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Master1022

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- TL;DR Summary
- Question about change of variables for the Diffusion PDE

Hi,

I understand the underlying concept of changing variables in PDEs (so that we can reduce it to a simpler form), however, I am just not completely clear on the mathematics of it so I have a quick question about it.

For example, if we have the transmission line equation [tex] \frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial x^2} [/tex]

with initial condition: [itex] V(0,x) = 0 [/itex] and boundary conditions: [itex] V(t,0) = V_0 [/itex] and [itex] V(t,L) = 0 [/itex]. Now we want to change variables to reduce the problem into a simpler form without any free parameters.

I let [itex] \xi = \frac{x}{L} [/itex]

[tex] \frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial x^2} [/tex] [tex] \frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial \left( L \xi \right)^2} [/tex] [tex] RC \frac{\partial V}{\partial t} = \frac{\partial ^2 V}{\partial \xi^2} [/tex]

then I let [itex] \tau = \frac{t}{RC} [/itex], therefore [tex] \frac{\partial V}{\partial \tau} = \frac{\partial ^2 V}{\partial \xi^2} [/tex]

So far, I am happy with this. Now this is where I start to have some questions

We want to simplify the boundary condition [itex] V(t, 0) = V_0 [/itex] and we therefore let [itex] U = \frac{V(t, 0)}{V_0} [/itex]. So now that boundary condition is = 1.

However, does making this change not introduce a new free parameter back into the geral expression?

[tex] \frac{\partial \left(U V_0 \right)}{\partial \tau} = \frac{\partial ^2 \left(U V_0 \right)}{\partial \xi^2} [/tex]

The source I am using seems to suggest that step 3) does not change the GE in the way I suggested, but instead changes V to U directly, but I cannot see why.

I see that we are non-dimensionalising it, but we did the same thing with both of the other variables and we subsituted their new values into the GE.

Thank you, any help is appreciated.

I understand the underlying concept of changing variables in PDEs (so that we can reduce it to a simpler form), however, I am just not completely clear on the mathematics of it so I have a quick question about it.

For example, if we have the transmission line equation [tex] \frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial x^2} [/tex]

with initial condition: [itex] V(0,x) = 0 [/itex] and boundary conditions: [itex] V(t,0) = V_0 [/itex] and [itex] V(t,L) = 0 [/itex]. Now we want to change variables to reduce the problem into a simpler form without any free parameters.

**The Method:**

1) Rescale the length variable so that it ranges over one1) Rescale the length variable so that it ranges over one

I let [itex] \xi = \frac{x}{L} [/itex]

**2) Rescale the other independent variable to remove free parameters from the general expression**[tex] \frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial x^2} [/tex] [tex] \frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial \left( L \xi \right)^2} [/tex] [tex] RC \frac{\partial V}{\partial t} = \frac{\partial ^2 V}{\partial \xi^2} [/tex]

then I let [itex] \tau = \frac{t}{RC} [/itex], therefore [tex] \frac{\partial V}{\partial \tau} = \frac{\partial ^2 V}{\partial \xi^2} [/tex]

So far, I am happy with this. Now this is where I start to have some questions

**3) Redefine the independent variable to non-dimensionalize and simplify B/ICs**We want to simplify the boundary condition [itex] V(t, 0) = V_0 [/itex] and we therefore let [itex] U = \frac{V(t, 0)}{V_0} [/itex]. So now that boundary condition is = 1.

However, does making this change not introduce a new free parameter back into the geral expression?

[tex] \frac{\partial \left(U V_0 \right)}{\partial \tau} = \frac{\partial ^2 \left(U V_0 \right)}{\partial \xi^2} [/tex]

The source I am using seems to suggest that step 3) does not change the GE in the way I suggested, but instead changes V to U directly, but I cannot see why.

I see that we are non-dimensionalising it, but we did the same thing with both of the other variables and we subsituted their new values into the GE.

Thank you, any help is appreciated.