I want to solve the following $$\frac{\partial^2 T}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial T}{\partial \rho}=\frac{1}{\alpha^2}\frac{\partial T}{\partial t}$$ with boundary conditions $$T(\rho, 0)=f(\rho) \qquad 0 \leq \rho \leq 1$$ and $$T(1,t)=0$$

Using Mathematica.

I tried

DSolve[D[T[r, t], {r, 2}] + (1/r)*D[T[r, t], r] = (1/a)*D[T[r, t], t], T[r, t], {r, t}]

but it gives me error.


Try this

PDE = D[T[r, t], {r, 2}] + (1/r)*D[T[r, t], r] == (1/a)*D[T[r, t], t];

DSolve[{PDE, T[r, 0] == f[r], T[1, t] == 0}, T[r, t], {r, t}]

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