Laplace transformation theories

M.G. Smith, Laplace Transform Theory, Van Nostrand, London, 1966. 

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... 

W.T. Thomson, Laplace Transformation, Theory and Engineering Application, Longmans, Green and CO, London, New York, Toronto, 1957. 

Schiff, J. L., The Laplace Transform Theory and Application, Springer, New York, 1999. 

Elimination of Ci and C3 from these equations will result in the desired relation between inlet Cj and outlet Co concentrations, although not in an exphcit form except for zero or first-order reactions. Alternatively, the Laplace transform could be found, inverted and used to evaluate segregated or max mixed conversions that are defined later. Inversion of a transform hke that of Fig. 23-8 is facilitated after replacing the exponential by some ratio of polynomials, a Pade approximation, as explained in books on hnear control theory. Numerical inversion is always possible. 

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

The solution of this equation has been discussed by DANCKWERTSt28), and here a solution will be obtained using the Laplace transform method for a semi-infinite liquid initially free of solute. On the assumption that the liquid is in contact with pure solute gas, the concentration Cm at the liquid interface will be constant and equal to the saturation value. The boundary conditions will be those applicable to the penetration theory, that is ... 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

Its poles are determined to any order of by expansion of M. However, even in the lowest order in the inverse Laplace transformation, which restores the time kinetics of Kemni, keeps all powers to Jf (t/xj. This is why the theory expounded in the preceding section described the long-time kinetics of the process, while the conventional time-dependent perturbation theory of Dirac [121] holds only in a short time interval after interaction has been switched on. By keeping terms of higher order in i, we describe the whole time evolution to a better accuracy. 

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. 

Since we rely on a look-up table to do reverse Laplace transform, we need the skill to reduce a complex function down to simpler parts that match our table. In theory, we should be able to "break up" a ratio of two polynomials in 5 into simpler partial fractions. If the polynomial in the denominator, p(s), is of an order higher than the numerator, q(s), we can derive 1... 

Since Laplace transform can only be applied to a linear differential equation, we must "fix" a nonlinear equation. The goal of control is to keep a process running at a specified condition (the steady state). For the most part, if we do a good job, the system should only be slightly perturbed from the steady state such that the dynamics of returning to the steady state is a first order decay, i.e., a linear process. This is the cornerstone of classical control theory. 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

Spiegel, Murray R., Schaum s Outline of Theory and Problems of Laplace Transforms, McGraw-Hill Book Company, New York (1965). 

Model representations in Laplace transform form are mainly used in control theory. This approach is limited to linear differential equation systems or their... 

Spiegel MR (1965) Theory and problems of Laplace transforms. McGraw-Hill, New York Nicholson RS, Olmstead ML (1972) Numerical solutions of integral equations. In Matson JS, Mark HB, MacDonald HC (eds) Electrochemistry calculations, simulations and instrumentation, vol 2. Marcel Dekker, New York, p 119... 

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

To answer this question an excursion to field theory is appropriate. We carry through a Laplace transform, defining... 

Laplace transforms, introducing the conjugate variables rm, rn = 1,..., M (In field theory a variable of the type rfn would be addressed as a mass1.) After the Laplace transform a propagator of momentum k in the m-th polymer line yields a factor G7jF(k rm). All segment integrations are eliminated. 

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. 

According to the superposition theorem of system theory for linear responses, this response to a step-function in the current can be employed to deduce the impedance behavior. As regards a qualitative discussion, one can adopt the above description by just replacing short/long times by high/small frequencies. Quantitatively the impedance is given by a Laplace transformation of Eq. (64) (or equivalently by applying Kirchhoff s laws to the equivalent circuit (Eq. (63))) with the result... 

Although small, this is a principal disadvantage of the simplest integral theory. The near-contact density of the products nonlinear in c is lost in the lowest-order approximation to this parameter. However, the nonzero contribution to this region is provided by a modified encounter theory outlined in Section XII. The chief merit of MET is that the argument of the Laplace transformation of n r,t) in (3.311) is shifted from 1 /td to 1/xd + ck. As a result, in the limit xD = oo we have instead of (3.313) [133] ... 

Taking into account the definitions given for the Laplace transforms of the kernels in Eqs. (3.382), (3.386), and (3.389), we can put the stationary IET equations in exactly the same form as in the Markovian theory ... 

Here pss is the Laplace transformation of the solution to the coupled equations (3.593), where ka + kc is substituted for kc in the boundary conditions. In the spinless theory they reduce to a single equation that is the contact analog of Eq. (3.234) for G = pSs-... 

It is remarkable that the integral equation (3.693) proved to be formally exact [51], so that all theories differ only in their definition of the kernel, given as E(f) or V(.v). Sometimes the kernel is explicitly defined in the original works, but more often the reduction to the integral form and extraction of the kernel is a separate problem solved in Ref. 46. In a few cases this procedure was nontrivial and required rather long and sophisticated calculations that are of no interest except for the final results represented by the Laplace transformed kernels S in Table V. 

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