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Heatequation_exampleB.gif (288 × 177 pixels, file size: 30 KB, MIME type: image/gif, looped, 121 frames, 13 s)

Summary

Description
English: General audience description:

Here the horizontal axis represents the location along a bar of metal and the graph records the temperature at that location. It begins with an initial temperature which is hot at one side and cool at the other, and then shows how the temperature of the bar approaches an equilibrium. It is assumed that no heat is lost from the bar and that there are no heat sources. This demonstrates two key properties of the heat equation: approaching an equilibrium, and the maximum principle. The maximum principle says that the temperature will always have a maximum either earlier in time or at the ends of the bar.

Description:
Graphical representation of the solution to the heat equation for an "infinite slab" of width 1 given by:

where k = .061644 subject to the boundary conditions:

and with the initial heat distribution given by:

In this case, the left face (x=0) and the right face (x=1) are perfectly insulated. This image shows how the heat redistributes, flowing from the warmer left edge to the cooler right edge, then equalizing to a constant temperature throughout. This temperature happens to be the average value of cos(2x) over [0,1], as one might expect.

The solution:

where:

Solution details:

This solution was obtained using separation of variables.
Source Own work
Author User:Wtt
Other versions
File:Heatequation exampleB frames.svg
Source code
InfoField
Mathematica Source:
j = 10;
k = .061644;

A[n_] := (-4 (-1)^n* Sin[2])/(-4 + n^2*Pi^2);

u[x_, t_] := Sin[2]/2 + Sum[A[n]*Cos[n*Pi*x]*Exp[-k(n*Pi)^2*t], {n, 1, j}];

For[i = 0, i <= 12, i += .1,
  Print@Plot[u[x, i], {x, 0, 1},
    Prolog -> {Line[{{0, 1}, {1, 1}, {1, 1}, {1, 0}}]},
    PlotRange -> {0, 1},
    AxesLabel -> {"x", "u=temp"},
    PlotLabel -> {i}
    ]
]

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
This licensing tag was added to this file as part of the GFDL licensing update.
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic, 2.0 Generic and 1.0 Generic license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
You may select the license of your choice.

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Date/TimeThumbnailDimensionsUserComment
current22:01, 17 April 2007Thumbnail for version as of 22:01, 17 April 2007288 × 177 (30 KB)Wtt~commonswiki
21:35, 13 April 2007Thumbnail for version as of 21:35, 13 April 2007288 × 177 (29 KB)Wtt~commonswiki== Summary == Graphical representation of the solution to the heat equation for a "slab" of width 1 given by: :<math>\ u_t = ku_{xx}</math> subject to the boundary conditions: :<math>u_x(0,t) = 0,\ \ u_x(1,t)=0</math> and with the initial heat distribu

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