Abstract
The one-dimensional diffusion equation is the linear second order partial differential equation
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Notes
- 1.
In the theory of stochastic processes, \( \frac{1}{2}\varDelta \) represents the infinitesimal generator of the Brownian motion.
- 2.
Dimensions of r: [r] = [cat] × [time]-1 × [mass]-1 .
- 3.
Assuming that the time derivative can be carried inside the integral.
- 4.
[ q ] = [cat] × [lenght] -1 × [time] - 1.
- 5.
[k] = [cat] × [deg]-1 × [time]-1 × [length]-1 (deg stays for degree, Celsius or Kelvin).
- 6.
[c v ] = [cal] × [deg]-1 × [mass]-1 .
- 7.
Remember that the bar has perfect lateral thermal insulation.
- 8.
Formula ( 3.10 ) is based on Newton’s law of cooling: the heat loss from the surface of a body is a linear function of the temperature drop U −u from the surroudings to the surface. It represents a good approximation to the radiative loss from a body when |U — u| /u ≪ 1.
- 9.
Appendix A.
- 10.
It is also true that U (z, s) → y in the pointwise sense, when y ≠ 1 and (z, s) →
(y, 0). We omit the proof.
11 Appendix A.
- 11.
Appendix A
- 12.
The Weierstrass test works here for t ≥ t0 > 0.
- 13.
Recall that by domain we mean an open connected set in ℝn.
14 We can also allow boundaries with corner points, like squares, cones, or edges,
like cubes. It is enough that the set of points where the tangent plane does not
exist has zero surface measure (zero length in two dimensions). Lipschitz domains
have this property.
- 14.
We can also allow boundaries with corner points, like squares, cones, or edges, like cubes. It is enough that the set of points where the tangent plane does not exist has zero surface measure (zero length in two dimensions). Lipschitz domains have this property.
- 15.
Linear Newton law of cooling.
- 16.
We omit the rather long proof.
- 17.
A solution of a particular evolution problem is a similarity or self-similar solution if its spatial configuration (graph) remains similar to itself at all times during the evolution. In one space dimension, self-similar solutions have the general form
$$ u\left(x,t\right)=a(t)F\left(x\Big/b(t)\right) $$where, preferably, u/a and x/b are dimensionless quantity.
- 18.
Rigorously, the precise conditions are:
liminf U (x) = 0.
x—
- 19.
Recall that
I dz = \pK.
Jr.
- 20.
The first integral in (3.64) is a Riemann-Stieltjes integral, that formally can be written as
$$ {\displaystyle \int \varphi (x){\mathcal{H}}^{\prime }(x) dx} $$and interpreted as the action of the generalized function H on the test function p.
- 21.
[γ] = [time]−1 .
- 22.
As in dimension n = 1, in (3.70) the integral has a symbolic meaning only.
- 23.
We omit the long and technical proof.
- 24.
Negative production (f < 0) means removal.
- 25.
For further details we refer to Quarteroni, Sacco, Saleri (2007).
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Salsa, S., Vegni, F.M.G., Zaretti, A., Zunino, P. (2013). Diffusion. In: A Primer on PDEs. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2862-3_3
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