Maximum principle

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can you please help me give an example of an illustration of the maximum principle of the heat equation (using partial differential equations)

thank you..
 

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  • #2
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em..how about a trivial example that the heat equation reduces to the Laplace equation, and as we know every solution to Laplace equation satisfies maximum principle.
 
  • #3
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thank you. how about other examples?
 
  • #4
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do you want a real-life example or an actual math problem that uses the maximum principle?
 
  • #5
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an actual math problem.. thank you..
 
  • #6
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an actual math problem.. thank you..
Check it out: http://www.math.ucsb.edu/~grigoryan/124A/lecs/lec8.pdf

It's great. It has the proof of the maximum principle (its weaker form) in detail. I think you should find it useful.
Honestly, there are plenty of problems online and in textbooks (you can use google books). Here is one I found in mine:

Consider the diffusion equation [tex] u_t = u_{xx} [/tex] in (0 < x < 1, 0 < t < [tex] \infty[/tex]) with u(0, t) =u(1, t) =0 and u(x, 0) =4x(1 - x).

(a) Show that 0 < u(x,t) < 1 for all t>0 and 0<x< 1.

(b) Show that u(x, t) = u(1 - x, t) for all t [tex]\geq[/tex] 0 and 0 [tex]\leq[/tex] X [tex]\leq[/tex] 1.

(c) Use the energy method to show that [tex] \int u^2 dx [/tex] is a strictly decreasing function of t.

parts a and b emphasize more on the maximum principle.
 

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