Inverse Laplace transform technique

In this chapter, analytical solutions were obtained for parabolic and elliptic partial differential equations in semi-infinite domains. In section 4.2, the given linear parabolic partial differential equations were converted to an ordinary differential equation boundary value problem in the Laplace domain. The dependent variable was then solved in the Laplace domain using Maple s dsolve command. The solution obtained in the Laplace domain was then converted to the time domain using Maple s inverse Laplace transform technique. Maple is not capable of inverting complicated functions. Two such examples were illustrated in section 4.3. As shown in section 4.3, even when Maple fails, one can arrive at the transient solution by simplifying the integrals using standard Laplace transform formulae. 

In principle, (p(E) can be obtained by inverse Laplace transform techniques [43,44], The precision required for experimental data over a very extended time domain is, however, so high that this method is of little practical use. It is thus more convenient to parametrize (p(E) in a suitable analytic form, to insert it into Eq. (62), and to calculate numerically the... 

Experimentally, the absorbance A(5) of a band is measured as a function of the angle of incidence B and thus of S. Two techniques can be used to determine a(z). A functional form can be assumed for a(z) and Eqs. 2 and 3 used to calculate the Laplace transform A(5) as a function of 8 [4]. Variable parameters in the assumed form of a(z) are adjusted to obtain the best fit of A(5) to the experimental data. Another approach is to directly compute the inverse Laplace transform of A(5) [3,5]. Programs to compute inverse Laplace transforms are available [6]. 

All the system response curves in frequency and time domains were calculated numerically from equations that are much too involved to reproduce in detail here. Transfer functions in Laplace transform notation are easily defined for the potentiostat and cell of Figure 7.1. Appropriate combinations of these functions then yield system transfer functions that may be cast into time- or frequency-dependent equations by inverse Laplace transformation or by using complex number manipulation techniques. These methods have become rather common in electrochemical literature and are not described here. The interested reader will find several citations in the bibliography to be helpful in clarifying details. 

While polydisperse model systems can nicely be resolved, the reconstruction of a broad and skewed molar mass distribution is only possible within certain limits. At this point, experimental techniques in which only a nonexponential time signal or some other integral quantity is measured and the underlying distribution is obtained from e.g. an inverse Laplace transform are inferior to fractionating techniques, like size exclusion chromatography or the field-flow fractionation techniques. The latter suffer, however, from other problems, like calibration or column-solute interaction. 

Evaluate in the time domain the time-amount course by applying traditional inverse Laplace transforms [359] or numerical inversion techniques [353,360],... 

Fig. 2,17. Pressure and temperature dependence of reaction (24). The solid lines are a calculated fit using an inverse Laplace transform/Master Equation technique described in...

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). 

This does not mean that the inverse Laplace transform does not exist instead, one has to use advanced techniques for finding the desired inverse Laplace transform (see chapter 8 for details). 

In example 8.9 the Laplace transform technique was used to solve a time dependent problem. Inversing the Laplace transform is not straightforward. For complicated time dependent boundary conditions the convolution theorem can be used to find the inverse Laplace transform efficiently. If H(s) is the solution obtained in the Laplace domain, H(s) is represented as a product of two functions ... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

The last term in Eq. (A-3) is the Laplace transform of E(a) Aweq(a), where a plays the role of t in Eq. (A-4) and t plays the role of s in Eq. (A-4). Thus the problem is reduced to this experimental observations give us the Laplace transform of the desired distribution function [modified by multiplication by Aweq(A)], We want the distribution function. The required calculation is a numerical inverse Laplace transform. This is clearly feasible, and numerical techniques are discussed in the literature 56). It is not a simple matter to carry out, however, and the accuracy requirements on the data are likely to be stringent. No direct method of computing the distribution from frequency response is known, although the step response can be computed from the frequency response by standard techniques. In view of the foregoing discussion, it appears that in principle, at least, the distribution can be computed from the frequency response. 

Site heterogeneity was investigated using a deconvolution technique based on inverse Laplace transform (ILT) method. As could be seen from the active site distribution shown in Fig. 11, the unpromoted catalyst had essentially only one kind of site (3). K-promotion increased the average... 

The solution can be found conveniently by using Laplace transform technique with respect to time, but the solution takes somewhat complicated form, since the inversion integral includes the branch points corresponding to the critical frequency. The detail of the solution has been described by Toma Morioka [4]. Here we say only that the present result, generalized for arbitrary propagation direction, can be obtained by replacing M in [4] to m = ucos6/c. 

A method was proposed for the parameterization of impedance based models in the time domain, by deriving the corresponding time domain model equation with inverse Laplace transform of the frequency domain model equation assuming a current step excitation. This excitation signal has been chosen, since it can be easily applied to a Li-ion cell in an experiment, allows the analytical calculation of the time domain model equation and is included in the definition of the inner resistance. The voltage step responses of model elements were presented for lumped elements and derived for distributed model elements that have underlying fractional differential equations using fractional calculus. The determination of the inner resistance from an impedance spectrum was proposed as a possible application for this method. Tests on measurement data showed that this method works well for temperatures around room temperature and current excitation amplitudes up to 10 C. This technique can be used for comparisons of measured impedance spectra with conventionally determined inner resistances. 

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. 

After Kc is specified, y t) can be determined from the inverse Laplace transform of Eq. 11-76. But first the roots of the cubic polynomial in s must be determined before performing the partial fraction expansion. These roots can be j calculated using standard root-finding techniques (( and Canale, 2010). Figure 11.23 demonstrates that] increases, the response becomes more oscillatory unstable for Kc = 15, More details on the actual stabilit limit of this control system are given in Example 11.10. 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

In classical control theory, we make extensive use of Laplace transform to analyze the dynamics of a system. The key point (and at this moment the trick) is that we will try to predict the time response without doing the inverse transformation. Later, we will see that the answer lies in the roots of the characteristic equation. This is the basis of classical control analyses. Hence, in going through Laplace transform again, it is not so much that we need a remedial course. Your old differential equation textbook would do fine. The key task here is to pitch this mathematical technique in light that may help us to apply it to control problems. 

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. 

Eq. 4 is amenable to solution techniques based on the numerical inversion of Laplace-transformed equations these calculations can be performed rapidly and are therefore suitable for calibration. In Figure 1, typical soil/bentonite column predictions are shown to highlight the effect of the influent mixing zone on the spatial contaminant distributions for low-flow systems. The simulation results, which were generated for column conditions described by Khandelwal et al. (1998), indicate that the mixing zone has a significant influence on the shape of the spatial contaminant distribution and, therefore should be considered explicitly in estimating sorption parameters from spatial column data. 

When we have found a solution for the Laplace transformed function, then we need to make an inverse transformation to find the solution in terms of time and coordinates. There are elegant techniques for doing this based on the theory of complex functions, but often these are not necessary since there exist extensive tables in mathematical handbooks of functions and their Laplace transformed functions. Only in cases where the relevant functions have not been tabulated will it be necessary to carry out the inverse transformation using these techniques. 

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