Multiple solutions of a singularly perturbed boundary value problem arising in chemical reactor theory

Abstract

Boundary value problems of the form $$\left\{ \begin{gathered}\varepsilon u'' + u' = g(t,u),\,\,\,\,\,\,\,\,\,\,\,\,\,0 \leqslant t \leqslant 1 \hfill \\u'(0) - au(0) = A \hfill \\u'(1) - bu(1) = B \hfill \\\end{gathered} \right.$$ ((1)) arise in the study of adiabatic tubular chemical flow reactors with axial diffusion (cf., e.g., Raymond and Amundson (7) and Burghardt and Zaleski (1)). We are interested in obtaining asymptotic solutions u(t,ɛ) of (1) as the positive parameter ɛ tends to zero. This corresponds to the Peclet number becoming large. Progress toward solving the problem has been made by Cohen (2), Keller (3), and Parter (6). We have succeeded in obtaining asymptotic solutions to this and certain other problems of the form $$\left\{ \begin{gathered}\varepsilon u'' + f(t,u,\varepsilon )u' = g(t,u,\varepsilon ) \hfill \\m\left( {u(0),\,u'(0),\varepsilon } \right) = 0 \hfill \\n \left( {u(1),u'(1),\varepsilon } \right) = 0 \hfill \\\end{gathered} \right.$$but shall consider only the physical problem (1) in this short note. We shall assume that g(t,u) is infinitely differentiable in both arguments.