I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann boundary condition is specified at the outlet and Dirichlet boundary condition at the inlet.
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x} + Q $$ with initial condition $$c(x,0) = C_0$$ Neumann boundary condition at the right boundary $$\frac{\partial C}{\partial x}=0\text{ at }t>0.$$
Dirichlet boundary condition at the left boundary $$c(0,t) = C_L$$
The method to obtain a numerical solution for this problem is answered here and the Mathematica code is also posted.
I would like to compare the numerical solution with the analytical solution. Are there some references in which I can find an analytical solution for this problem? Could someone please suggest a reference or share provide an answer?