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Vuldoraq
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Hi, Please could someone help me with the following? Sorry for the extra long post!
a) Using the Maximum/Minimum principle, find the maximum and minimum points of the solution of the following mixed problem;
u_t - c^2u_{xx} = 0, \ \ 0 < x < l, \ \ t>0
u(0,x) = x(x-l) \ \ 0< x< l
u(t,0) = 0 = u(t,l) \ \ t > o
and
b) Let u, belonging to the set of continuously differentiable once and continuously differentiable twice functions, be the solution to the following mixed problem;
u_t - u_{xx} =0, \ \ 0 < x < 2, \ \ 0 < t < T
u(0,x) = 0
u(t,0) = f(t)
u(t,2) = 0
where 0 \le f(t) \le 1
Prove that,
0 \leq u(t,x) \leq 1 - \frac{x}{2}
hint: Write the initial boundary value problem for w(t,x) = u(t,x) - v(t,x) where v(t,x) = 1 - x/2 and use the maximum principle.
None?
I'm really stuck on these. For a start I don't really understand the Maximum principle, so any information would be most helpful. I think it means that if you can find a maximum on the boundary of a function, then you have found the maximum for the whole space. The same is true for the minimum.
So is the boundary defined by our boundary/initial conditions? And how would you then find the maximum/minimum on that boundary?
Here is what I have for part a)
The only boundary that gives us any information is the one where u(0,x) = x(x-l) \ \ 0< x< l. The minimum of this boundary must be zero, since x lies between zero and l. The maximum can be found by differentiating once and setting equal to zero. This gives x=l/2. Plugging this back into the original boundary equation gives me a maximum of l/4. Therefore we have,
0 \le u(x,t) \le \frac{l}{4}
Is this the right thing to do? It doesn't seem very rigorous to me, so I think that I have done something wrong.
For part b)
We get an equation for w,
w_t - w_{xx} = 0
w(0,x) = \frac{x}{2} - 1
w(t,0) = f(t) -1
w(t,2) = 0
I find the minimum on the boundary of w to be minus one, because the lowest of f(t)-1 is when f(t)=0. I also get the maximum of w to be zero, corresponding to when x=2 or f(t)=1. Therefore we have the following,
w(t,x) \ge -1
w(t,x) \le 0
since w = u-v we have,
u-v \ge -1
u-v \le 0
which implies,
u \ge v - 1 = -x/2
u \le v = 1-x/2
Therfore I get, -x/2 \le u(t,x) \le 1-x/2 which is clearly wrong!
Please could someone help me make sense of all this? I have spent hours and hours trying to work it out and I must be missing something obvious. Thanks for your time!
Homework Statement
a) Using the Maximum/Minimum principle, find the maximum and minimum points of the solution of the following mixed problem;
u_t - c^2u_{xx} = 0, \ \ 0 < x < l, \ \ t>0
u(0,x) = x(x-l) \ \ 0< x< l
u(t,0) = 0 = u(t,l) \ \ t > o
and
b) Let u, belonging to the set of continuously differentiable once and continuously differentiable twice functions, be the solution to the following mixed problem;
u_t - u_{xx} =0, \ \ 0 < x < 2, \ \ 0 < t < T
u(0,x) = 0
u(t,0) = f(t)
u(t,2) = 0
where 0 \le f(t) \le 1
Prove that,
0 \leq u(t,x) \leq 1 - \frac{x}{2}
hint: Write the initial boundary value problem for w(t,x) = u(t,x) - v(t,x) where v(t,x) = 1 - x/2 and use the maximum principle.
Homework Equations
None?
The Attempt at a Solution
I'm really stuck on these. For a start I don't really understand the Maximum principle, so any information would be most helpful. I think it means that if you can find a maximum on the boundary of a function, then you have found the maximum for the whole space. The same is true for the minimum.
So is the boundary defined by our boundary/initial conditions? And how would you then find the maximum/minimum on that boundary?
Here is what I have for part a)
The only boundary that gives us any information is the one where u(0,x) = x(x-l) \ \ 0< x< l. The minimum of this boundary must be zero, since x lies between zero and l. The maximum can be found by differentiating once and setting equal to zero. This gives x=l/2. Plugging this back into the original boundary equation gives me a maximum of l/4. Therefore we have,
0 \le u(x,t) \le \frac{l}{4}
Is this the right thing to do? It doesn't seem very rigorous to me, so I think that I have done something wrong.
For part b)
We get an equation for w,
w_t - w_{xx} = 0
w(0,x) = \frac{x}{2} - 1
w(t,0) = f(t) -1
w(t,2) = 0
I find the minimum on the boundary of w to be minus one, because the lowest of f(t)-1 is when f(t)=0. I also get the maximum of w to be zero, corresponding to when x=2 or f(t)=1. Therefore we have the following,
w(t,x) \ge -1
w(t,x) \le 0
since w = u-v we have,
u-v \ge -1
u-v \le 0
which implies,
u \ge v - 1 = -x/2
u \le v = 1-x/2
Therfore I get, -x/2 \le u(t,x) \le 1-x/2 which is clearly wrong!
Please could someone help me make sense of all this? I have spent hours and hours trying to work it out and I must be missing something obvious. Thanks for your time!
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