How to obtain a solution for the following IBVP?

I am trying to solve the following advection-diffusion equation for transient flow conditions for radial flow.

The governing equation is as follows.
$$frac{partial T}{partial t} = frac{partial^2 T}{partial x^2} + frac{1-2v(t)}{x} frac{partial T}{partial x}$$
$$frac{partial T}{partial t} = frac{partial^2 T}{partial x^2} + frac{f(t)}{x} frac{partial T}{partial x}$$
where
$$f(t)=1 -2 v(t)$$
Initial condition
$$T(x,t=0)=0$$
BCs.
$$T(x=0,t)=1$$
$$lim_{x to infty} T(x,t)=0$$
I have tried to solve the problem using the following solution procedure.

Assume the solution takes the following form.
$$T(x,t)=left ( e^{-frac{x^2}{4t}}right) F(t)$$
The similarity variable $-frac{x^2}{4t}$ is appropriate selection for solving diffusion equation for radial flow.

The partial derivatives of $T(x,t)$ are as follows.
$$frac{partial T}{partial x} =-frac{x}{2t}left ( e^{-frac{x^2}{4t}}right) F(t)$$
$$frac{partial^2 T}{partial x^2} =F(t)left( -frac{1}{2t}left ( e^{-frac{x^2}{4t}}right) + left( frac{x}{2t} right)^2 left ( e^{-frac{x^2}{4t}}right) right)$$
$$frac{partial T}{partial t} = left (left( frac{x}{2t} right)^2 e^{-frac{x^2}{4t}}right)F(t) + left ( e^{-frac{x^2}{4t}}right)frac{partial F}{partial t}$$
By substituting into the governing equation, the following ODE in $F(t)$ is obtained.
$$frac{dF}{dt}=-left ( frac{1 + F(t)}{2t}right)F(t)$$
The solution of the ODE is as follow.
$$F(t) = expleft ( -int_{0} ^{t} left ( frac{1 + F(u)}{2u}right) , duright)$$
Finally, the solution of the governing is as follow.
$$T(x,t) =left ( e^{-frac{x^2}{4t}}right) expleft ( -int_{0} ^{t} left ( frac{1 + F(u)}{2u}right) , duright)$$
This solution is the same as that given in Handbook of Linear Partial Differential Equations for Engineers and Scientists – Page 367 (when $a=1$)(https://www.taylorfrancis.com/books/9780429166464). Unfortunately, this solution satisfies the initial condition as well as the outer BC, however it doesn’t satisfy the inner BC. When x is put equals to zero, the resulting solution will be as follows.
$$T(x = 0,t) = expleft ( -int_{0} ^{t} left ( frac{1 + F(u)}{2u}right) , duright)$$
My question is how to use the given solution to obtain a solution that satisfies the governing equations, initial conditions, and all boundary conditions of my problem. The resulting solution seems to be a solution of the same problem, however with time dependent inner BC. Duhamel’s integral can be used to obtain a solution for time-dependent BC problem given the corresponding solution for constant BC problem, however the problem here seems to the opposite. Can anyone give me a hint of how to proceed to obtain the solution that satisfies the inner BC?