Fast Laplace Transform

Application of Fast Fourier Transform FFT Fast Laplace Transform (FLT)... 

An important technical development of the PFG and STD experiments was introduced at the beginning of the 1990s the Diffusion Ordered Spectroscopy, that is DOSY.69 70 It provides a convenient way of displaying the molecular self-diffusion information in a bi-dimensional array, with the NMR spectrum in one dimension and the self-diffusion coefficient in the other. While the chemical-shift information is obtained by Fast Fourier Transformation (FFT) of the time domain data, the diffusion information is obtained by an Inverse Laplace Transformation (ILT) of the signal decay data. The goal of DOSY experiment is to separate species spectroscopically (not physically) present in a mixture of compounds for this reason, DOSY is also known as "NMR chromatography."... 

When diffusion is not fast enough to avoid some spatial inhomogeneity, the value of the kernel must be considered. In general, eqn. (224) is too awkward to invert, so that the limiting value for long times can be determined using the final value theorem of Laplace, transforms, wherein g(s) is the Laplace transform of f(f)... 

Whatever the excitation, the transformation of the response from the frequency to the time domain (Fig. 11.21) is done with the inverse Fourier transform, normally as the FFT (fast Fourier transform) algorithm, just as for spectra of electromagnetic radiation. Remembering that the Fourier transform is a special case of the Laplace transform with... 

Block Diagram Analysis One shortcoming of this feedforward design procedure is that it is based on the steady-state characteristics of the process and, as such, neglects process (Ramies (i.e., how fast the controlled variable responds to changes in the load and manipulated variables). Thus, it is often necessary to include dynamic compensation in the feedforward controller. The most direct method of designing the FF dynamic compensator is to use a block dir rram of a general process, as shown in Fig. 8-34, where G, represents the disturbance transmitter, (iis the feedforward controller, Cj relates the disturbance to the controlled variable, G is the valve, Gp is the process, G is the output transmitter, and G is the feedback controller. All blocks correspond to transfer fimetions (via Laplace transforms). 

Another approach presented by Yamaoka, Yano, and Tanaka, uses a fast inverse Laplace transform to generate the integrated equation data. Thus the model is described in terms of the Laplace transform equations and solved numerically. 

Yano, Y. Yamaoka, K. Tanaka, H. A non-linear least squares program, MULTI(FILT), based on fast inverse laplace transform for microcomputers. Chem. Pharm. Bull. 1989, 37, 1035-1038. 

The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected i.e., Dj, = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen s model is equivalent to Carta s [34]. 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

We assume that the reaction term is of the KPP type, F(p) = rp l — p). Our main goal is to find the front velocity for a general memory kernel in terms of its Laplace transform, K, and to show that fronts travel with a finite velocity in the fast reaction limit only if the initial value of the memory kernel is positive definite. This general result will be applied to some specific memory kernels. 

Except for short times, the series on the RHS of Eq. 12.3 converges quite fast, hence, only a few terms are needed. For short times, the following solution, obtained from the Laplace transform analysis, is particularly useful... 

Special linear systems arise from the Poisson equation, d uldx + d uldy = f x, y) on a rectangle, 0 Laplace equation of Section II.A is a special case where fix, y) = 0.] If finite differences with N points per variable replace the partial derivatives, the resulting linear system has equations. Such systems can be solved in 0(N log N) flops with small overhead by special methods using fast Fourier transform (FFT) versus an order of AC flops, which would be required by Gaussian elimination for that special system. Storage space also decreases from 2N to units. Similar saving of time and space from O(N ) flops, 2N space units to 0(N log N) flops and space units is due to the application of FFT to the solution of Poisson equations on a three-dimensional box. 

As was mentioned in Sects. 2.8.1 and 2.8.2, application of the Laplace transform to the transient potential and current permits determination of the operational impedance. Such a method was initially introduced by Pilla [90-93] and applied to studies using mercury electrodes. Using a fast potentiostat a small potential step was applied, and both voltage and current transients were measured. Of course, because of the nonideal potentiostat response, the potential increase was not a rectangular step but occiured more slowly. Examples of the measured potential and current transients are shown in Fig. 3.6. Such data acquisition was extended to longer times and then extrapolated as the integration had to be continued to inlinity. 

First, electrical circuits containing resistances only are presented, followed by circuits containing R, C, and L elements in transient and ac conditions. To understand the concept of impedance, the notions of Laplace and Fourier transforms are presented and must be understood thoroughly. In this chapter, impedance plots are also presented, along with several examples for various circuits. Next, methods for determining impedances, including fast Fourier transform-based techniques, are discussed. 

Considerably less work exists addressing Leveque s problem for inlet channel flow with wall reaction. Carslaw and Jaeger [58] and Petersen [68] presented solutions for plug-flow conditions, using the Laplace transform. Pancharatnam and Homsy [69] used the same technique for laminar flow. The inversion of the transformed solution is given in terms of an infinite summation with coefficients given by recurrence relations (first 24 out of 50 coefficients are tabulated). Ghez [70] considered a first-order reversible reaction with the same solution method. Moreover, asymptotic expansions in the limits of fast and slow reactions were presented. 

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