Please help is solving the non homogeneous heat problem.

[yungman]

Homework Statement

Find solution of a nonhomogeneous heat problem:

$$\frac{\partial U}{\partial t} = c^2( \frac{\partial^2 U}{\partial r^2} + \frac{1}{r}\frac{\partial U}{\partial r} + \frac{1}{r^2}\frac{\partial^2 U}{\partial \theta^2} + g(r,\theta,t)$$

With boundary condition: $U(a,\theta, t) = 0$

Initial condition: $U(r,\theta,0) = f(r,\theta)$



2. The attempt at a solution 1

The associate homogeneous equation is:

$$U(r,\theta,t)=R\Theta T \;\;\;\;\Rightarrow\;\;\;\; R\Theta T' + c^2(R''\Theta T + \frac{1}{r}R'\Theta T + \frac{1}{r^2}R\Theta'' T)= g(r,\theta,t)$$

Where $$U_c(r,\theta,t) = \sum_{m=0}^{\infty}\sum_{n=1}^{\infty}J_m(\lambda_{mn}r)[A_{mn}cos(m\theta) + B_{mn} sin(m\theta)]e^{-c^2\lambda^2_{mn}t}$$

I don't know how to solve for particular solution.

 

[yungman]

Anyone please? I just want some guidance how to approach this problem.