# Question:Can this system of non-linear coupled PDE's be solved?

## Question:Can this system of non-linear coupled PDE's be solved?

Maple 2019

Dill_ABC_Model_PDE_System_Solution.mw

The attached Maple 2019  document attempts to solve a non-linear system of two coupled, time-dependent first-order PDE's, given a list of initial and boundary conditions.  The system models the optical transmittance through a thin photoresist layer whose transmittance changes upon exposure to the incident exposure energy, and hence, the cumulative transmittance through the layer is itself a function of both the exposure time and the distance traveled through the resist layer.  The list of fixed parameters, P, defines the characteristics of a particular photoresist (hereafter "pr") and an assumed exposure irradiance.

My first attempt towards a general solution without initial or boundary conditions (hereafter "ics" & "bcs") apparently "succeeds" (in that no error messages are thrown), however, the form of the solution is quite complicated and difficult (for me at least) to interpret.  I think I understand that the _Cn are undefined constants that require supplying ics & bcs to determine the solutions for the transmitted intensity I(z,t) & the normalized molar fraction of the photo-active component in the pr, M(z,t).  However I do not understand what the symbol _f refers to in the returned solutions.

I make a second attempt to solve the system numerically, supplying a list of the [ics,bcs] as arguments to Pdesolve, however the error message "Error, (in pdsolve/numeric/process_PDEs) PDEs can only contain dependent variables with direct dependence on the independent variables of the problem, got {Iota(0, t), Iota(z, 0), Mu(0, t), Mu(z, 0)}" raises the question of whether I have misunderstood the required syntax in using Pdesolve or that the system as posed is in fact insoluble by Maple.

I would appreciate any insights that readers of this post can contribute, as my experience using Maple and PDesolve in particular must be considered embryonic at best.

Scott Milligan

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