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First integrals and bifurcations of a Lane-Emden equation of the second kind. (English) Zbl 1143.80005
Summary: First integrals admitted by an approximate Lane-Emden equation modelling a thermal explosion in a rectangular slab and cylindrical vessel are investigated. By imposing the boundary conditions on the first integrals we obtain a nonlinear relationship between the temperature at the center of the vessel and the temperature gradient at the wall of the vessel. For a rectangular slab the presence of a bifurcation indicates multivalued solutions for the temperature at the center of the vessel when the temperature gradient at the wall is fixed. For a cylindrical vessel we find a bifurcation indicating multivalued solutions for the temperature gradient at the walls of the vessel when the temperature at the center of the vessel is fixed.
MSC:
80A20Heat and mass transfer, heat flow
80A32Chemically reacting flows (thermodynamic aspects)
34C23Bifurcation (ODE)
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