# Thread: Partial Differential Equations

1. ## Partial Differential Equations

Please see attachment for question

2. Originally Posted by sssouljah
Please see attachment for question
This is a nice exercise on the Fourier series solution of PDE. Do you know this theory because it is absolutely necessary if you want to find the solution. If so, can you explain where exactly you are stuck?

Coomast

3. Originally Posted by sssouljah
Please see attachment for question
$
u(x,t)=X(x)\;T(t)
$

$
u_t=XT_t,\ \ \ \ u_{xx}=TX_{xx}, \ \ \ \ u_t-Du_{xx}-\alpha u=0
$

$
\frac{T_t}{T}-D\frac{X_{xx}}{X}-\alpha = 0
$

$
D\frac{X_{xx}}{X}=\frac{T_t}{T}-\alpha=-C\text{ (constant)}
$

$
u(x,t)=e^{(\alpha-C)t}\left[A\cos\left(\sqrt{\frac{C}{D}}\;x\right)+B\sin\left (\sqrt{\frac{C}{D}}\;x\right)\right]
$

$
u_x(0,t)=0 \Rightarrow B=0
$

$
u_x(L,t)=0 \Rightarrow \sin\left(\sqrt{\frac{C}{D}}\;L\right) = 0 \Rightarrow C_n=\frac{n^2\pi^2D}{L^2}, \ n=1, 2, 3, ...
$

$
u(x,0)=\cos\left(\frac{\pi x}{L}\right)+\cos\left(\frac{2\pi x}{L}\right) \Rightarrow n=1, 2, \ A_1=A_2 =1
$

- $
\boxed{u(x,t)=e^{\left(\alpha-\tfrac{\pi^2D}{L^2}\right)t}\cos\left(\frac{\pi x}{L}\;\right)+e^{\left(\alpha-\tfrac{4\pi^2}{L^2}\right)t}\cos\left(\frac{2\pi x}{L}\;\right)}
$