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Unsaturated horizontal water flow in 2 layer soil:

we have, M(for Moisture), K (for hydraulic conductivity), h (for hydraulic head), x , t (for time).

M(x,t,h) , K(x,t,h) , h(x,t).

PDE is of the form: ∂M/∂t = ∂/∂x (K ∂h/∂x ) .

on the domain (x=[0,xL] ) is divided into 2 layers (x1 & x2) each with its own M and K (2 soil type).

So I have M1, K1 for first layer and M2 & K2 for the second. Instead of solving these two PDE, by applying some linear transformations of the kind:

M*= k1.M ; K*= k2.K ; h*= k3.h ; x1*=k4.x1 ; x2*= k5.x2 ; t*= t .

I'm solving one unique PDE* in the domain (x1* [itex]\cup[/itex] x2*) with transformed boundary(at the upper of 1st layer and lower of 2nd layer) and initial conditions. Now here is my problem: when I get PDE* solutions and then transforms theme back, I get answers different from when not applying this whole transformation procedure.

I guess the question is: when I do a series of [linear] transformations to a PDE is it true to transform the solution back using the inverse of that transformations?

tried my best to express the problem,

thanks