Exponential function, Laplace transform

Exchange frequency, 167 Exchange integral, 194 Exergonic reaction, 223 Exponential function, Laplace transform of, 83... 

Sufficient Conditions for the Existence of Laplace Transform Suppose/ is a function which is (1) piecewise continuous on eveiy finite intei val 0 < t exponential growth at infinity, and (3) Jo l/t)l dt exist (finite) for every finite 6 > 0. Then the Laplace transform of/exists for all complex numbers. s with sufficiently large real part. 

Inverse Laplace transforms have been tabulated for most analytical functions, including power, exponential, trigonometric, hyperbolic and other functions. In this context we require only the inverse Laplace transform which yields a simple exponential ... 

From Eq. (2-18) on page 2-7, the Laplace transform of a time delay is an exponential function. For example, first and second order models with dead time will appear as... 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

Note also that Eq. (5.2) is equivalent to the common Laplace transform. A comparison of double-exponential and distributional analyses is represented in Figure 5.1. The distribution function shows width about central values which the double-exponential fit cannot express because of its mathematical form. Here the appearance of central values may partially be a consequence of the model functions assumed in the solution. Nevertheless, width directly... 

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. 

Let us now apply the definition of the Laplace transformation to some important time functions steps, ramps, exponential, sines, etc. 

Since we found in Chap. 6 that the responses of linear systems are a series of exponential terms, the Laplace transformation of the exponential function is the most important of any of the functions. 

Remember that the Laplace transformation of a function multiplied by an exponential e" is simply the Laplace transform of the function with s — a substituted for s. 

Remember that the Laplace transformation of the exponential was K/(s + a). So the (s -I- a) term in the denominator of a Laplace transformation is similar to the (z — term in a z transformation. Both indicate an exponential function. In the s plane, we have a pole at s = —a. In the z plane we will find later in this chapter that we have a pole at z = So we can imme-... 

For present purposes, the functions of time, f(f), which will be encountered will be piecewise continuous, of less than exponential order and defined for all positive values of time this ensures that the transforms defined by eqn. (A.l) do actually exist. Table 9 presents functional and graphical forms of f(t) together with corresponding Laplace transforms. The simpler of these forms can be readily verified using eqn. (A.l), but as extensive tables of functions and their transforms are available, derivation is seldom necessary, (see, for instance, ref. 75). A simple introduction to the Laplace transform, to some of its properties and to its use in solving linear differential equations, is presented in Chaps. 2—4 of ref. 76, whilst a more complete coverage is available in ref. 77. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

Both forms have a distribution of relaxation times about t0 which contribute to s(Laplace transform can be obtained numerically, and in this case C(t) can be well fit to a stretched exponential function [46] ... 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

It should be pointed out that p > a in order to avoid exponential solutions that increase with time. The inverse Laplace transform of x( ) can be achieved by decomposing x(5) into simple functions... 

Because st is dimensionless, s has the units of reciprocal time and is commonly expressed as = 1 /x. The parameter x, called relaxation time, is taken to be the time at which t = x, that is, the time at which G(t) = Gq/c. This approximation, however, fails even for rather simple viscoelastic systems. In fact, the curve obtained for G(t) using Eq. (9.2) drops much more rapidly than that corresponding to real systems (see Fig. 9.1). A better description of these systems is achieved by using a sum of exponentials (1-5), for example, Gi exp(-5 t). By assuming that the relaxation times of the viscoelastic mechanisms involved in the relaxation process vary continuously between 0 and oo, the sum can be replaced by an integral in such a way that G t) may be considered the Laplace transform of an unknown function N s) (2). According to this,... 

Pfalzner and March [14] have performed numerically the Laplace transform inversion referred to above to obtain the density p( ) from the Slater sum in Eq. (10). Below, we shall rather restrict ourselves to the extreme high field limit of Eq. (10), where analytical progress is again possible. Using units in which the Bohr magneton is put equal to unity, the extreme high field limit amounts to the replacement of the sinh function in Eq. (10) by a single exponential term, to yield... 

Once the Laplace transform u(x,s) of the temperature () (x, /,) which fits the initial and boundary conditions has been found, the back-transformation or so-called inverse transformation must be carried out. The easiest method for this is to use a table of correspondences, for example Table 2.3, from which the desired temperature distribution can be simply read off. However frequently u(x,s) is not present in such a table. In these cases the Laplace transformation theory gives an inversion theorem which can be used to find the required solution. The temperature distribution appears as a complex integral which can be evaluated using Cauchy s theorem. The required temperature distribution is yielded as an infinite series of exponential functions fading with time. We will not deal with the application of the inversion theorem, and so limit ourselves to cases where the inverse transformation is possible using the correspondence tables. Applications of... 

Here Qa is the mean value of property Q averaged over basin a (at energy ), and (X) is the spectral weight in the continuum limit of the modes with exponential decay constant X. If 2(0 in fact has the stretched exponential form, then (X) will be proportional to the Laplace transform F(X), for which both numerical (Lindsey and Patterson, 1980) and analytical (Helfand, 1983) studies are available. In the simple exponential decay limit= 1, F(X) reduces to an infinitely narrow Dirac delta function but it broadens as p decreases toward the lower limit to involve a wide range of simple exponential relaxation rates. 

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

The existence of the transform is conditional. It requires (a) that F(t) be bounded at all interior points on the interval t < oo (b) that it have a finite number of discontinuities and (c) that it be of exponential order. This last condition means that e F t) must be bounded for some constant a as oo thus, it requires the function s magnitude to rise more slowly than some exponential e as t becomes very large. It is plain that e is of exponential order, whereas e is not. The first condition clearly rules out t — 1) as having a transform, but it says nothing about t ot t. It turns out that F t) may possess an infinite discontinuity at f = 0 if t F t) is bounded there for some positive value of n less than unity. Thus, t does have a Laplace transform, but t does not. In practical applications, conditions (a) and (c) do occasionally offer obstacles, but (b) rarely does. 

Remark. It is important to notice that the Laplace transforms of all the basic functions examined in this section and of additional functions shown in Table 7.1 are ratios of two polynomials in s. The only exceptions are the Laplace transforms of functions translated in time, which include the exponential terms e cS. Therefore, for any function f(t) (not including a time-translated term), we will have... 

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