# PDEs: Diffusion Equation Change of Variables

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• Master1022
In summary, the conversation discusses the method of changing variables in PDEs in order to simplify the problem. The steps involved include rescaling the length and time variables, as well as redefining the independent variable to non-dimensionalize and simplify the boundary condition. However, there is confusion about whether this step introduces a new free parameter. Ultimately, the misunderstanding is cleared up and it is concluded that the boundary condition is cancelled out by the rescaled variables.
Master1022
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 $$\frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial x^2}$$
with initial condition: $V(0,x) = 0$ and boundary conditions: $V(t,0) = V_0$ and $V(t,L) = 0$. 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 one

I let $\xi = \frac{x}{L}$

2) Rescale the other independent variable to remove free parameters from the general expression
$$\frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial x^2}$$ $$\frac{\partial V}{\partial t} = \frac{L^2}{RC} \frac{\partial ^2 V}{\partial \left( L \xi \right)^2}$$ $$RC \frac{\partial V}{\partial t} = \frac{\partial ^2 V}{\partial \xi^2}$$
then I let $\tau = \frac{t}{RC}$, therefore $$\frac{\partial V}{\partial \tau} = \frac{\partial ^2 V}{\partial \xi^2}$$

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 $V(t, 0) = V_0$ and we therefore let $U = \frac{V(t, 0)}{V_0}$. So now that boundary condition is = 1.

However, does making this change not introduce a new free parameter back into the geral expression?
$$\frac{\partial \left(U V_0 \right)}{\partial \tau} = \frac{\partial ^2 \left(U V_0 \right)}{\partial \xi^2}$$
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 have solved my misunderstanding, when we apply the partial derivative twice on the RHS, we are not squaring the numerator and hence there will only be one $V_0$. Therefore, the $V_0$ (to the power 1) in the numerator of both the LHS and RHS of the equation will cancel out.

BvU

## 1. What is a PDE and how does it relate to the Diffusion Equation?

A PDE, or partial differential equation, is a mathematical equation that involves partial derivatives of an unknown function with respect to multiple independent variables. The Diffusion Equation is a specific type of PDE that describes the behavior of a diffusing substance, such as heat or chemical concentration, over time and space.

## 2. How do you solve the Diffusion Equation using change of variables?

The Diffusion Equation can be solved using the method of change of variables, which involves substituting a new set of variables into the equation. This can help simplify the equation and make it easier to solve. The most common change of variables used for the Diffusion Equation is the separation of variables method, where the solution is expressed as a product of two functions of different variables.

## 3. What are the benefits of using change of variables to solve the Diffusion Equation?

Change of variables can help make the Diffusion Equation easier to solve by reducing it to a simpler form. It can also help identify patterns and relationships in the solution that may not be apparent in the original equation. This method can also be used to solve more complex boundary value problems that may not have an analytical solution.

## 4. Are there any limitations to using change of variables for the Diffusion Equation?

While change of variables can be a powerful tool for solving the Diffusion Equation, it may not always be applicable or effective. In some cases, the change of variables may not lead to a simpler equation or may not be possible to perform. Additionally, this method may not work for more complex PDEs with non-linear terms.

## 5. How is the Diffusion Equation used in real-world applications?

The Diffusion Equation has many practical applications in fields such as physics, chemistry, biology, and engineering. It is used to model the behavior of diffusing substances, such as heat transfer in materials, chemical reactions in a solution, and the spread of pollutants in the environment. It is also used in financial mathematics to model the diffusion of stock prices over time.

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