# Simple looking but hard to solve nonlinear PDE

1. Jul 8, 2008

### smallphi

I am solving the following simple looking nonlinear PDE:

$$(\partial f / \partial t)^2 - (\partial f / \partial r)^2 = 1$$

Using different tricks and ansatzs I've obtained the following analytic solutions so far:

$$f(r,t) = a\, t + b\, r + c, \,\,\,\, a^2 - b^2=1.$$

$$f(r,t) = \sqrt{(t-r-c)(t+r+d)} .$$

where a,b,c,d are constants.

Note that if one changes the variables to u = (t+r)/2, v=(t-r)/2 then the PDE looks even more ridiculously simple:

$$\frac{\partial f }{\partial u } \,\, \frac{ \partial f }{ \partial v}= 1.$$

The above two solutions were obtained by trying additive and multiplicative separation of the (u,v) variables.

Given the apparent simplicity of the equation is there a general way to approach it and generate more solutions?

Last edited: Jul 8, 2008
2. Aug 1, 2008

### Defconist

Yes, there is a general solution for any first order nonlinear equation. Method of characteristics, usually this methods requires initial condition, but you can supply something general like u(x,0) = f(x). This way you get either a general solution or something that is pretty close. I myself haven't studied the nonlinear case yet, you have to study it up, good luck.

Defconist
p.s. Try out this free book: http://www.freescience.info/go.php?id=1493&pagename=books

3. Aug 2, 2008

### smallphi

4. Oct 16, 2009

Method of Characteristics and Nonlinear PDE

The Method of Characteristics is more complex for Fully Nonlinear PDE. This Stanford PDF derives the system of ODEs used to solve them. The two example PDEs are very much like the ones in your question.

I'm able to follow up to the Quasi-linear and Semi-linear cases. Burger's Equation is one of those. I can't yet solve the Fully Nonlinear PDE, with the this method.