An important property of the Laplace transform is that it is a linear operator, and contribution of individual terms can simply be added together (superimposed) ... 

The main properties of the Laplace transform that can be used in solving diffusion equations are as follows ... 

It is also worthwhile to notice that the overall order of reaction is 2 when a = 1, that is, with an exponential distribution. The only value of a for which (f>(a,0) is finite is a = 1. This result can be generalized to any initial distribution by simply making use of the limit properties of the Laplace transform C(t) at large times becomes proportional to t if and only if c(x,0) at small x becomes proportional to x h > 0. Now if 0 < f2 < 1, c(0,0) = and if 13 > 1 c(0,0) = 0. It follows that c(0,0) is finite only if f3 = 1, in which case C(t) at large times becomes proportional to 1/t, which is the behavior typical of sound-order kinetics (Kram-beck, 1984b). The question of the large-time asymptotic behavior of the overall kinetics is discussed later in more general terms. 

Thus it is obvious by the properties of the Laplace transform that the second term corresponds to the asymptotic behavior given by Eq. (47). If we now... 

These simple properties of the Laplace transform make it a very convenient tool for solving systems of first-order linear differential equations, such as the equations for growth and decay of nuclides in radioactive disintegrations and neutron inadiation. They permit these differential equations to be treated as if they were systems of simple transformed linear equations without derivatives. 

One of the most important properties of the Laplace transformation is that it is linear. 

The Laplace transform, which is a linear operator, is frequently used as a mathematical tool when dealing with linear systems. The Laplace transform is often useful in dealing with more complex convolution relationships. For example, consider the following property of the Laplace transform operation L ... 

In Table 11.8 we show some properties of the Laplace transform. 

We usually use tables and some properties to get the Laplaee transforms. A table of the most important Laplace transforms is given in Appendix D. In order to cover a wider range of functions using a limited table like the one in Appendix D, we must utilize some important properties of the Laplace transformation. The main properties of Laplace transform are as follows ... 

In that case the property of the Laplace transformation of a differential term becomes ... 

This chapter will analyze the mixing process in more detail. The process was already introduced in chapter 4, but some special properties of the Laplace transform and some special cases of the mixing process will be reviewed. In subsequent chapters other types of processes will be analyzed for their dynamic behavior. The purpose of the Laplace transform is to analyze how the process output of interest changes if the process input is changed. This will result in knowledge about the behavioral properties of the system, such as order, stability, integrating or non-minimum phase response behavior. 

One of the important properties of the Laplace transform and the inverse Laplace transform is that they are linear operators a linear operator satisfies the superposition principle ... 

In this chapter we have considered the application of Laplace transform techniques to solve linear differential equations. Although this material may be a review for some readers, an attempt has been made to concentrate on the important properties of the Laplace transform and its inverse, and to point out the techniques that make manipulation of transforms easier and less prone to error. 

Here the output voltage and input displacement Xi are written in uppercase to emphasize that this transfer function operates in the frequency domain. It is a property of the Laplace transform that multiplication by the (complex) frequency s in the frequency domain is equivalent to differentiation in the time domain, so sX, is the input velocity. Another property of the Laplace transform is that to evaluate the (complex) transfer function at a given frequency f in Hz or angular frequency co = 2nf in rad/s, one makes the substitution s = jm. 

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