what i got here,
but i have no idea how to solve.
now by defining several dimensionless variables, we got another form of diffusion equation:
X,T,U here are all dimensionless.
U with initial conditions:
and boundary conditions :
find the analytic solution of this problem.
you would use a separation of variables and ultimately lead to
Since you have a source term in your new equation
then find a Fourier sine series for the source term of the form
and seek a solution of your porblem in the form
where is to be determined (this should give rise to an ODE for ).