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# Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory

Cohen, Donald S. (1971) Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory. SIAM Journal on Applied Mathematics, 20 (1). pp. 1-13. ISSN 0036-1399. https://resolver.caltech.edu/CaltechAUTHORS:COHsiamjam71a

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## Abstract

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

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http://dx.doi.org/10.1137/0120001DOIUNSPECIFIED
Additional Information:© 1971 Society for Industrial and Applied Mathematics. Received by the editors January 5, 1970. This work was performed at the Technological University of Denmark and was supported in part by a grant from Statens naturvidenskabelige Forskningsraad. The author would like to express his appreciation to Professor Erik B. Hansen at the Technical University of Denmark for the many discussions and critical readings of this manuscript which greatly facilitated and improved the paper.
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