An alternative approach to the heat equation
Tóm tắt
The technique of conformal transformations of the metric, widely used in general relativity and in cosmology, is applied to the analysis of heat conduction in an anisotropic medium, in which the thermal conductivity is described by a tensor instead of a scalar. The anisotropic medium is reduced to an effective equivalent one, which is isotropic. The simplification is achieved for a particular conformal factor of the transformation, uniquely determined under physically reasonable assumptions on the thermal conductivity, density, and specific heat of the medium. Another application consists in the formal elimination of source or sink terms from the heat equation by using a suitable conformal transformation.
Tài liệu tham khảo
Wald RM (1984) General Relativity. Chicago: The University of Chicago Press
Landau LD; Lifshitz EM (1989) The Classical Theory of Fields, fourth revised edition. Oxford: Pergamon Press
Weinberg S (1972) Gravitation and Cosmology. New York: Wiley
Kolb EW; Turner MS (1994) The Early Universe. Reading, MA: Addison-Wesley
Faraoni V et al (1999) Conformal transformations in classical gravitational theories and in cosmology. Fund. Cosmic Phys. 20: 121
Kobayashi S; Nomizu K (1963) Foundations of Differential Geometry. New York: Interscience
Abe N (1984) Notes on Kolmogorov’s remark concerning classical dynamical systems on closed surfaces. In Geometry of Geodesics and Related Topics, K. Shiohama ed., Advanced Studies in Pure Mathematics 3. Amsterdam: North Holland
Biesiada M; Rugh SE (1994) Maupertuis principle, Wheeler’s superspace and an invariant criterion for local instability in general relativity. Preprint gr-qc/9408030 (available at http://xxx.lanl.gov)
Faraoni V; Faraoni DM (2002) Elimination of the potential by means of conformal transformations in the Schrödinger and Klein-Gordon equations. Found Phys 32: 773
Landau LD; Lifshitz EM (1976) Mechanics. Oxford: Pergamon Press
Goldstein H (1980) Classical Mechanics. Reading, MA: Addison-Wesley