Mathematical methods Laplace transforms

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. 

First of all, the mathematical background will be developed for the case of a simple electrode reaction O + n e = R. In this treatment, contrasts like potential versus current perturbation, large amplitude versus small amplitude, and single step versus periodical perturbation are emphasized. While discussing these principles, the most common methods derived from them will be briefly mentioned. On the other hand, it will be shown that, by virtue of the method of Laplace transformation, these methods have much in common and contain, in principle, the same information if the detected cell response is of the same order. 

When DJuL is found to be large and the tracer response curve is skewed, as in Fig. 2.23b, but without a significant delay, a continuous stirred-tanks in series model (Section 2.3.2), may be found to be more appropriate. The tracer response curve will then resemble one of those in Fig. 2.8 or Fig. 2.9. The variance a2 of such a curve with a mean of tc is related to the number of tanks / by the expression a2 = t2/i (which can be shown for example by the Laplace transform method 7 from the equations set out in Section 2.3.2). Calculations of the mean and variance of an experimental curve can be used to determine either a dispersion coefficient Dl or a number of tanks i. Thus each of the models can be described as a one parameter model , the parameter being DL in the one case and i in the other. It should be noted that the value of i calculated in this way will not necessarily be integral but this can be accommodated in the more mathematically general form of the tanks-in-series model as described by Nauman and Buffham 7 . 

Duffy, D., On the numerical inversion of Laplace transforms Comparison of three new methods on characteristic problems from applications, ACM Transactions on Mathematical Software, Vol. 19, No. 3, 1993, pp. 333-359. 

One may conclude therefore that the solution of Pick s second law (a partial differential equation) would proceed smoothly if some mathematical device could be utilized to convert it into the form of a total differential equation. The Laplace transformation method is often used as such a device. 

Mayersohn, M. Gibaldi, M. Mathematical methods in pharmacokinetics. I. Use of the laplace transform for solving differential rate equations. Am. J. Pharm. Ed. 1970, 34, 608-614. 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

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. 

This system of equations may be solved by several methods including Laplace transformation and matrix algebra. The mathematical details have been given by Connors [G2]. 

In this appendix some important mathematical methods are briefly outlined. These include Laplace and Fourier transformations which are often used in the solution of ordinary and partial differential equations. Some basic operations with complex numbers and functions are also outlined. Power series, which are useful in making approximations, are summarized. Vector calculus, a subject which is important in electricity and magnetism, is dealt with in appendix B. The material given here is intended to provide only a brief introduction. The interested reader is referred to the monograph by Kreyszig [1] for further details. Extensive tables relevant to these topics are available in the handbook by Abramowitz and Stegun [2]. 

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 integrals in Eq. [12] are not tabulated in widely used mathematical ref erences (Petit Bois, 1961, Abramowitz Stegun, 1965 Gradshteyn Ryzhik, 1980, p. 128-129). They appear to present an insurmountable difficulty to solution of the problem of describing the tracer concentration in a soil column due to a sinusoidal loading boundary concentration by applying the Laplace transform method with the readily applied convolution rule. 

The use of Laplace transforms offers a very simple and elegant method of solving linear or linearized differential equations which result from the mathematical modeling of chemical processes. 

The classical mathematical method for inverting a z transform is to use the linearity theorem [Eq. (14.36)]. We expand the function F(r) into a sum of simple terms and invert each individually. This is completely analogous to Laplace transformation inversion. Let F( ) be a ratio of polynomials in z, Mth-order in the numerator and Nth-order in the denominator. We factor the denominator into its N roots Pi, Pi, P3, , Pn-... 

Such a general equation can only be solved if the boundary conditions are known. By the selection of the boundary conditions one can adapt Eq. (5.13) for selected conditions. The principal mathematical method to solve the equation is Laplace transformation. In the following some simple results will be described. 

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. 

The mathematical treatment of stochastic models of bicomponential reactions is rather difficult. The reactions X Yand X Y Z were investigated by Renyi (1953) using Laplace transformation. The method of the generating function does not operate very well in the general case, since it leads to higher-order partial differential equations. In principle chemical... 

The equation can be solved by the method of Boltzmann transformation, as in Understanding Voltammetry, or alternatively by a mathematical method known as Laplace transformation, in which an integral transform is used to convert a partial differential equation into an ordinary differential equation. The transformation is ... 

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