Reverse Laplace transform

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... 

Instead of spacing out in the Laplace-domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. But wouldn t it be nice if we could actually trace the time profile without having to do the reverse Laplace transform ourselves Especially the response with respect to step and impulse inputs Plots of time domain dynamic calculations are extremely instructive and a useful learning tool.1... 

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... 

During the reverse scan, the Laplace transformation is based on t-t as the time variable. Equation (1.10) then becomes... 

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

In chemical degradation kinetics and pharmacokinetics, the methods of eigenvalue and Laplace transform have been employed for complex systems, and a choice between two methods is up to the individual and dependent upon the algebraic steps required to obtain the final solution. The eigenvalue method and the Laplace transform method derive the general solution from various possible cases, and then the specific case is applied to the general solution. When the specific problem is complicated, the Laplace transform method is easy to use. The reversible and consecutive series reactions described in Section 5.6 can be easily solved by the Laplace transform method ... 

In a theoretical treatment of time domain responses, van Gemert and de Graan used a reverse procedure, calculating P(<) from a known c(/(u) behaviour. This treatment is idealized because it was necessary to assume a perfect step pulse with zero rise-time. Using Laplace transform... 

A useful technique for treating reversal methods in chronopotentiometry (and other techniques in electrochemistry) is based on the response function principle (2, 17). This method, which is also used to treat electrical circuits, considers the system s response to a perturbation or excitation signal, as applied in Laplace transform space. One can write the general equation (2)... 

Application of the Laplace transformation method63 to this bounded diffusion problem results, in the case of reversible charge transfer ... 

Considerably less work exists addressing Leveque s problem for inlet channel flow with wall reaction. Carslaw and Jaeger [58] and Petersen [68] presented solutions for plug-flow conditions, using the Laplace transform. Pancharatnam and Homsy [69] used the same technique for laminar flow. The inversion of the transformed solution is given in terms of an infinite summation with coefficients given by recurrence relations (first 24 out of 50 coefficients are tabulated). Ghez [70] considered a first-order reversible reaction with the same solution method. Moreover, asymptotic expansions in the limits of fast and slow reactions were presented. 

The classical flutter is more likely to occur when the coefficient does not reverse its sign as the reduced velocity 27t/K, increases. Basically, the first flexural and torsional modes have major contributions to the state instability. Furthermore, it is more convenient to deal with the equations of motion in the Laplace transform domain. 

The general boundary conditions, Eq. (142), also called the kinetic boundary conditions, can be exploited for solving the particle adsorption problem under the pure diffusion transport conditions when Eq. (138) applies. Using the Laplace transformation method, analytical results have been derived for the spherical and planar interfaces, both for irreversible and reversible adsorptions [2,113,114]. The effect of the finite volume also has been considered in an exact... 

One could also try to take the reverse route, starting from equation (35) (which implies ((y)) = I, not important for the present discussion). Then one derives Jfcoo(T) from detailed balance and the statistical thermodynamical expression for the equilibrium constant. Finally, one might obtain specific rate constants k(E) by an inverse Laplace transformation (operator in equation 73), with = (kT) ... 

The temprature dependence of the friction can be obtained by reverse Fourier transform of Eq. (134), which after the Laplace transform gives... 

As an example of application of Laplace transforms to complex reaction schemes, consider the consecntive first-order reversible reactions... 

The authors solved this set of equations with the appropriate initial and boundary conditions for constant dyeing conditions, and uniform liquor fiow without reversal. The Laplace transformation method was used for the solution, and the transformed equations were solved explicitly. 

Application of the Laplace transformation method to this bounded diffusion problem results, in the case of reversible charge transfer through the layer of surface area A (at a sweep rate v), in a current-potential relation of the form... 

The analytical solution of the general rate model imder linear conditions is discussed elsewhere (Chapter 6). It can be derived in the Laplace domain but the reverse transform of the analytical solution into the time domain is far too complex to be useful. However, the moments of this analytical solution can easily be derived in the time domain. For example, when using the Suzuki model (2.45a), the first moment is given by... 

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