Laplace transform properties

Table F.l Some Laplace transform properties and pairs of functions used as probability density functions for semi-Markov modeling.

In this section, we consider several Laplace transform properties that are useful in process dynamics and control. 

Application of the definition shows that the Laplace transform is a linear oper-ator " this property is represented in Eqs. (3-67) and (3-68). 

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 ... 

The properties of Laplace transform and the transforms of some common functions. 

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) ... 

This approach works because of the linear property of Laplace transform. 

The convolution and general properties of the Fourier transform, as presented in Section 11.1, are equally applicable to the Laplace transform. Thus,... 

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see Integral Transforms (Operational Methods) ]. The one-sided Laplace transform indicated by L[f(t)] is defined by the equation L[f(t)] = Jo f t)e st dt. It has numerous important properties. The ones of interest here are L[f(t)] = sL[f(t)] -/(0) L[f"(t) = /(0) f( 0) ... 

A special situation arises in the limit of small scavenger concentration. Mozumder (1971) collected evidence from diverse experiments, ranging from thermal to photochemical to radiation-chemical, to show that in all these cases the scavenging probability varied as cs1/2 in the limit of small scavenger concentration. Thus, importantly, the square root law has nothing to do with the specificity of the reaction, but is a general property of diffusion-dominated reaction. For the case of an isolated e-ion pair, comparing the t—°° limit of Eq. (7.28) followed by Laplace transformation with the cs 0 limit of the WAS Eq. (7.26), Mozumder derived... 

For a model with a known transfer function the several moments can be obtained directly without need for inversion of the transform. This a consequence of a property of the derivative of the Laplace transform, namely certain limits as s= 0 ... 

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

There are good reasons for choosing to carry out certain operations in the Laplace domain rather than performing equivalent operations in the time domain. In particular, integration of a function with respect to time is equivalent in the Laplace domain to division by the Laplace variable s. Conversely, differentiation corresponds to multiplication by s. This latter property enables differential equations to be Laplace transformed and then solved by algebraic means. These Laplace domain operations are all more simple than their time domain counterparts. In addition convolution in the time domain is equivcdent to multiplication in the Laplace domain. Formally, this may be represented by eqn. (24), the left-hand side of which is termed the convolution integral. 

For present purposes, the functions of time, f(f), which will be encountered will be piecewise continuous, of less than exponential order and defined for all positive values of time this ensures that the transforms defined by eqn. (A.l) do actually exist. Table 9 presents functional and graphical forms of f(t) together with corresponding Laplace transforms. The simpler of these forms can be readily verified using eqn. (A.l), but as extensive tables of functions and their transforms are available, derivation is seldom necessary, (see, for instance, ref. 75). A simple introduction to the Laplace transform, to some of its properties and to its use in solving linear differential equations, is presented in Chaps. 2—4 of ref. 76, whilst a more complete coverage is available in ref. 77. 

Let us briefly examine the importance of the Mittag-Leffier function in relaxation modelling. The mathematical properties of the Mittag-Leffier function are compiled in Appendix B. Besides via the series representation, the Mittag-Leffier function is defined through its Laplace transform... 

The use of Cole-Cole plots is not very developed in practice, despite the fact that they open the way for the modeling of the viscoelastic behavior in dynamic as well as in static loading cases (through Laplace transform). By contrast, these plots could be interesting from the fundamental point of view if certain parameters would reveal a clear dependence with the crosslink density. The effects of crosslinking are difficult to detect on the usual viscoelastic properties, except for the variation of the rubbery modulus E0. 

The formal properties of calculus integrals and the integration by parts formula lead, among others, to the following rules of the Laplace transform ... 

Two important properties of Laplace transforms are their scaling and linearity as ... 

The Laplace transform has some properties which are extremely useful in aiding in the resolution of equations in electrochemistry and other branches of science. 

Of course, k has dimensions such that the product kx is dimensionless. The Laplace transform has the following useful properties ... 

These properties allow one to solve the integration problem in transform space and then back-transform the result into real space. There are extensive tables of Laplace transforms, some of which are given in Table 2.9. 

The traditional way is to measure the impedance curve, Z(co), point-after-point, i.e., by measuring the response to each individual sinusoidal perturbation with a frequency, to. Recently, nonconventional approaches to measure the impedance function, Z(a>), have been developed based on the simultaneous imposition of a set of various sinusoidal harmonics, or noise, or a small-amplitude potential step etc, with subsequent Fourier- and Laplace transform data analysis. The self-consistency of the measured spectra is tested with the use of the Kramers-Kronig transformations [iii, iv] whose violation testifies in favor of a non-steady state character of the studied system (e.g., in corrosion). An alternative development is in the area of impedance spectroscopy for nonstationary systems in which the properties of the system change with time. 

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