In the HW section, someone proposed:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]u^2\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0;\quad u(x,0)=x[/tex]

As per "Basic PDEs" by Bleecker and Csordas", treating this as:

[tex]F(x,y,u,p,q)=0\quad\text{with}\quad \frac{\partial u}{\partial x}=p\quad\text{and}\quad\frac{\partial u}{\partial y}=q[/tex]

and solving for parametric functions as a function of two variables t and s such that:

[tex]\frac{dx(s,t)}{dt}=\frac{\partial F}{\partial p}[/tex]

[tex]\frac{dy(s,t)}{dt}=\frac{\partial F}{\partial q}[/tex]

[tex]\frac{du(s,t)}{dt}=p\frac{\partial F}{\partial p}+q\frac{\partial F}{\partial q}[/tex]

Solving the ODEs subject to boundary conditions:

[tex]u(x,0)=u(f(s),g(s))=s [/tex]

yields:

[tex]x(s,t)=s^2t[/tex]

[tex]y(s,t)=t[/tex]

[tex]u(s,t)=s[/tex]

Solving for u:

[tex]u(x,y)=\pm\sqrt{\frac{x}{y}}[/tex]

To what extent can I rely upon this method to solve more complex non-linear PDEs? Guess that would involve some analysis of sorts limited by the ability to affect some integration or another. Some limitations I can think of include:

1. I just glossed-over the need sometimes to solve a set of 5 equations instead of the simpler 3 sets like above. Seems though should be able to numerically integrate the 5 equations (or 3) no matter how complex and thus arrive at least in principle to a parametric solution described above.

2. u(x,y) cannot allways be determined explicitly in terms of x and y although the parametric forms are equally valid albeit a bit more difficult to manipulate. Suppose I can work on them a bit but I mean, that could take a whole semester to fully analyze:

[tex]F(x,y,u,p,q)=0[/tex]

Probably been done already anyway.

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# An interesting PDE perhaps?

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