I am trying to solve the convection-diffusion equation for radial flow in a semi-infinite system by using separation of variables technique. I have obtained the following spatial solution and I am trying to get the particular solution (C1 & C2) by using the boundary condition, however, I faced some problems.

The governing PDE is:

$$ \frac{∂^2 T}{∂ r^2} - (β / r) \frac{∂T}{∂r} = θ^2 \frac{∂T}{∂t}$$

The boundary and Initial conditions are as follow:

$$ T(r,0) = 1$$ $$ T(0,t) = 0$$ $$ T(∞,t) = 1$$

My solution is as follow:

$$ T(r,t) = τ (t) R(r)$$ $$ τ (t) = exp (- \frac{γ^2}{θ^2})$$ $$ R(r) = r^{0.5 (β+1)} (C_1 J_m (γ r) + C_2 Y_m (γ r))$$ Where: $$ m = -0.5 (β+1) $$ γ^2 is the separation constant


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