- #1
yungman
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Homework Statement
Find solution of a nonhomogeneous heat problem:
[tex] \frac{\partial U}{\partial t} = c^2( \frac{\partial^2 U}{\partial r^2} + \frac{1}{r}\frac{\partial U}{\partial r} + \frac{1}{r^2}\frac{\partial^2 U}{\partial \theta^2} + g(r,\theta,t) [/tex]
With boundary condition: [itex] U(a,\theta, t) = 0 [/itex]
Initial condition: [itex] U(r,\theta,0) = f(r,\theta)[/itex]
2. The attempt at a solution 1
The associate homogeneous equation is:
[tex] U(r,\theta,t)=R\Theta T \;\;\;\;\Rightarrow\;\;\;\; R\Theta T' + c^2(R''\Theta T + \frac{1}{r}R'\Theta T + \frac{1}{r^2}R\Theta'' T)= g(r,\theta,t)[/tex]
Where [tex] U_c(r,\theta,t) = \sum_{m=0}^{\infty}\sum_{n=1}^{\infty}J_m(\lambda_{mn}r)[A_{mn}cos(m\theta) + B_{mn} sin(m\theta)]e^{-c^2\lambda^2_{mn}t}[/tex]
I don't know how to solve for particular solution.
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