# Maximum principle

1. Aug 20, 2010

### aa1174

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..

2. Aug 20, 2010

### kof9595995

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. Aug 22, 2010

### aa1174

thank you. how about other examples?

4. Aug 22, 2010

### aq1q

do you want a real-life example or an actual math problem that uses the maximum principle?

5. Aug 22, 2010

### aa1174

an actual math problem.. thank you..

6. Aug 22, 2010

### aq1q

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 $$u_t = u_{xx}$$ in (0 < x < 1, 0 < t < $$\infty$$) 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 $$\geq$$ 0 and 0 $$\leq$$ X $$\leq$$ 1.

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

parts a and b emphasize more on the maximum principle.

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