# Understanding the change of variables for PDEs

• I
• Isaac0427
In summary, changing variables in partial differential equations involves using the chain rule and other differentiation rules from calculus. This can be done by rotating coordinates and diagonalizing the matrix A. This allows for easier solving of equations with second derivatives. However, there are still questions about handling non-constant coefficients and non-zero values for B and C that need to be addressed.
Isaac0427
TL;DR Summary
I have been trying to put together an understanding of change of variables for PDEs, and I am wondering if I am on the right track.
I've been trying to get change of variables in PDEs down (I don't particularly like my textbook or professor's approach to it), and I want to ask here if I am getting this right. Let ##\vec{x}=(x_1,x_2,...,x_n)^T## and ##\partial_\vec{x}=(\partial_{x_1},\partial_{x_2},...,\partial_{x_n})^T##. I believe that a second-order linear PDE with constant coefficients (in n dimensions) can be written in the form
$$\left( \partial_\vec{x}^TA\partial_\vec{x}+B\partial_\vec{x}+C \right) u=f(\vec{x})$$
where ##A## is an nxn matrix, ##B## is an n-dimensional row vector, and ##C## is a scalar. Now, we can change coordinates to ##\vec{y}=P\vec{x}## where ##P## is an nxn matrix. From my understanding, this means ##\partial_\vec{x}=P\partial_\vec{y}##. Now, though for a partial differential equation the choice of ##A## is not unique, there is a unique symmetric choice of ##A##. Assuming ##A## is diagonalizable, we choose ##P## such that it diagonalizes ##A##-- this also makes it an orthogonal matrix, and ##P^{-1}=P^T##. So, changing variables gives us
$$\left( \partial_\vec{y}^TP^TAP\partial_\vec{y}+BP\partial_\vec{y}+C \right) u=f(P^T\vec{y}).$$

This gets rid of the cross terms for the second derivatives, which would then make life much easier when solving these equations. I did compare this for the ##A=0## case with problems done using a different method, and the answers were the same (this time though choosing ##P## such that ##BP=(1,0)##).

If everything above is correct, this leaves me with two more questions:

1. I know that when ##B=\vec{0}^T## and ##C=0##, in the 2-D case, we are left with either the diffusion equation, Laplace's equation, or the heat equation. How does one handle solving the equation when ##B## and/or ##C## are nonzero?

2. Is there a way to generalize this for non-constant coefficients?

Thank you very much in advance!

Last edited:
changing variables in PDE is nothing more than the chain rule and other differentiation rules from calculus
if ##y^i=f^i(x)## then ##\partial_{x^i}=\frac{\partial f^s}{\partial x^i}\Big|_{x\mapsto y}\partial_{y^s}##
In fact that is all

Last edited:
wrobel said:
changing variables in PDE is nothing more than the chain rule and other differentiation rules from calculus
if ##y^i=f^i(x)## then ##\partial_{x^i}=\frac{\partial f^s}{\partial x^i}\Big|_{x\mapsto y}\partial_{y^s}##
In fact that is all
Right, but I'm trying to figure out this with matrices/rotating coordinates to diagonalize ##A##. My understanding of what you wrote is that if you have ##\vec{y}=P\vec{x}##, this means (assuming ##P## is a constant nxn matrix) ##\partial_\vec{x}=P\partial_\vec{y}##. Is that correct? And if so, I believe that means the rest of the first part of my post is correct, and the other two questions still stand.

## 1. What is the purpose of changing variables in PDEs?

The purpose of changing variables in PDEs is to simplify the equation and make it easier to solve. By substituting new variables, the PDE can be transformed into a simpler form that can be solved using known techniques.

## 2. How do you choose the appropriate change of variables for a PDE?

The appropriate change of variables for a PDE depends on the specific equation and the desired outcome. It is important to choose a change of variables that will simplify the equation and make it easier to solve. This can be done by analyzing the structure of the PDE and identifying any patterns or symmetries.

## 3. Can changing variables affect the solution of a PDE?

Yes, changing variables can affect the solution of a PDE. In some cases, it can lead to a more general or simpler solution. However, in other cases, it may not have a significant impact on the solution.

## 4. What are the common techniques used for changing variables in PDEs?

Some common techniques used for changing variables in PDEs include substitution, scaling, and transformation. These techniques involve replacing the independent and/or dependent variables with new variables that can simplify the equation.

## 5. Are there any limitations to using change of variables in PDEs?

Yes, there are limitations to using change of variables in PDEs. In some cases, the transformation may not be possible or may lead to a more complicated equation. It is important to carefully consider the choice of variables and their impact on the PDE before making any substitutions.

Replies
1
Views
1K
Replies
1
Views
2K
Replies
15
Views
2K
Replies
3
Views
979
Replies
7
Views
2K
Replies
16
Views
1K
Replies
7
Views
616
Replies
24
Views
897
Replies
65
Views
3K
Replies
4
Views
1K