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## Non Linear PDE in 2 dimensions

Hi all. I'm trying to solve this PDE but I really can't figure how. The equation is
[tex]
f(x,y) + \partial_x f(x,y) - 4 \partial_x f(x,y) \partial_y f(x,y) = 0
[/tex]
As a first approximation I think it would be possible to consider $$\partial_y f$$ a function of only y and $$\partial_x f$$ a function of only x but even in this case I couldn't find a general solution.
Any idea?

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Quote by L0r3n20 					 				 				Hi all. I'm trying to solve this PDE but I really can't figure how. The equation is $$f(x,y) + \partial_x f(x,y) - 4 \partial_x f(x,y) \partial_y f(x,y) = 0$$ As a first approximation I think it would be possible to consider $$\partial_y f$$ a function of only y and $$\partial_x f$$ a function of only x but even in this case I couldn't find a general solution. Any idea?
Your equation needs to be supplemented by a boundary condition: say $f(x,g(x)) = h(x)$ for suitable $g(x)$.

The method of characteristics looks like a good bet.

By the chain rule,
[tex]
\frac{\mathrm{d}f}{\mathrm{d}t} = \frac{\partial f}{\partial x}\frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial f}{\partial y}\frac{\mathrm{d}y}{\mathrm{d}t}
[/tex]
which by comparison with your equation gives the following system:
[tex]
\frac{\mathrm{d}x}{\mathrm{d}t} = 1 \\
\frac{\mathrm{d}y}{\mathrm{d}t} = -4\frac{\partial f}{\partial x} \\
\frac{\mathrm{d}f}{\mathrm{d}t} = - f
[/tex]
subject to the initial conditions $f(0) = f_0 = h(x_0)$, $x(0) = x_0$, $y(0) = y_0 = g(x_0)$ so that $f(x_0,g(x_0)) = h(x_0)$.

Solving the first equation gives $x = t + x_0$, and the third gives $f = f_0e^{-t} = f_0e^{x_0-x}$. Substituting these into the second gives
[tex]
\frac{\mathrm{d}y}{\mathrm{d}t} = 4f \\
[/tex]
so that $y = y_0 + 4f_0(1 - e^{-t})$.

Therefore given a characteristic starting at $(x_0,g(x_0))$, the value of the function at $(x,y) = (x_0, g(x_0) + 4h(x_0)(1 - e^{-t}))$ is $h(x_0)e^{-t}$.

It is of vital importance that the curve $(x,g(x))$ on which the boundary condition is given is not a characteristic (ie a curve $(x(t),y(t)$) for some $(x_0,y_0)$). There may also be a problem if characteristics intersect.

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