# Maximum principle

## Main Question or Discussion Point

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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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.

thank you. how about other examples?

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

an actual math problem.. thank you..

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 $$u_t = u_{xx}$$ 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.