Abstract
The archetypal parabolic equation is the diffusion equation, or heat equation, in one spatial dimension. Because it involves a time derivative of odd order, it is essentially irreversible in time, in sharp distinction with the wave equation. In physical terms one may say that the diffusion equation entails an arrow of time, a concept related to the Second Law of Thermodynamics. On the other hand, many of the solution techniques already developed for hyperbolic equations are also applicable for the parabolic case, and vice-versa, as will become clear in this chapter.
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Notes
- 1.
This section is largely a more detailed repetition of Sect. 2.4.2.
- 2.
In the more general three-dimensional context, the production term p is measured per unit volume (rather than length) and the flux term q is measured per unit area. Since the cross section has been assumed to be constant, we did not bother to effect the formal passage to one dimension.
- 3.
With \(D=k/c\).
- 4.
This section follows [5].
- 5.
As already pointed out, any solution is already smooth in the interior of the domain. Continuity refers, therefore, to the data specified on (part of) the boundary.
- 6.
Recall that a continuous function defined over a compact (closed and bounded) subset of \({\mathbb R}^n\) attains its maximum and minimum values at one or more points of its domain.
- 7.
Note that the temperature appearing in the heat equation is not necessarily the absolute thermodynamic temperature.
- 8.
See [4], p. 211.
- 9.
As suggested in [6], p. 605.
- 10.
See [1], p. 78.
- 11.
For technical reasons, the space of functions over which these functionals are defined consists of the so-called space of test functions. Each test function is of class \(C^\infty \) and has compact support (that is, it vanishes outside a closed and bounded subset of \(\mathbb R\)). The graph of a test function can be described as a smooth ‘bump’.
- 12.
We must now use the fact that the function space consisted of functions with compact support, so that they, and all their derivatives, vanish at infinity.
- 13.
Here again we are using the compact support property.
- 14.
Although we are presenting the principle in the context of an infinite rod, the same idea can be applied to the case of the finite rod.
- 15.
This is a somewhat different interpretation from that of Sect. 8.9.
- 16.
See [2], p. 40.
- 17.
See [3], p. 71.
References
Courant R, Hilbert D (1962) Methods of mathematical physics, vol I. Interscience, Wiley, New York
Epstein M (2012) The elements of continuum biomechanics. Wiley, London
Farlow SJ (1993) Partial differential equations for scientists and engineers. Dover, New York
John F (1982) Partial differential equations. Springer, Berlin
Petrovsky IG (1954) Lectures on partial differential equations. Interscience, New York (Reprinted by Dover (1991))
Zauderer E (1998) Partial differential equations of applied mathematics, 2nd edn. Interscience, Wiley, New York
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Epstein, M. (2017). The Diffusion Equation. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_10
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DOI: https://doi.org/10.1007/978-3-319-55212-5_10
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