Transforming a Second-Order PDE into Canonical Form: Tips and Techniques

[r4nd0m]

How do I transform a second-order PDE with constant coefficients into the canonical form?

I tried to solve this problem:
u_xx + 13u_yy + 14u_zz - 6u_xy + 6u_yz + 2u_xz -u_x +2u_y = 0

I wrote the bilinear form of the second order derivatives and diagonalized it. I found out that it is a hyperbolic equation. Now the problem is how to write it into the canonical form.

What I tried is I wrote it as:
u_aa + u_bb + u_cc + ...(first order derivatives) = 0
where a,b,c are the new variables (in which the matrix is diagonal) and computed the first order derivatives.
Is this a good approach or something else should be done?

 

[HallsofIvy]

Do you know how to convert a general conic section equation to its "cononical form"? It's really the same method. Replace the partial derivatives with x, x², y, y², etc. and convert that equation. Then change back to the partial derivatives.

 

[tacman]

The approach you have taken is a good start in transforming the given PDE into canonical form. Here are some additional tips and techniques that can help you in this process:

1. Identify the type of PDE: Before attempting to transform the PDE into canonical form, it is important to identify the type of PDE it is. In this case, as you have correctly identified, it is a hyperbolic equation.

2. Write the bilinear form: Writing the bilinear form of the PDE helps in identifying the coefficients of the second-order derivatives. This is the first step in diagonalizing the matrix.

3. Diagonalize the matrix: Diagonalizing the matrix involves finding the eigenvalues and eigenvectors of the coefficient matrix of the second-order derivatives. This helps in transforming the PDE into a simpler form.

4. Substitute new variables: Once the matrix is diagonalized, you can substitute new variables in place of the original variables. This helps in simplifying the PDE and writing it in canonical form.

5. Use the characteristic variables: For hyperbolic equations, using the characteristic variables is a useful technique in transforming the PDE into canonical form. These variables are obtained from the eigenvalues and eigenvectors of the coefficient matrix.

6. Solve for the first-order derivatives: Once you have substituted the new variables and used the characteristic variables, you can solve for the first-order derivatives of the transformed PDE.

Overall, your approach of writing the PDE in bilinear form and diagonalizing the matrix is a good start. You can then follow these additional tips and techniques to transform it into canonical form.