Time-pi t profiles using Laplace transform

Instead of the gamma single-passage distribution for the peripheral compartment, Wise [298] proposed the mixed random walk in series distribution, 

We propose the retention-time distributions A Exp( ) and A2 Erl(A, v) for the first and second compartments, respectively. The peripheral compartment 2 is then constituted by the v pseudocompartments that are required to express Erl(A, v). It follows that 

The system now becomes an to = u+l compartment model and the probabilistic transfer differential equations are 

In the above equations, pi represents the probability that a molecule starting in compartment 1 is in compartment i at time t. By using t = At and p = n/A, one obtains the dimensionless system of differential equations 

This model is a special case of the model studied by Matis and Wehrly [369] in which Ai Erl(Ai, u ) and A2 Ev A2.V2) retention-time distributions are associated with the first and second compartments, respectively. The analysis of the characteristic polynomial of this model implies that there are at least two complex eigenvalues, except for the case v = 2 with parameters satisfying the condition 

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