Critical length of a transport process in rod geometry

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Abstract
In this paper, we study a two‐point boundary‐value problem which governs the transport of n different type of particles in a rod of finite length. Through the construction of an upper solution we establish a simple relation between the rod length and the physical parameters of the transport medium under which the maximal and the minimal sequences obtained in an earlier paper converge, respectively, to a maximal and a minimal solution of the problem. This relation leads to a lower bound for the critical length of the rod when fission occurs in the system. The convergence of the constructed sequences gives a mathematical justification for the existence of a physically meaningful solution to the system. It is also shown, under a slightly stronger condition on the rod length, that the maximal solution coincides with the minimal solution and the physical system is subcritical. In addition, an explicit recursion formula for the calculation of approximate solutions is given.
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