(adsbygoogle = window.adsbygoogle || []).push({}); help w/ diffusion equation on semi-infinite domain 0<x<infinity

Woo! First post! And I'm trying out/learning the latex code which is really neato!

Okay, so...please help!

I'm trying to solve

[tex]\frac{\partial^{2}T}{\partial x^{2}} + \frac{1}{x}\frac{\partial T}{\partial x} = \frac{1}{\alpha}\frac{\partial T}{\partial t}

[/tex]

for [tex] 0 < x < \infty [/tex]

with initial condition such as [tex] T(x,0) = g(x) [/tex]

and [tex] T(\infty,t) = C_{1} [/tex]

and [tex] T(0,t) = f(t)[/tex]

Is this achievable with separation of variables? I get stuck with the spatial problem and the B.Cs.

The two equations i got using separation of variables were:

let

[tex]

T(x,t) = U(x)V(t)

[/tex]

then

[tex] U''V + \frac{1}{x}U'V = \frac{1}{\alpha}UV' [/tex]

[tex] V(U''+\frac{1}{x}U') = \frac{1}{\alpha}UV' [/tex]

[tex]\frac{V'}{V} = \frac{\alpha}{U}(U''+\frac{U'}{x}) = -\lambda [/tex]

so the spatial problem I get is [tex]U''+\frac{1}{x}U'+\frac{\lambda}{\alpha}U = 0[/tex]

I am unsure of the boundary conditions for the spatial problem

time problem I get is [tex]V' = -\lambda V [/tex]

Can this be solved with these B.Cs? I dunno cuz its non homogeneous B.Cs and now im stuck. I've tried a forum search but haven't had any luck.

Any help or guidance would be appreciated. Let me know if anything is unclear.

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# Diffusion equation, semi-infinite solution

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