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On transformations of the biharmonic equation

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  26 February 2010

George W. Bluman  and
R. Douglas Gregory
George W. Bluman
Affiliation:
Department of Mathematics and Institute of Applied Mathematics and Statistics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Y4.
R. Douglas Gregory
Affiliation:
Department of Mathematics, University of Manchester, Manchester. M13 9PL
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Abstract

Consider a point transformation of the biharmonic equation

namely a coordinate transformation

together with a change of dependent variable given by

for some multiplier F(ξ, η).

MSC classification

Secondary: 35J30: Higher-order elliptic equations
Type
Research Article
Information
Mathematika , Volume 32 , Issue 1 , June 1985 , pp. 118 - 130
Copyright
Copyright © University College London 1985

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References

1.Bluman, G.. Construction of Solutions to Partial Differential Equations by the Use of Transformation Groups. Ph.D. Thesis (California Institute of Technology, 1967).Google Scholar
2.Bluman, G.. On mapping linear partial differential equations to constant coefficient equations. SIAM J. Appl. Math., 43 (1983), 1259–1273.Google Scholar
3.Bluman, G. and Cole, J. D.. The general similarity solution of the heat equation. J. Math. Mech., 18 (1969), 1025–1042.Google Scholar
4.Bluman, G. and Cole, J. D.. Similarity Methods for Differential Equations (Springer, New York, Heidelberg, Berlin, 1974).Google Scholar
5.Jeffery, G. B.. Plane stress and plane strain in bipolar coordinates. Phil. Trans. Roy. Soc. A, 221 (1921), 265–293.Google Scholar
6.Lie, S.. Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch. Math. Naturvidensk, 6 (1881), 328–368.Google Scholar
7.Michell, J. H.. The inversion of plane stress. Proc. Lond. Math. Soc, 34 (1901), 134–142.Google Scholar
8.Ovsiannikov, L. V.. Group Analysis of Differential Equations (Academic Press, New York, 1982).Google Scholar
9.Sternberg, E. and Eubanks, R. A.. On the method of inversion in the two-dimensional theory of elasticity. Quart. Appl. Math., 8 (1951), 392–395.CrossRefGoogle Scholar
10.Ahlbrandt, C. D., Hinton, D. B. and Lewis, R. T.. Inversion in the unit sphere for powers of the Laplacian. Ordinary Differential Equations and Operators. Lecture Notes in Mathematics, 1032 (Springer, 1983), 1–8.Google Scholar