Laplace’s transformation

This being stated, applying Laplace s transform one obtains from Fick s second law that the maximum current (i.e. the current at the potential corresponding to the maximum of the peak) for a planar electrode is expressed by ... 

Appendix B describes in detail the solution of this problem as an application of the Laplace s Transform method and Eqs. (2.142)-(2.144) have also been deduced. 

For ideal flow models such as perfect mixing flow, plug flow and all other ideal models, a combination of functions E(t) and F(t) can be obtained directly or indirectly using the model transfer function T(p). Before obtaining an expression for E(t) for the perfect mixing flow, we notice that the transfer function of a flow model is in fact the Laplace s transformation of the associated E(t) function ... 

The Laplace s transformation of the differential equation (3.70) gives relation (3.71) where p is the Laplace s argument ... 

Now, we can write the plug flow model transfer function. With the Laplace s transformation of relation (3.73) and with Eqs. (3.74)-(3.76) we have ... 

In this case we can derive a relation on the basis of Laplace s transformation for a similar unavailability function... 

It is very difficult to obtain analytical solutions, except for a few sets of simple initial and boundary conditions. The use of Laplace s transform is, in principle, appealing. However, it can only be recommended if one can entrust some pal with the inverse Laplace transform of the solution, in order to revert to the real space, or if the case is simple enough for the inverse Laplace transform to be listed in available textbooks. 

Applying Laplace s transform to the system of equations (10) with the initial conditions (11), we obtain... 

After application of inverse Laplace s transform to above equation, we have... 

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

This equation can be solved by separation of variables, provided the potential is either a constant or a pure radial function, which requires that the Lapla-cian operator be specified in spherical polar coordinates. This transformation and solution of Laplace s equation, V2 / = 0, are well-known mathematical procedures, closely followed in solution of the wave equation. The details will not be repeated here, but serious students of quantum theory should familiarize themselves with the procedures [15]. 

Few articles on receptivity present a qualitative view of particular transition routes created by not so well-defined excitation field (see e.g. Saric et al. (1999)). Such approaches do not demonstrate complete theoretical and /or experimental evidence connecting the cause (excitation field) and its effect(s) (response field). Here, a model based on linearized Navier- Stokes equation is presented to show the receptivity route for excitation applied at the wall. This requires a dynamical system approach to explain the response of the system with the help of Laplace-Fourier transform. 

The Laplace integral transformation, used in Section 4.3.1, allows the indentifica-tion of its kernel as K(z,p) = K(z,s) = e . It corresponds to the case when we produce a transformation with time. So, for this case, we particularize the relation (4.151) as ... 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

Newman2°° 2° modeled the transient response of a disk electrode to step changes in current. The solution to Laplace s equation was performed using a transformation to rotational elliptic coordinates and a series expansion in terms of Lengendre polynomials. Antohi and Scherson expanded the solution to the transient problem by expanding the number of terms used in the series expansion. ... 

Apply the Laplace transformation to the equations using the command laplace( equations ,t,s) which converts the equations from time domain to the Laplace (s) domain. 

Hence, appears as Laplace s specific transform of correlation function 11 for a free polymer segment. Go should be replaced by G, in Rule 1, and 2sq can be regarded as the minimal area between two interaction points along the continuous chain. 

Here, the partition function for the end-to-end distanc,< is represented through Laplace s inverse transform. 

The theory behind conformal transformations is elegant and simple and it turns out that the basic notions can be introduced using no more than undergraduate math. To keep the ideas elementary, we deal with a simple fluid model first. Consider the two-dimensional, planar, steady flow of a constant density liquid in an isotropic, homogeneous medium, satisfying Laplace s equation... 

In the language of mathematics, Laplace s equation remains invariant under conformal transformation, which is equivalent to saying that harmonic functions stay harmonic. There are also geometric interpretations associated with Equations 5-12 and 5-13 discussed elsewhere, for example, Churchill... 

Pressure-pressure formulations. In the steady, areal flow limit, both time and vertieal (z) derivatives vanish, leaving a transformed pressure whose (m+l) power satisfies Laplace s equation. 

From Equation 9-25, the boundary value problem governing E(x,y) is still formulated in (x,y) Cartesian coordinates. However, the same transformation used for the pressure equation will lead to Laplace s equation... 

There exist powerful simulation tools such as the EMTP and SPICE. Those tools, however, involve a number of assumpfions and the limit of the applications, which are not easy for a user to understand. Thus, it happens quite often that a simulation result is not correct due to the user s misunderstanding of fhe applicafion limifs related to the assumptions of the tools. To avoid this kind of an incorrecf simulation, the best way is to develop a user s own simulation tool. For this purpose, an FD method of transient simulations is the easiest one, because the method is entirely based on the theory explained in Chapter 1 and Section 1.5, and requires only numerical transformation of a frequency response into a time response by adopting Fourier/Laplace inverse transform. The theory of a distributed-parameter circuit, a transient analysis in a lumped-parameter circuit, and the Fourier/Laplace transform is given as an undergraduate course in electrical engineering department in most universities all over the world. This section explains how to develop a computer code of the FD transient simulations. 

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