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J-B. J. FOURIER'S immensely influential treatise Theorie Analytique de la Chaleur [21J, and the subsequent developments and refinements of FOURIER's ideas and methods at the hands of many authors, provide a highly successful theory of heat conduction. According to that theory, the growth or decay of the temperature e in a conducting body is governed by the heat equation, that is, by the parabolic partial differential equation Such has been the influence of FOURIER'S theory, which must forever remain the classical theory in that it sets the standard against which all other theories are to be measured, that the mathematical investigation of heat conduction has come to be regarded as being almost identicalt with the study of the heat equation, and the reader will not need to be reminded that intensive analytical study has t But not entirely; witness, for example, those theories which would replace the heat equation by an equation which implies a finite speed of propagation for the temperature. The reader is referred to the article [9] of COLEMAN, FABRIZIO, and OWEN for the derivation of such an equation from modern Continuum Thermody namics and for references to earlier work in this direction. viii Introduction amply demonstrated that the heat equation enjoys many properties of great interest and elegance.
List of contents
1 Preliminaries.-
1.1 One-dimensional linear thermoelasticity.-
1.2 An energy integral.- 2 The Coupled and Quasi-static Approximation.-
2.1 An integro-differential equation.-
2.2 Construction of solutions.-
2.3 Failure of the Maximum Principle.-
2.4 Behaviour of the kernel.-
2.5 Initial sensitivity to the boundary.-
2.6 A monotone property of the entropy.- 3 Trigonometric Solutions of the Integro-differential Equation.-
3.1 Maximum Principles for the pointwise mean total energy density and the pointwise mean square heat flux.-
3.2 The effect of coupling on trigonometric solutions.- 4 Approximation by Way of the Heat Equation or the Integro-differential Equation.-
4.1 Status of the heat equation.-
4.2 Comments on Theorem 13.-
4.3 Proof of Theorem 13.-
4.4 Mean and recurrence properties of the temperature.-
4.5 Status of the integro-differential equation.- 5 Maximum and Minimum Properties of the Temperature Within the Dynamic Theory.-
5.1 Maximum and minimum properties with prescribed heat fluxes.-
5.2 Maximum and minimum properties with prescribed temperatures.- References.
