Hi again(adsbygoogle = window.adsbygoogle || []).push({});

I am studying PDEs and came across a solved problem in my textbook, which describes the transformation of a parabolic second order PDE to canonical form. I want to know how to find the second canonical substitution when one has been computed from the characteristic equation.

(PS--This is not a homework problem.)

For instance, suppose the given equation is

[tex]

y^{2}u_{xx} - 2xyu_{xy} + x^{2}u_{yy} = \frac{y^2}{x}u_{x} + \frac{x^2}{y}u_{y}

[/tex]

The solution is as follows:

Compare it with the 'standard' semi-linear second order PDE:

[tex]a(x,y)u_{xx} + 2b(x,y)u_{xy} + c(x,y)u_{yy} = \phi(x,y,u,u_{x},u_{y})[/tex]

to get [itex]a(x,y) = y^{2}[/itex], [itex]b(x,y) = -xy[/itex], [itex]c(x,y) = x^{2}[/itex]. Since [itex]b^{2}-ac = 0[/itex], the equation is parabolic. Considering level curves

[tex]\zeta(x,y) = c_{1}[/tex]

[tex]\eta(x,y) = c_{2}[/tex]

corresponding to the new independent variables [itex](\zeta,\eta)[/itex], the characteristic equation is

[tex]a\left(\frac{dy}{dx}\right)^{2} - 2b\left(\frac{dy}{dx}\right) + c = 0[/tex]

It has a double root [itex]y^{2}+x^{2} = c_{1}[/itex]. Thus

[tex]\zeta(x,y) = x^{2} + y^{2}[/tex]

But this determines only one of the canonical variables. The only condition on [itex]\eta[/itex] is that

[tex]\frac{\partial(\zeta,\eta)}{\partial(x,y)} \neq 0[/tex]

which means that [itex]\eta[/itex] should not be explicitly dependent on [itex]\zeta[/itex] or conversely.

Here, it seems "natural" to take [itex]\eta(x,y) = x^{2}-y^{2}[/itex]. But how does one find a [itex]\eta[/itex] in the general case?

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Canonical Transformation of Parabolic PDEs

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**