The mesh has points included at 1/3 and 2/3. This is where we will position the point sources later . I also refined the mesh, because I am going to add a convection term which will make this PDE convection dominant (large Peclet number) . We set up the dependent variable, the mass concentration c, time variable t and the spatial independent variable x . Also some material properties are defined .
For the point sources the idea is to use a regularized delta function . The Dirac delta function poses a problem in numerical simulations with FEM implementation that is not prepared to take it as an input . This is because the Dirac delta function is singular at the source location
and needs special treatment that differs from how FEM treats other terms . This can be avoided with an approximation to the Dirac delta function . The process of approximating the Dirac delta function is called regularization .
More information about this can be found in the Heat Transfer tutorial (www.reference.wolfram.com/language/PDEModels/tutorial/HeatTransfer/HeatTransfer.html#945943234) . The process for mass transport is the same. So if you also want to volume sources you can proceed in the same way as mentioned in that tutorial .
Qp is the point source strength . The unit would be [mol/s]. The first point source is active all the time while the second only starts at a specific time but is stronger . We set up the PDE and solve it .