PDEPE Solver: implement boundary condition with time derivative

I have a partial differential equation which I would like to solve with the pdepe solver of Matlab R2019a. The equations, initial and boundary conditions can be found below. For the sake of readibility, I simplified the equations a bit. In these equations are constants.

The boundary conditions are:

With the initial condition:

My problem is implementing the boundary condition at in pdepe. The shape of the boundary condition of pdepe does not allow a time derivative. I tried to rewrite the boundary condition by, substituting the expression for from the PDE in the boundary condition which results in:

However, this results in a boundary condition with a second order derivative, which is also not allowed by the boundary condition shape of the pdepe solver.

What would be the best and most elegant way to implement this boundary condition in a shape so it can be solved by pdepe? Hopefully someone has some experience with this. Thanks in advance!

Answer 1

[Roshni Garnayak]

As of now boundary conditions of these form are not supported by ‘pdepe’ function.