ABSTRACT
This is a generalized Barbashin equation of the form we have studied in Subsection 16~1.
19.3. Systems with substantially distributed parameters As we have shown in the previous subsection, integro-differential equations of Barbashin type, both linear and nonlinear, naturally arise from systems of n differential equations after passing to the limit n --+ 00 or, what is essentially the same, after replacing discrete indices by continuous variables. While this way of obtaining Barbashin equations may seem rather "academic", it has in fact a reasonable physical background which we will discuss in this subsection (see BRACK [1985]). Consider, for instance, the system
= Aq(t) +Bu(t) Cq(t) +Du(t),{~;x(t) =(19.27)
where the values of A, B, C, and Dare m x m-matrices, and the values of q, x, and u are m X n-matrices. The system (19.27) represents the nonstationary balance equations for certain quantities (called
substances) over given balance spaces, and the state vector q represents a distribution density /unction for these substances. Physically, n denotes the number of independent balance spoces, while m is the number of independent substances. Here passing to the limit n -+ 00 or m -+ 00 has a natural physical meaning. In fact, n -+ 00 means that no more discrete balance spaces can be distinguished which occurs in systems with spotially distributed porometers. The mathematical model leads here to a system of m partial differential equations. On the other hand, m -+ 00 means that one has to deal with an arbitrarily large number of substancesj the corresponding systems are then called systems with essentially distributed porometers (BRACK [1985]). In this case one is led to the system
(19.28) 8q~, t) =a(s)q(s,t)+
16 A(s,O')q(O',t)dO'+16 B(s, O')u(O',t) dO' which consists of n integra-differential equations of Barbashin type. If B is a diagonal matrix, the system (19.28) may be written in the form
(19.29) 8q~,t) = a(s)q(s, t)+
16 A(s,O')q(O',t)d<T+ b(s)u(s,t). The variable" E [a, b) in (19.28) or (19.29) may have different meanings, according to the physical setting. For example, s may be the number of C -atoms in chemical compouds, the chain length in p0lymers, or the boiling temperoture of a multicomponent mizture (see e.g. RoTH [1972], RATZSCH-KEBLEN [1984], GILLES-KNOPP [1967]). For fixed t, the function q(', t) may be regarded as density function of the distribution of the material.
