I am looking for some guidance on how to go about solving the following problem

$$\frac{\partial v}{\partial t}=k\frac{\partial^2 v}{\partial r^2} - \phi'(t)$$ Boundary conditions:

\begin{align} v(0,t) &= 0 \\ \quad v(a,t) &= 0\end{align} Initial conditions: $$v(r,0) = -w(r,0)$$ Any help would be appreciated. Thanks!


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