Laplace transform inversion

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. 

B. PARTIAL-FRACTIONS EXPANSION. The linearity theorem [Eq. (18.36)] permits us to expand the function 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 iVth-order in the denominator. We factor the denominator into its N roots pi, P2, Ps,... 

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... 

Freedom to rescale the coordinates and time variable has been used to set the force constants and one of the masses equal to unity (with no loss of generality). These equations can be converted to purely algebraic equations by Laplace transformation. Inverse Laplace transformation yield trajectories in the form,... 

The final step is to perform inverse Laplace transformation on c x, s) to obtain c x, t). Using the table of Laplace transformations, inverse transformation of equation (6.5.27) results in... 

The auxiliary function, h(t), in this formula, called the distribution function, is obtained from the following Laplace transform inversion ... 

The third chapter addresses linear second-order ordinary differential equations. A brief discourse, it reviews elementary differential equations, and the chapter serves as an important basis to the solution techniques of partial differential equations discussed in Chapter 6. An applications section is also included with ten worked-out examples covering heat transfer, fluid flow, and simultaneous diffusion and chemical reaction. In addition, the residue theorem as an alternative method for Laplace transform inversion is introduced. 

The inversion of Eq. 9 from the complex s plane to the time plane can be made by a numerical Laplace transform inversion method. However, for high values of R, the dimensionless concentration in the reservoir is given by [3]... 

By the elementary rule of Laplace transform inversion we have ... 

As an alternative procedure to using the D operator in the development above, it could instead combine with the governing equation in Laplace transform space. Doing so would result in an equation identical to Equation 4.34 except for the fact that operator D would be replaced by the transform parameters. Equation 4.35 would then be obtained by a Laplace transform inversion. 

Cost, T.L. and Becker, E.B., A multidata method of approximate Laplace transform inversion , Int.l J.lfor Numerical Methods in Engineering 2, 1970,p. 207-219. 

Upon Laplace transform inversion via the convolution theorem, one obtains... 

Bertero, M., Boccacci, P., Pike, E. R., On the Recovery and Resolution of Exponential Relaxation Rates from Experimental Data A Singular-value Analysis of the Laplace Transform Inversion in the Presence of Noise, Proc. R Soc. London, Ser. A., 1982, 383, 15-29. 

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed tables. NOTE F(/i -1- 1) = Iq x e dx (gamma function) /(f) = Bessel function of the first land of order n. 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

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 practice, inverse transformation is most easily achieved by using partial fractions to break down solutions into standard components, and then use tables of Laplace transform pairs, as given in Table 3.1. 

In this expression, z is the distance from the surface into the sample, a(z) is the absorption coefficient, and S, the depth of penetration, is given by Eq. 2. A depth profile can be obtained for a given functional group by determining a(z), which is the inverse Laplace transform of A(S), for an absorption band characteristic of that functional group. 

Experimentally, the absorbance A(5) of a band is measured as a function of the angle of incidence B and thus of S. Two techniques can be used to determine a(z). A functional form can be assumed for a(z) and Eqs. 2 and 3 used to calculate the Laplace transform A(5) as a function of 8 [4]. Variable parameters in the assumed form of a(z) are adjusted to obtain the best fit of A(5) to the experimental data. Another approach is to directly compute the inverse Laplace transform of A(5) [3,5]. Programs to compute inverse Laplace transforms are available [6]. 

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

This is the procedure From the postulated kinetic scheme we write the differential rate equations. Take the Laplace transforms of the differential equations. Solve the resulting set of algebraie equations for the transforms of the concentrations. Then take the inverse transforms to obtain the coneentrations as funetions of time. 

Solutions derived by Laplace transformation are in terms of the complex variable s. In some cases, it is necessary to retransform the solution in terms of time, performing an inverse transformation... 

Thus, using the modified kernel, the set of transport equations given by (236) can be solved for any real system, and again the inverse Laplace transformation becomes an unnecessary procedure. This removes a considerable... 

The temperature 0 is then obtained from the tables of inverse Laplace transforms in the Appendix (Table 12, No 83) and is given by. 

It is interesting to note that independent, direct calculations of the PMC transients by Ramakrishna and Rangarajan (the time-dependent generation term considered in the transport equation and solved by Laplace transformation) have yielded an analogous inverse root dependence of the PMC transient lifetime on the electrode potential.37 This shows that our simple derivation from stationary equations is sufficiently reliable. It is interesting that these authors do not discuss a lifetime maximum for their formula, such as that observed near the onset of photocurrents (Fig. 22). Their complicated formula may still contain this information for certain parameter constellations, but it is applicable only for moderate flash intensities. 

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. 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

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 ... 

The plasma concentration function Cp in the time domain is obtained by applying the inverse Laplace transform to the two rational functions in the expression for Cp in eq. (39.58) ... 

The inverse Laplace transform can be obtained again by means of the method of indeterminate coefficients. In this case the coefficients A, B and C must be solved by equating the corresponding terms in the numerators of the left- and right-hand parts of the expression ... 

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