Mathematical methods Laplace transform technique

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 flow under consideration is mathematically more simple in a plane duct than in an axisymmetric round tube. However, no readily solution for smooth walls, i.e. in the case where the EPR is absent, is given in the literature. The formulations and solutions of the problem of a pulsating viscous flow to be taken for a comparison have been presented only for round tubes. The known solutions employ mainly the Laplace transformation and small-parameter methods. The present investigation follows the standard technique of mathematical physics of finding the periodic solutions described particularly by Schlichting [566] with references to some original sources. Their solution for smooth walls will serve for a comparison. 

The goal of the electrochemical modelhng in this chapter is to solve the mathematical model developed in the previous chapter in order to obtain the form of the algebraic (containing no derivatives) function C X,T), i.e., to determine how the concentration of the chemical species varies in space and in time. From this, other information, such as the current passed at the electrode, can be inferred. A munber of analytical techniques exist that may be used for solving partial differential equations (PDEs) of the type encountered in electrochemical problems, including integral transform methods such as the Laplace transform, and the method of separation of variables. Unfortunately these techniques are not applicable in all cases and so it is often necessary to resort to the use of numerical methods to find a solution. 

However, the authors do not claim that these three main strategic lines in company of CETO functions constitute the unique way nor the best path to solve the molecular integral problem directed to find plausible substitutes of GTO functions. Other integration methods to deal with the present discussion can be used and analyzed, for instance Fourier, Laplace or Gauss transform methodology or any other possible choices and techniques available in the modem mathematical panoply. 

In Chapter 14 we define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary), develop transfer functions in the z domain, and discuss stability. Design of digital controllers is studied in Chapter 15 using root locus and frequency response methods in the z plane. We use practically all the stability analysis and controller design techniques that we introduced in the Laplace and frequency domains, now applying them in the z domain for sampled-data systems. 

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