I am trying to reproduce solution of the following differential equation.

I know its solution is

I tried the following code.

equ = NDSolve[{D[u[\[Tau], r], {\[Tau], 1}] - 
     D[u[\[Tau], r], {r, 2}] - ((1/r)*D[u[\[Tau], r], {r, 1}]) == 0}, 
  Derivative[0, 1][u][\[Tau], 0] == 0, 
  Derivative[0, 1][u][\[Tau], RC] == -h*u, u[0, r1] = u0, 
  u[0, r2] = 0, {0 <= r1 >= R}, {R <= r1 >= RC}, {r, 0, RC}, {\[Tau], 
   0, Infinity}]

but ended with the error "NDSolve::dsvar:$u^{0,1}[\tau,RC]=-hu$ cannot be used as a variable. I want to solve this equation keeping both $`h`$ and $`u`$ as variables and reproduced the given solution. Currently, I am out of idea how to deal with it.

Waiting for some valuable directions and suggestions.

I am trying to reproduce solution of the following differential equation.

I know its solution is

I tried the following code.

equ = NDSolve[{D[u[\[Tau], r], {\[Tau], 1}] - 
     D[u[\[Tau], r], {r, 2}] - ((1/r)*D[u[\[Tau], r], {r, 1}]) == 0}, 
  Derivative[0, 1][u][\[Tau], 0] == 0, 
  Derivative[0, 1][u][\[Tau], RC] == -h*u, u[0, r1] = u0, 
  u[0, r2] = 0, {0 <= r1 >= R}, {R <= r1 >= RC}, {r, 0, RC}, {\[Tau], 
   0, Infinity}]

but ended with the error "NDSolve::dsvar:$u^{0,1}[\tau,RC]=-hu$ cannot be used as a variable. I want to solve this equation keeping both $`h`$ and $`u`$ as variables and reproduced the given solution. Currently, I am out of idea how to deal with it.

Waiting for some valuable directions and suggestions.

Note: In my code a=R and b=RC.