Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-18T23:33:57.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Degenerate forms of Maxwell's equations

Published online by Cambridge University Press:  17 February 2009

A. L. Carey  and
K. McNamara
A. L. Carey
Affiliation:
Department of Pure Mathematics, The University of Adelaide, Adelaide 5001, South Australia.
K. McNamara
Affiliation:
Department of Pure Mathematics, The University of Adelaide, Adelaide 5001, South Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies degenerate forms of Maxwell's equations which arise from approximations suggested by geophysical modelling problems. The approximations reduce Maxwell's equations to degenerate elliptic/parabolic ones. Here we consider the questions of existence, uniqueness and regularity of solutions for these equations and address the problem of showing that the solutions of the degenerate equations do approximate those of the genuine Maxwell equations.

Type
Research Article
Information
The ANZIAM Journal , Volume 31 , Issue 3 , January 1990 , pp. 277 - 300
Copyright
Copyright © Australian Mathematical Society 1990

References

[1] Carey, A. L. and O'Brien, D. M., “Existence and regularity results for Maxwell's equations in the quasi-static limit”, J. Austral. Math. Soc. B 27 (1986) 402–415.CrossRefGoogle Scholar
[2] Cheng, D. H. S., “On the formulation of the dyadic Green's function in a layered medium”, Electromagnetics 6 (1986) 171–182.CrossRefGoogle Scholar
[3] Colins, R. E., “Dyadic Green's function expansions in spherical co-ordinates”, Electromagnetics 6 (1986) 183–207.Google Scholar
[4] Gradshteyn, H. and Ryzhik, G., Table of integrals, series, and products (Academic Press, New York, 1987).Google Scholar
[5] Hohmann, G. W., “Three-dimensional induced polarisation and electromagnetic modeling”, Geophysics 40 (1975) 309–324.CrossRefGoogle Scholar
[6] Hohman, G. W. and Filipo, W. A. San, “Integral equation solution for the transient electromagnetic response of a three-dimensional body in a conductive half-space”, Geophysics 50 (1985) 798–809.Google Scholar
[7] Joly, P., “Propagation of an electromagnetic field in a conductive half-space”, (Boundary Elements 7, Eds. Brebbia, G. A. and Maier, G., Springer-Verlag, Berlin, pp. 3–53–3–68, 1985) and Y. Goldman, P. Joly, M. Kern, “The electric field in a conductive half space as a model for petroleum and mining prospection” to appear in Math. Meth. in Applied Sciences and references therein.Google Scholar
[8] Lions, J. L. and Magnes, E., Non-homogeneous boundary value problems and applications, Vol. 2, (Springer-Verlag, Berlin, 1972).Google Scholar
[9] McNamara, K., M. Sc. Thesis, University of Adelaide, 1988.Google Scholar
[10] Nedelec, J. C., “Mixed finite elements in R3”,Numerische Mathematik 35, (1980) 315–341.CrossRefGoogle Scholar
[11] Oberhettinger, F. and Badii, L., Tables of Laplace transforms (Springer-Verlag, 1973).CrossRefGoogle Scholar
[12] O'Brien, D. M. and Smith, R. S., “Transient electromagnetic response of a layered conductin medium at asymptotically late times”, J. Austral. Math. Soc. B 27 (1985) 1–30.CrossRefGoogle Scholar
[13] Olver, F. W. J., Asymptotics and special functions (Academic Press, New York, 1974).Google Scholar
[14] Showalter, R. E., “Initial and final value problems for degenerate parabolic systems”, Indiana Univ. Math. Journal, 28 (1979) 883–894.CrossRefGoogle Scholar
[15] Tai, Chen-To, Dyadic Green's functions in electromagnetic theory (Intext, Scranton, Pennsylvania, 1971).Google Scholar
[16] Treves, F., Basic linear partial differential equations (Academic Press, New York, 1975).Google Scholar
[17] Van Bladel, J., Electromagnetic fields (New York, McGraw-Hill, 1964).Google Scholar