Numerical Fourier/Laplace transform

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. 

A numerical calculation code of the Fourier/Laplace transform is prepared in commercial software such as MATLAB, MAPLE, or even Excel. Therefore, it is easy to carry out an inverse transform provided that all frequency responses are given by the user. Similarly, if the user can prepare the time response of a transient voltage, for example, as digital data of a measured result, then the user can easily obtain its frequency response using the software. However, it is better to first understand the basic theory of the Fourier/Laplace transform. 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

DISTRIBUTION THEORY AND TRANSFORM ANALYSIS An Introduction to Generalized Functions, with Applications, A.H. Zemanian. Provides basics of distribution theory, describes generalized Fourier and Laplace transformations. Numerous problems. 384pp. 5b 8b. 65479-6 Pa. 8.95... 

Theret et al. [1988] analyzed the micropipette experiment with endothelial cell. The cell was interpreted as a linear elastic isotropic half-space, and the pipette was considered as an axisymmetric rigid ptmch. This approach was later extended to a viscoelastic material of the cell and to the model of the cell as a deformable layer. The solutions were obtained both analytically by using the Laplace transform and numerically by using the finite element method. Spector et al. [ 1998] analyzed the application of the micropipette to a cylindrical cochlear outer hair cell. The cell composite membrane (wall) was treated as an orthotropic elastic shell, and the corresponding problem was solved in terms of Fourier series. Recently, Hochmuth [2000] reviewed the micropipette technique applied to the analysis of the cellular properties. 

Previous sections dealt with the analytical development of Laplace transform and the inversion process. The method of residues is popular in the inversion of Laplace transforms for many applications in chemical engineering. However, there are cases where the Laplace transform functions are very complicated and for these cases the inversion of Laplace transforms can be more effectively done via a numerical procedure. This section will deal with two numerical methods of inverting Laplace transforms. One was developed by Zakian (1969), and the other method uses a Fourier series approximation (Crump 1976). Interested readers may also wish to perform transforms using a symbolic algebra language such as Maple (Heck 1993). 

Another method of getting the numerical inverse is by the Fourier series approximation. The Laplace transform pairs are given as in Eqs. 9.1 and 9.3, written again for convenience... 

Crump, K. S. Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation, J. Assoc. Comput. Machinery, 23, 89-96 (1976). 

Based on the MT model a method has been developed to calculate the distribution of capture rates from the measured photocurrent [68, 69]. In contrast to Ref [34], the method is based on a Fourier transform tecbnique with better numerical stability as compared to the inverse Laplace transform used in Ref [34]. 

The time response of the frequency dependence explained in Section 1.5.2 is calculated by a numerical Fourier or Laplace inverse transform in the following form [1,20] ... 

To practitioners in reservoir engineering and well test analysis, the state-of-the-art has bifurcated into two divergent paths. The first searches for simple closed-form solutions. These are naturally restricted to simplified geometries and boundary conditions, but analytical solutions, many employing method of images techniques, nonetheless involve cumbersome infinite series. More recent solutions for transient pressure analysis, given in terms of Laplace and Fourier transforms, tend to be more computational than analytical they require complicated numerical inversion, and hence, shed little insight on the physics. 

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