Differential Equation for the Generating Function

The differential equation for the generating function in equilibrium, Eq. (17.19), is obtained by inserting the generating functions for V and DL and is 

The equation is strictly a priori valid only at x = 1. However, we insert for general X and perform the mathematical manipulations needed. We think that if the process is valid for general x, then it should hold also for a special value of x  

The equation can be normalized, because F(l) will refer to the sum of the partial pressures. The equilibrium constant is dependent on the temperature. 

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Exercise. Find a differential equation for the generating function F(z, t) supposing that rn and gn are polynomials in n. 

A recent attempt at a direct stochastic theory by Weinberger [94] using the deterministic flow term as an external (precomputed) constraint should be mentioned here. The intractability of a large coupled system of second-order partial differential equations for the generating function is then reduced to a (nonlinearly coupled) system of ordinary differential equations. The price is the loss of proper population regulation and possible extinction. 

This results in a differential equation for the generating function... 

It may be noted that Stockmayer s differential equation for his generating function follows immediately as a special case of eqn (4) by putting 3 / t =0 (corresponding, of course, to the... 

The methods start from an evolution equation describing the internal fluctuation. In the next step a fixed value of an external parameter, namely the infinitesimal transition probability, is substituted by a stochastic process. This procedure can be done in the master equation or in the equation for the generating function. The first case leads to an integrodifferential equation, while the second model leads to a stochastic partial differential equation. 

A very useful tool for finding analytically the distribution of Nft) is to obtain and solve partial differential equations for the associated cumulant generating functions. The moment generating function, denoted by Ai (0, t), is defined for a multivariate integer-valued variable N (t) as... 

The set (25) can be converted into a single differential equation (27) for the generating function g(z)... 

It is interesting to note that the partial differential Equation (6.7) for the generating function can be derived directly by invariant imbedding methods as is shown in 5 of [8]. This is an illustration of the general notion that whenever an equation for the expected value can be derived, an equation for the more meaningful generating function can also be obtained. 

For the generating function Fof the stationary distribution we can write the following ordinary differential equation ... 

A generating function is defined by equation 2.5-47. To illustrate it use. Table 2.> 2 gives the generating function for an exponential distribution as -A/(0-X). Each moment i.s obtained by successive differentiations. Equation 2.5-48 shows how to obtain the first moment. By taking the limit of higher derivatives higher moments are found. 

In the case of classic chemical kinetics equations, one can get in a few cases analytical solution for the set of differential equations in the form of explicit expressions for the number or weight fractions of i-mcrs (cf. also treatment of distribution of an ideal hyperbranched polymer). Alternatively, the distribution is stored in the form of generating functions from which the moments of the distribution can be extracted. In the latter case, when the rate constant is not directly proportional to number of unreacted functional groups, or the mass action law are not obeyed, Monte-Carlo simulation techniques can be used (cf. e.g. [2,3,47-52]). This technique was also used for simulation of distribution of hyperbranched polymers [21, 51, 52],... 

The finite difference method (FDM) is probably the easiest and oldest method to solve partial differential equations. For many simple applications it requires minimum theory, it is simple and it is fast. When a higher accuracy is desired, however, it requires more sophisticated methods, some of which will be presented in this chapter. The first step to be taken for a finite difference procedure is to replace the continuous domain by a finite difference mesh or grid. For example, if we want to solve partial differential equations (PDE) for two functions 4> x) and w(x, y) in a ID and 2D domain, respectively, we must generate a grid on the domain and replace the functions by functions evaluated at the discrete locations, iAx and jAy, (iAx) and u(iAx,jAy), or 4>i and u%3. Figure 8.1 illustrates typical ID and 2D FDM grids. 

The boundary condition for this partial differential equation is obtained from (9.36). Multiplying both sides of this relationship by A x and using the definition of the cumulant generating function, the partial differential equation of the... 

Batch Reactors. One of the classic works in this area is by Gee and Melville (21), based on the PSSA for chain reaction with termination. Realistic mechanisms of termination, disproportionation, and combination, are treated with a variety of initiation kinetics, and analytical solutions are obtained. Liu and Amundson (37) solved the simultaneous differential equations for batch and transient stirred tank reactors by using digital computer without the PSSA. The degree of polymerization was limited to 100 the kinetic constants used were not typical and led to radical lifetimes of hours and to the conclusion that the PSSA is not accurate in the early stages of polymerization. In 1962 Liu and Amundson used the generating function approach and obtained a complex iterated integral which was later termed inconvenient for computation (37). The example treated was monomer termination. 

These conditions are used to generate results on the cost relationships. These results are obtained by solving the partial differential equations for different amounts loaded, column length and plate count to obtain chromatograms. The yield is calculated from each chromatogram. A surface of yield versus the amount loaded and the number of plates, table or surface is prepared. Then the flow rate, column length and amount loaded are optimized to the objective function 174]. No solvent recycling is assumed. 

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