Abstract
Boundary value problems of the form
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
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.
This author's work was supported in part by Air Force Office of Scientific Research under grant number AFOSR-67-1062D.
This author's work was supported in part by the Science Research Council of the United Kingdom while he was on leave from New York University.
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References
A. Burghardt and T. Zaleski, "Longitudinal Dispersion at Small and Large Peclet Numbers in Chemical Flow Reactors," Chem. Eng. Sci. 23(1968), 575–591.
D.S. Cohen, "Multiple Solutions of Singular Perturbation Problems", SIAM J. Math. Anal., to appear
H.B. Keller, "Existence Theory of Multiple Solutions of a Singular Perturbation Problem," to appear.
R.E. O'Malley, Jr., "Singular Perturbation of a Boundary Value Problem for a System of Nonlinear Differential Equations," J. Diff. Eq. 8(1970), 431–447.
R.E. O'Malley, Jr., "On Initial Value Problems for a Nonlinear System of Differential Equations with Two Small Parameters," Arch. Rational Mech. Anal. 40(1971), 209–222.
S.V. Parter, "Remarks on the Existence Theory for Multiple Solutions of a Singular Perturbation Problem," to appear.
L.R. Raymond and N.R. Amundson, "Some Observations on Tubular Reactor Stability," Can. J. Chem. Eng. 42(1964), 173–177.
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Inter science, New York, 1965.
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Chen, J., O'Malley, R.E. (1972). Multiple solutions of a singularly perturbed boundary value problem arising in chemical reactor theory. In: Everitt, W.N., Sleeman, B.D. (eds) Conference on the Theory of Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol 280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066949
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DOI: https://doi.org/10.1007/BFb0066949
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