- #1
lostidentity
- 18
- 0
I have the following diffusion equation
[tex]
\frac{\partial^{2}c}{\partial r^{2}} + \frac{2}{r}\frac{\partial c}{\partial r} = \frac{1}{\alpha}\frac{\partial c}{\partial t}
[/tex]
where [tex]\alpha[/tex] is the diffusivity. The solution progresses in a finite domain where [tex]0 < r < b[/tex], with initial condition
[tex] c(r,0) = g(r) [/tex]
and the boundary conditions
[tex]
c(b,t) = 1
[/tex]
[tex]
c(0,t) = 0
[/tex]
How will I proceed with this using the separation of variables?
I think the time-dependent part is straight forward after separation of variables. But how will I deal with the spatial part where Bessel functions have to be dealt with?
Thanks.
[tex]
\frac{\partial^{2}c}{\partial r^{2}} + \frac{2}{r}\frac{\partial c}{\partial r} = \frac{1}{\alpha}\frac{\partial c}{\partial t}
[/tex]
where [tex]\alpha[/tex] is the diffusivity. The solution progresses in a finite domain where [tex]0 < r < b[/tex], with initial condition
[tex] c(r,0) = g(r) [/tex]
and the boundary conditions
[tex]
c(b,t) = 1
[/tex]
[tex]
c(0,t) = 0
[/tex]
How will I proceed with this using the separation of variables?
I think the time-dependent part is straight forward after separation of variables. But how will I deal with the spatial part where Bessel functions have to be dealt with?
Thanks.
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