Total solutions, Laplace transform

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

For the very low density varieties of the cases shown in Figure 2 and, more particularly, Figure 4 (curve 7), for which initiation is slow compared to both termination (release from end of template) and polymerization, a simpler treatment, in which the interference of one ribosome with another is totally neglected, should suffice. In this case an equation of the form of Eq. (1), herein only applied to the problem of DNA synthesis, should be valid, but Eqs. (2) and (3) should be modified to account for repetitive initiation at site 1 and continuing release from site K, respectively Eqs. (4) and (6) will not apply. In the even more restricted (but perhaps biochemically relevant) case in which, in addition to neglecting ribosome interference, one may also neglect the back reaction (kb x 0), one may solve this system of equations (Eq. (1), plus Eqs. (2) and (3) modified as described) very easily by taking Laplace transforms.13 This is the only case with repetitive initiation for which we have been able to find solutions for the transient, as well as steady state, behavior. 

The solution of Fick s second law gives the variation of flux, and thence diffusion-limited current, with time, it being important to specify the conditions necessary to define the behaviour of the system (boundary conditions). Since the second law is a partial differential equation it has to be transformed into a total differential equation, solved, and the transform inverted1. The Laplace transform permits this (Appendix 1). 

The Laplace transform is essential in order to transform a partial differential equation into a total differential equation. After solving the equation the transform is inverted in order to obtain the solution to the mathematical problem in real time and space. 

The quasireversible LSV case was treated by Matsuda and Ayabe [389], who used a series sum as an approximation to the integral equation obtained from the Laplace-transform solution of the problem. The result depends on the heterogeneous rate constant, both the peak current and the peak potential varying with this parameter. Basha et al. [82] tried to improve on the results but it seems that those of Nicholson and Shain [417] were better. These also provided results for the totally irreversible case, first described by Delahay [199]. For this, the y(ai)-function has a constant maximum, given to four figures in [73,74], 0.4958, from the tables in [417]. Peak potential varies with rate constant, as with the quasireversible case. 

The most common approach to solution of partial differential equations of the type represented by (7) involves the use of Laplace transformation (Crank, 1957). The method involves transforming the partial differential equation into a total differential equation in a single independent variable. After solving the total differential equation inverse transformation of the solution can be carried out in order to reintroduce the second independent variable. Standard Laplace transforms are collected in tables. 

One may conclude therefore that the solution of Pick s second law (a partial differential equation) would proceed smoothly if some mathematical device could be utilized to convert it into the form of a total differential equation. The Laplace transformation method is often used as such a device. 

The solution of Pick s second law is facilitated by the use of Laplace transforms, which convert the partial differential equation into an easily integrable total differential equation. By utilizing Laplace transforms, the concentration of diffusing species as a function of time and distance from the diffusion sink when a constant normalized current, or flux, is switched on at f = 0 was shown to be... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

This problem has been solved by Laplace transforms to obtain the solution for the total number density as... 

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