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A monotone iterative technique for solution of p th order ( p < 0) reaction‐diffusion problems in permeable catalysis
Author(s) -
Perry W. L.
Publication year - 1984
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/jcc.540050412
Subject(s) - dimensionless quantity , reaction–diffusion system , monotone polygon , mathematics , monotonic function , boundary value problem , eigenvalues and eigenvectors , exponent , mathematical analysis , steady state (chemistry) , diffusion , order (exchange) , nonlinear system , thermodynamics , chemistry , geometry , physics , quantum mechanics , linguistics , philosophy , finance , economics
The problem of determining the amount of reactant present in the steady state of a reaction‐diffusion problem with p th order reaction kinetics and slab geometry can be achieved by solving the boundary value problem\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} u''(x) = \phi ^2 u^p (x),\,\,\,\,\,\,\,\,\,\,\,\,{\rm 0 < }x{\rm < 1} \\\\ u'(0) = 0{\rm }\,\,\,\,\,\,\,\,\,\,\,\,u(1) = 1 \\ \end{array} $$\end{document}where u is a normalized (dimensionless) concentration, and ϕ is the Thiele modulus. By considering the related nonlinear eigenvalue problem\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} y''(x) = \lambda y^p (x),\,\,\,\,\,\,\,\,\,\,\,\,{\rm 0 < }x{\rm < 1, }\,\,\,\,\,\,\,p{\rm < 0} \\\\ y'(0) = 0,{\rm }\,\,\,\,\,\,\,\,\,\,\,\,y(1) - y'(1) = 0 \\ \end{array} $$\end{document}and constructing a sequence of functions that converges monotonically to y ( x ), solutions of the original boundary value problem are obtained for the negative exponent case.