# Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?

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?

• ias.ac.in/article/fulltext/jess/118/05/0539-0549 Jun 22 at 15:37
• @MaximUmansky Thanks for the reference. Please correct me if I am wrong. In the article, I could find solutions for convection-diffusion physics but not for convection-diffusion with point sources. Jun 22 at 16:15
• Oh, you are right, there is no source term there. But there are other references where sources seem to be included, e.g., nature.com/articles/s41598-020-63982-w. Jun 22 at 20:29
• @MaximUmansky, but I think that the OP is about finding the Green function, isn't it? Jun 22 at 23:18