Laplace space

In the limiting case where we have a thin-gap thin-ring electrode and consequently normal convection may be neglected, this problem can, in principle, be solved analytically. However, inversion from Laplace space is difficult and polynomial expansion is necessary except for small k,. If we define... 

Note that here z is not the usual Laplace frequency (used in rest of the review), but it is a Fourier frequency. Equation (226) can be written in the Fourier-Laplace space (z) as... 

For reversible reactions, K is constructed from two parts that are responsible for the forward (kall(Q)/uA) and backward (L/L(0)/mc) processes. Both of them are proportional to E(0), which can also serve as a standard for comparison of the different theoretical approaches. The majority of them provide L(,v) when two lifetimes are equal, uA =uc. In this case the formal solution is the same as that for the reversible reaction between the ground-state particles but multiplied by exp(—uAt). In the Laplace space it leads to a shift in the Laplace variable, s —> s + uA. 

Since the second term of the right-hand side does not satisfy the second boundary condition (jt— oo, y- 0), then B (s) = 0. The third boundary condition in Laplace space is... 

Skinner and Wolynes came back to the problem of Eq. (1.8) and solved it with a projection method in Laplace space. An interesting aspect of their work is the development of the contracted distribution function a(a t) (see also Section II) inside the time-convolution integral. They pointed out that this provides perturbation terms neglected erroneously by Brinkman. This interesting feature of their approach is included in the AEP illustrated in this chapter. They explicitly evaluated correction terms up to order The projection technique has also been used by Chaturvedi and Shibata, who used a memoiyless equation as the starting point of their treatment. 

Other problems of transient system response may be solved in a similar way. More complex examples are presented, for example, in Refs. 33-34. It should be added that an arbitrary signal may be applied to the system and if the Laplace transforms of the potential and current are determined, for example, by numerical transform calculations, the system impedance is determined. In the Laplace space the equations [e.g., Eqs. (9) and (11)] are much simpler than those in the time space [e.g. Eqs. (10) and (12)] and analysis in the frequency space 5 allows the determination of the system parameters. This analysis is especially important when an ideal potential step caimot be applied to the system because of the bandwidth limitations of the potentiostat. In this case it is sufficient to know i(t) and the real value of the potential applied to the electrodes by the potentiostat, E(t), which allows numerical Laplace transformations to be carried out and the system impedance obtained. 

Next, transform equation (2-50) into Laplace space in the same manner to obtain... 

Starting off with equation (p), Appendix 1 of Chapter 2, an expression for the Boltzmann principle in Laplace space,... 

This gives us suitable expressions for both Cj and c2 in Laplace space [eqns. (214) and (222), respectively]. Now, the transport-limited current at the detector is given by... 

Hence, in order to calculate the current, the appropriate functions to invert from Laplace space are... 

The generalized reaction-diffusion equation (2.82) can be written in a form using fractional derivatives for subdiffusive transport, where the waiting PDF of species i is given in Laplace space by (2.52), (i) 1 —. In that case... 

The rules listed above for (p t) should reproduce this behavior. For this purpose we need to work in the Laplace space. Let s) be the Laplace transform of 4>(t). The rules (i)-(iii) lead to the expression... 

The model formulated in Sect. 8.3.1 predicts this subdiffusive behavior. For simplicity we assume that the waiting time PDF for the dendrite (pyit) is exponential. Experimental evidence suggests a power-law distribution for the waiting time PDF < 2(0 as i 00, which reads in Laplace space 2 s) = 1 - (tj) ... 

Here the subscripts denote the successive points along the thick lines while the two sets of chmns are distinguished by the prime. In the Laplace space this becomes... 

After the solution is obtained in the Laplace space, a numerical transform inversion (r, t) = L [7 (r, r)] is then performed to convert the solution to the original time domain. 

Coen et al (1987) report an approach, new to electrochemistry, to reduce the computation time required for a 2-D system (the current at a micro-band electrode). The diffusion equation is Laplace transformed, converted to an integral equation and solved, still in Laplace space, for the concentration gradient at the boundaries. They developed an efficient algorithm for the inverse Laplace transformation, which then yields the current as a function of time. Clearly, this is not for everyone at least one of the authors is a mathematician. The method has been used previously (Rizzo and Shippy, 1970) to simulate heat conduction. 

The Laplace-space method of Rizzo and Shippy (1971) and Coen et al (1986) was, in fact applied by Coen et al to a two-dimensional simulation and appeared to be a success. As mentioned in Chapt. 5, the method has the decided disadvantage of being mathematically demanding. 

For a finite chain of iVsites Eq. 21 takes in the Laplace space the form [3, 13] ... 

To avoid problems for larger values of p, we have substituted the steady state solution which is easily given by solving the stationary equation dy/dx = 0 (in LAPLACE space). This problem can now be solved by using a classical Runge-Kutta-Wes algorithm. 

The next step is to contract the description, that is, solve Equation 1.22 for 8y,(k, t) and insert the solution in the continuity equation, to obtain an equation for 8 (k, t) alone. In the resulting equation, we must take the limit of overdamping, t > Xg = M/ °, since we are interested in describing only the diffusive regime. The resulting equation is also an equation for F(k, t), which in Laplace space reads... 

The exact memory function expressions for F(k, t) and F Hk, t) in Equations 1.23 and 1.24 were extended to colloidal mixtures in reference [20]. Written in matrix form and in Laplace space, these exact expressions for the matrices F(]c, t) and F k, t) (when convenient, their A -dependence will be explicitly exhibited) in terms of the corresponding irreducible memory function matrices C(k, t) and 0 k, t) read... 

Steady state scavenging rate constant for ions The full time dependent scavenging rate constant for a constant Vc in the Laplace space is analytically known to be [3]... 

---