MaplePrimes - answers and comments on Question, pdsolve command

nonlinear equation

Robert Israel — Wed, 08 Dec 2010 05:26:06 Z

That square-root term makes a big difference: in particular, it makes your equation nonlinear.
Actually, you can get solutions if you express your pde in a way that eliminates the square root:

> pde:= (-alpha^2*f(x,y,z)-diff(diff(f(x,y,z),x),x)-diff(diff(f(x,y,z),y),y)-diff(diff(f(x,y,z),z),z))^2 = a^2*(diff(f(x,y,z),x)^2+diff(f(x,y,z),y)^2+diff(f(x,y,z),z)^2)

> pdsolve(pde);

nonlinear equation

tsunamiBTP — Sun, 26 Dec 2010 00:08:46 Z

Robert,

OK, I agree the eq is nonlinear, but what if I need to consider that term of the gradient magnitude. Would it be more appropriate to move the LaPlacian & the zeroth order term of X to the other side of the eq & square it to get rid of the sq root? Then you have a Helmholtz eq that is squared & you pick a value for the gradient that is considered a nominal value for the gradient magnitude for a defined region.

Then the Helmholtz eq is = to the sq root of some constant.

Can pdesolve work then?

I am not on a machine right now with MAPLE so I have not tried.

+ do you know any literature that is pretty standard on approaching such a problem? I am in the process of chasing something down via internet today on XMAS.

Solution

Robert Israel — Sun, 26 Dec 2010 11:05:15 Z

That's what I did to get rid of the square root. And the result is

Simpler...

Robert Israel — Sun, 26 Dec 2010 11:48:50 Z

Note that the rather complicated solution returned by pdsolve (which is not the general solution) only depends on the variables x,y,z through _C1*x + _C2*y + _C3*z. Here _C1, _C2 and _C3 are arbitrary constants. The pde is invariant under rotations, so we may as well look for solutions where f is a function of x.

> dsolve(eval(pde, f(x,y,z) = f(x)));

My output for f(x,y,z) appears different from yours?

tsunamiBTP — Tue, 28 Dec 2010 02:20:54 Z

I sometimes have difficulty with interpreting MAPLE output, but my result does not appear anything like yours. Maybe it is telling me the same thing, but am I in some ODD display mode that causes MAPLE to display the answer as it is below or is it indeed giving me something different from yours?

Look at pdsolve(Q2), I think I put in the correct equation since I copied it from your posting?

(1)

(2)

Loading PDEtools

(3)

(4)

(5)

(6)

(7)

(8)

(9)

Download pdsovle_output.mw

pdsovle_output.mw

Also I don't understand my output for Q either?

tsunamiBTP — Tue, 28 Dec 2010 02:25:39 Z

Your output I can interpret.