Thanks for your useful post.
However, could you please enlighten some parts of your post?
In fact, I think the "Equation 2: Time-varying non-linear PDEs: Fick's first law of diffusion" is actually the "linear" Heat PDE and it is not nonlinear!
The initial and boundary considitins in the NDSolve[] for Equation 2 are not consistent!
What is "x_0" there?
However, it is possible to correct all such typos, the main issue is that, where is the final function for "predicting"? I mean, give a very simple example in which by receiving a new input coefficient, then we can get the result (not just one figure).
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