smallphi
Jul8-08, 03:16 PM
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?
(\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?