Contour Problem

[JoanLat]

Hello everyone:

I was doing this exercise I found on the internet and I am having trouble trying to solve the final step (I have just begun with this type of exercises).

The problem is, once I have found the Eigenvalues: [math]n^2[/math] And the Eigenfunctions: [math]\ Cn* cos (nx)[/math]
I try to solve for Tn(t) and I have found the solution: [math]\ An * cos(nt) + \ Bn * sin(nt)[/math]
From here, I am having trouble trying to determine the constants (I think Cn is 1 for every n different from zero) but I am not sure. Maybe I am just sleepy and I cant find the mistake.

Thanks in advance for your help,


[math]\frac{\partial u^2(x,t)}{\partial t^2}-\frac{\partial u^2(x,t)}{\partial x^2}=0[/math] 0 < x < π t>=0

[math]\frac{\partial u(0,t)}{\partial x}=0[/math] t>0

[math]\frac{\partial u(π,t)}{\partial x}=0[/math] t>0

[math]{\ u(x,0)}=\frac{\ 3x^2-π^2}{\ 12}[/math] 0<=x<=π

[math]\frac{\partial u(x,0)}{\partial t}=1[/math] 0<=x<=π

 

[blamocur]

Maybe I am just sleepy...

Seems like you slept through the part where you wanted to post the problem's statement

 

[nasi112]

What makes you think that \(\displaystyle X(x) = Cn\cos nx\)?

 

[JoanLat]

Seems like you slept through the part where you wanted to post the problem's statement

Well maybe the problem´s statement is posted at the end of the post, where the only thing you are asked to solve is the contour problem I attached :

"Solve the following Contour Problem"

 

[JoanLat]

What makes you think that \(\displaystyle X(x) = Cn\cos nx\)?

Solving the Sturm-Liouville problem associated to the variable x, I have reached this eigenfunction, but maybe it is not correct at all.

 

[Deleted member 4993]

Well maybe the problem´s statement is posted at the end of the post, where the only thing you are asked to solve is the contour problem I attached :

"Solve the following Contour Problem"

Your OP lacks that statement.

Please post the complete problem as it was presented to you.

 

[JoanLat]

Your OP lacks that statement.

Please post the complete problem as it was presented to you.

"Solve the following Contour Problem"

[math]\frac{\partial u^2(x,t)}{\partial t^2}-\frac{\partial u^2(x,t)}{\partial x^2}=0[/math] 0 < x < π t>=0

[math]\frac{\partial u(0,t)}{\partial x}=0[/math] t>0

[math]\frac{\partial u(π,t)}{\partial x}=0[/math] t>0

[math]{\ u(x,0)}=\frac{\ 3x^2-π^2}{\ 12}[/math] 0<=x<=π

[math]\frac{\partial u(x,0)}{\partial t}=1[/math] 0<=x<=π

 

[nasi112]

And three days has passed......it seems neither the OP nor the professors are interested in contributing.

 

[nasi112]

Since no one is willing to help you, I have no other option, but to help you.

You did a great job in finding the functions \(\displaystyle X(x)\) and \(\displaystyle T(t)\).

The solution is now

\(\displaystyle u(x,t) = X(x) \ T(t) = \cos nx \ (A_n \cos nt + B_n \sin nt)\)

You don't need \(\displaystyle C_n\) because \(\displaystyle C_n \times A_n\) or \(\displaystyle C_n \times B_n\) is just another constant.

By applying the superposition principle, the solution will be

\(\displaystyle u(x,t) = A_0 + \sum_{n=1}^{\infty}\cos nx \ (A_n \cos nt + B_n \sin nt)\)

You posted this problem because you wanna know how to find the coefficients \(\displaystyle A_n\) and \(\displaystyle B_n\) plus now the new one \(\displaystyle A_0\).

Before finding them, let us define some functions. Let \(\displaystyle f(x) = \frac{3x^2 - \pi^2}{12}\) and \(\displaystyle g(x) = 1\).

Now we are ready to find the coefficients.

\(\displaystyle A_0 = \frac{1}{\pi}\int_0^\pi f(x) \ dx\)


\(\displaystyle A_n = \frac{2}{\pi}\int_0^\pi f(x) \cos nx \ dx\)


\(\displaystyle B_n = \frac{2}{\pi n}\int_0^\pi g(x) \cos nx \ dx\)


\(\displaystyle A_0\) and \(\displaystyle B_n\) will be 0.


The solution is now

\(\displaystyle u(x,t) = \sum_{n=1}^{\infty} A_n \cos nx \cos nt\)

In fact, to satisfy all the given conditions, the solution must be

\(\displaystyle u(x,t) = t + \sum_{n=1}^{\infty} A_n \cos nx \cos nt\)

To understand why we need \(\displaystyle t\) here, you will have to find a solution

\(\displaystyle u(x,t) = w(x,t) + v(t)\), where \(\displaystyle v(t)\) is gonna hold the burden of \(\displaystyle g(x) = 1\).