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Some comments on a singular elliptic problem associated with a rectangle*

Published online by Cambridge University Press:  14 November 2011

V. A. Nye
V. A. Nye
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow
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Synopsis

The construction of the approximate solution within a rectangle of a singular elliptic problem is discussed. It is found that, provided the boundary data satisfy certain continuity conditions at the corners of the rectangle, ordinary boundary layers and parabolic boundary layers only are necessary to describe the solution. A correction term, however, has to be added to the solution if the continuity conditions on the boundary data are not satisfied.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 3-4 , 1979 , pp. 283 - 296
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Cole, J. D.. Perturbation Methods in Applied Mathematics (Waltham, Mass.: Blaisdell, 1968).Google Scholar
2Cook, Pamela L. and Ludford, G. S. S.. The behaviour as ε → + 0 of solutions to in |y|≦1 for discontinuous boundary data. SIAM J. Math. Anal, 2 (1971), 567–594.CrossRefGoogle Scholar
3Cook, Pamela L., Ludford, G. S. S. and Walker, J. S.. Corner regions in the asymptotic solution of with reference to MHD duct flow. Proc. Camb. Phil. Soc. Math. Phys. Sci. 72 (1972), 117–122.CrossRefGoogle Scholar
4Cook, Pamela L. and Ludford, G. S. S.. The behaviour as ε → 0+ of solutions to on the rectangle 0≦x≦l,|y|≦. SIAM J. Math. Anal. 4 (1973), 161–184.Google Scholar
5Drake, David G. and Abu-Sitta, Ali M.. Magnetohydrodynamic Flow in a Rectangular Channel at High Hartmann Number. Z. Angew. Math. Phys. 17 (1966), 519–528.Google Scholar
6Eckhaus, W.. Matched Asymptotic Expansions and Singular Perturbations (Amsterdam: North Holland, 1973).Google Scholar
7Eckhaus, W. and de Jager, E. M.. Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type. Arch. Rational Mech. Anal. 23 (1966), 26–86.Google Scholar
8Fussey, D. E.. An experimental study of magneto-viscous effects in a combustion plasma. J. Phys. D. Appl. Phys. 4 (1971), 1913–1928.Google Scholar
9Grasman, J.. On singular perturbations and parabolic boundary layers. J. Engrg Math. 2 (1968), 163–172.CrossRefGoogle Scholar
10Matkowsky, B. J. and Reiss, E. L.. On the Asymptotic Theory of Dissipative Wave Motion. Arch. Rational Mech. Anal. 42 (1971), 194–212.CrossRefGoogle Scholar
11Nye, V. A.. Unsteady Magnetohydrodynamic flows. J. Inst. Math. Appi 20 (1977), 229–243.Google Scholar
12Protter, M. H. and Weinberger, H. F.. Maximum Principles in Differential Equations (Englewood Cliffs, N.J.: Prentice-Hall, 1965).Google Scholar
13Sloan, D. M.. Magnetohydrodynamic Flow in an Insulated Rectangular Duct with an oblique Transverse Magnetic Field. Z. Angew. Math. Mech. 47 (1967), 109–114.Google Scholar
14Temperley, D. J. and Todd, L.. The effects of wall conductivity in magnetohydrodynamic duct flow at high Hartmann numbers. Proc. Camb. Phil. Soc. Math. Phys. Sci. 69 (1971), 337–351.CrossRefGoogle Scholar
15Williams, W. E.. Magnetohydrodynamic flow in a rectangular duct at high Hartmann number. J. Fluid Mech. 16 (1963), 262–268.CrossRefGoogle Scholar