Is there a way to impose a no flux condition while still having a zero in the 'f' matrix?

I am currently attempting to simulate the 1D two-temperature model as described by Shin et. al. (https://doi.org/10.1016/j.nimb.2010.11.031).

In short, the PDEs are (tweaked slightly to fit my own experiment):

with the (suggested by Shin et. al.) boundary conditions:

What confuses me is that they imposed a no-flux condition for their electron energy (i.e. electron temperature) at the second boundary condition without having a second-order partial w.r.t. 'x' in the PDE relating to electron temperature. Is there a way to set this boundary condition in PDEPE? I am able to get a solution by tweaking around my mesh in 'x' and 't' (the results of which I have included here), but as you can see the temperature of the electrons is highly unstable when encountering the second boundary condition of 293.7K. Not to mention the lack of a flux condition causes some strange dip below 293.7K towards the front end (x=0).

My suspicion is that the flux condition will fix this instability, but I have no idea how to impose this condition when the matrix 'f' is forced to be zero in the PDEPE implementation. I also encounter the error "Warning: Failure at t=7.008307e-01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed" when I attempt to make the 'x' mesh finer or push the time mesh from 0.7:1:300 to 0:1:300 ps. I think this is related to the no flux condition as well, though I may be mistaken.

Sorry if this seems somewhat garbled. I am definitely willing to provide more information!

Edit: I attempted to put 1e-100*dudx(3) in the third slot of the 'f' matrix instead of 0 so I could impose this boundary condition artificially but it gave me the same result as before.

[Bill Greene]

First off, you have included a link to a paper hidden behind a pay wall; I can't access it.

What is ?

Because the third equation is first order in , I'm not sure what it means to impose a BC at each end, especially one involving .

[Daniel Du]

Unfortunately that is just the nature of academic papers. I took the relevant difeqs and showed them in the main post. is the input or source term for new charge carriers.

With regard to the boundary condition, that was my thinking as well. I am glad my mathematical intuition was not wrong. It is strange they would impose a first-order BC without a second-order difeq and I was incredibly confused when they wrote that in their paper.

Thanks!

[Daniel Du]

As an aside, would you know why I am getting the instabilities in Figure 3, with the odd spikes towards the bottom? I am currently attempting to troubleshoot that portion.