1. ## Partial Differential Equations

2. Originally Posted by sssouljah
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
$\displaystyle u(x,t)=X(x)\;T(t)$

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

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

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

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

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

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

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

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