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I am trying to solve the following initial-boundary value problem by using Hankel transformation:

$$ \frac{dT}{dt}= \frac{d^2T}{dr^2} + \frac{1}{r}\frac{dT}{dr} - ζ T + ψ\left(\frac{1}{t+t_{o}} exp\left(- \frac{r^2}{4R(t+t_{o})} \right) - \frac{1}{t} exp\left(- \frac{r^2}{4Rt} \right)\right) $$

The initial and boundary conditions are:

$$ T(r,0) = f(r)$$ $$ \lim_{r\to0}\frac{dT}{dr}=0$$ $$\lim_{r\to\infty}T(r,t)=0$$

My solution is as follow:

$$ T(r,t) = \frac{e{-ζ t}}{2 t} \int_{0}^{\infty} v f(v)\ e^{-\left (\frac{v^2 + r^2}{4t} \right)}\ I_0\left(\frac{vr}{2t} \right)dv\ -2 ψ R \int_{0}^{\infty} u J_0(u r)\left(\frac{(exp(-(u^2 + ζ) t) - exp(- R u^2 t)}{(u^2 + ζ) - R u^2} \right)(exp(- R u^2 t_{o})-1)\ du $$

I have obtained this solution manually through Hankel transformation. I need to make sure that the solution is correct. I have tried to substitute the solution back into the governing equation in Mathematica but it doesn't help as the solution is complex. Can any one help to assure that the solution is correct or not?

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