Question

can someone help me for following question
solve the following partial differential equation Using variable separable
(u)_t=3u_xx 00 (u)_x (0,t)=u_x (2,t)=0 u(x,0)=3x

Answer 1

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Answer 2

\[\lambda^2 U_{xx} = U_t\\3U_{xx}=U_t\rightarrow \lambda^2 = 3\\U(x,0) = f(x) = 3x\\U(x,t) = \sum_{n=1}^\infty C_n e^{-n^2\pi^2\alpha^2t/L^2 }*sin\frac{n\pi x}{L}\]
Is that what you need?
if so, your L =6 and just find out \(C_n\) in U(x,t) which is
\[C_n = \frac{2}{6}\int_0^6 3x sin \frac{n\pi x}{6}dx\]

Answer 3

@ybarrap hey, I forgot Fourier series formula for this problem. Can you check??

Answer 4

Is your problem
$$
\cfrac{\partial u}{\partial t}=3\cfrac{\partial^2 u }{\partial x^2}\\
\cfrac{\partial u}{\partial x}\left ( 0,t\right )=\cfrac{\partial u}{\partial x}\left ( 2,t\right )=0\\
u(x,0)=3x
$$
Where \( 0<x<2,~ t>0\)

Answer 5

he is not here , I just want to review what I learned. XD

Answer 6

The question is asking him to use "separation of variables" technique to solve.

Answer 7

I'm not sure how you would use Fourier series to solve, maybe Laplace, but not Fourier. Your equation for the coefficients looks right, but I don't see how to solve this problem using that technique.

Answer 8

ok, let mine aside, just show yours. hehee... I am crazy with this stuff.

Answer 9

ya!