- #1
geetar_king
- 26
- 0
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 don't know because its non homogeneous B.Cs and now I am 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.
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 don't know because its non homogeneous B.Cs and now I am 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.
Last edited: