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I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y = -m x -b are mapped to y = 1, and y = 0, which I already have a program to solve numerically. This will transform the PDE, but that is easier (I think) to work with..

My attempt:

I simplified the problem to first consider y = m x, and y = - m x.

The curve y = m x can be written as (x, m x), and y = - m x as (x, -m x).

The transformation, T, should satisfy T(x, m x) = (x, 1) and T(x, - m x) = (x,0). As far as I can tell, (x, m x), and (x, - m x) form a basis(?) and so this should completely determine the transformation, from what I recall in Linear Algebra (or maybe that only works for a linear transformation?). It looks like the transformation is non linear as well. I'm not quite sure where to go, I want to map from the coordinates (x,y) to (x',y') that transform the curves as described above. If I could find the relationships x = f(x',y') and y = g(x',y'), then I could transform the derivatives in the PDE. Any help would be appreciated! Thank you in advance..

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# Nonlinear coordinate transformation

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