Analytical Methods Of Solution

The theory developed in the previous chapters shows that the study of the linearized Boltzmann equation is a worthwhile undertaking and that many of the features of its solutions can be retained by using model equations. We can say more, that practically all the features are retained by a properly chosen model. The advantages offered by the models consist essentially in simplifying both the analytical and numerical procedures for solving boundary value problems of special interest. In particular, the use of models is invaluable in those cases when the solution of the latter is explicit (in terms of quadratures or functions whose qualitative behavior can be studied by analytical means). Accordingly, we shall devote this chapter to the analytical manipulations which can be used to obtain interesting information from the model equations. The method used throughout is the method of separation of variables already sketched in Chapter VI, Section 7. The first step is to construct a complete set of separated-variable solutions (“elementary solutions”) and then represent the general solution as a superposition of the elementary solutions; the second step is to use the boundary and initial conditions to determine the coefficients of the superposition. While the first problem can be solved for the model equations discussed in Chapter IV, the second problem can be solved exactly in only a few cases. The method retains its usefulness, however, even when the second problem is not solvable, or is only approximately solvable, because it is capable of providing an analytical representation of the solution and hence a picture of its qualitative behavior.