Laplace transform analysis, inverse

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

The determination of the microcanonical rate coefficient k E) is the subject of active research. A number of techniques have been proposed, and include RRKM theory (discussed in more detail in Section 2.4.4) and the derivatives of this such as Flexible Transition State theory. Phase Space Theory and the Statistical Adiabatic Channel Model. All of these techniques require a detailed knowledge of the potential energy surface (PES) on which the reaction takes place, which for most reactions is not known. As a consequence much effort has been devoted to more approximate techniques which depend only on specific PES features such as reaction threshold energies. These techniques often have a number of parameters whose values are determined by calibration with experimental data. Thus the analysis of the experimental data then becomes an exercise in the optimization of these parameters so as to reproduce the experimental data as closely as possible. One such technique is based on Inverse Laplace Transforms (ILT). 

Benet, L.Z. Turi, J.S. Use of general partial fraction theorem for obtaining inverse laplace transforms in pharmacokinetic analysis. J. Pharm. Sci. 1971, 60, 1593-1594. 

In the case of 2D diffusion-weighted NMR spectra a transformation exists, which is able to transform the second dimension directly into the desired diffusion coefficient (D). In this form, the spectra are known as 2D DOSY and the diffusion coefficients can be extracted directly without using further mathematical analysis. However, this transformation - the inverse Laplace transform - is ill-posed, i.e. the solution can be highly inaccurate without adequate regularisation. A wiser choice may be to analyse the original diffusion-weighted data using three-way methods, as exemplified in Section 3.3. 

The function (t) can be analysed by the method of cumulants [57] or by inverse Laplace transformation. These methods provide the mean relaxation rate T of the distribution function G(T) (z-average). For the second analysis procedure mentioned above, the FORTRAN program CONTIN is available [97,98]. It is sometimes difficult to avoid the presence of spurious amounts of dust particles or high molecular weight impurities that give small contributions to the long time tail of the experimental correlation functions. With CONTIN it is possible to discriminate these artifacts from the relevant relaxation mode contributing to (t). 

Suppose we suddenly increase the load current of a converter from 4 A to 5 A. This is a step load and is essentially a nonrepetitive stimulus. But by writing all the transfer functions in terms of s rather than just as a function of jco, we have created the framework for analyzing the response to such disturbances too. We will need to map the stimulus into the s-plane with the help of the Laplace transform, multiply it by the appropriate transfer function, and that will give us the response in the s-plane. We then apply the inverse Laplace transform and get the response with respect to time. This was the procedure symbolically indicated in Figure 7-3, and that is what we need to follow here too. However, we will not perform the detailed analysis for arbitrary load transients here, but simply provide the key equations required to do so. 

The inverse Laplace transforms of (s) in Eq. (A.35) are calculated using standard methods of complex analysis to yield ifi(T) ... 

Data analysis. In the analysis of the measured autocorrelation curve, an inverse Laplace transformation (ILT analysis) was performed employing the algorithm REPES [28] to obtain the distribution of relaxation times. This program is similar to... 

Inverse-Laplace Transform The distribution makes gi(T) deviate from a single exponential decay, as illustrated in Figure 3.17a. Conversely the analysis of the deviation allows estimation of G(F). Mathematically, gi(T) is the Laplace transform of G(F), as Eq. 3.59 shows. Then, the procedure to estimate G(F) from... 

S.3 Cumulant Expansion The inverse-Laplace transform is a convenient analysis method when the distribution is broad, especially bimodal or trimodal. When the distribution is narrow and gi(T) is close to a single exponential decay, a simpler analysis method, called a cumulant expansion, is more useful. In this... 

Several of the above-described publications extracted rotational spectra from inverse Laplace transforms of imaginary-time autocorrelation functions, quantities readily calculated with RQMC. The utility of defining a larger set of correlation functions, so-called symmetry-adapted imaginary-time autocorrelation functions was explored in a recent paper [50]. Computational efficiency in the calculation of weak spectral features was demonstrated by a study of He-CO binary complex. Some preliminary results of an analysis of a recently observed satellite band in the IR spectrum of CO2 doped He clusters were presented. 

Since is given by equations (32), (34) and (36), it can be determined for all z > 0 by performing the series analysis on J and K first to obtain these as functions valid for all values of their (complex) argument and then to perform the inverse Laplace transforms in equation (32) numerically. An alternative procedure is to perform a series analysis on equation (37) directly. There are two key features which are common to both methods. The first is that the methods build in the expected scaling behavior of the unknown function J and K or in the limit of large argument /y/E or z respectively. The second is that both methods use, as an independent variable, a variable... 

Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or linearized differential equation, dy/dt = f(y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y(s). The time domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. 

We now derive the time-domain solutions of first and second order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, reduced order model, and the effect of zeros will be discussed. 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

Equation (22) is particularly useful when a concentration gradient in depth exists. In this case, several spectra at different values of 9 are taken and the analysis is called angle resolved X-ray photoelectron spectroscopy. However, for a maximum efficiency, a flat surface (at an atomic level) is needed to avoid shade effects as shown by Fadley in his early works in the 1970s [44]. An additional problem exists the extraction of concentration profiles, cg(x), from Eq. (22) is an inverse problem the intensity as a function of the analysis angle is the Laplace transform of the composition depth profile of the sample [43] and does not have a unique solution. Several algorithms to solve the inversion problem were developed and tested [46]. They are all very unstable and sensitive to small statistical... 

At first sight the number of transforms can seem unnecessarily large, (recall that each has an inverse transform also.) Why do we need so many The differences are simply attributable to us wanting to use both continuous and discrete representations for time and frequency, and to make analysis easier (in the case of the z-transform and Laplace transform). The differences between them can be clearly shown in the following table ... 

There exist powerful simulation tools such as the EMTP [35]. These tools, however, involve a number of complex assumptions and application limits that are not easily understood by the user, and often lead to incorrect results. Quite often, a simulation result is not correct due to the user s misunderstanding of the application limits related to the assumptions of the tools. The best way to avoid this type of incorrect simulation is to develop a custom simulation tool. For this purpose, the FD method of transient simulations is recommended, because the method is entirely based on the theory explained in Section 2.5, and requires only numerical transformation of a frequency response into a time response using the inverse Fourier/Laplace transform [2,6,36, 37, 38, 39, 40, 41-42]. The theory of a distributed parameter circuit, transient analysis in a lumped parameter circuit, and the Fourier/Laplace transform are included in undergraduate course curricula in the electrical engineering department of most universities throughout the world. This section explains how to develop a computer code of the FD transient simulations. 

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