1
$\begingroup$

I am modelling the nonlinear behaviour of an bubble in hot water. I am trying to explain it's rotational, vibrational and translational motion in water with impurities and subject to varying temperatures. I got the following equation. How do I solve the following equation analytically?

$$x^2\frac{\partial^6x}{\partial y^6}+\frac32\left(\frac{\partial^2x}{\partial y^2}\right)^{2}+x\frac{\partial x}{\partial z}-\frac{Ax}{{\partial ^3x/\partial z^3}}+B~\frac{\partial^5 x}{\partial z^5 }+\frac{x^{3}e^{\operatorname{sinh}x}}{\partial^2 x/\partial y^2}+\frac{7}{11}\frac{e^{\frac{x^4}{\operatorname{sinh}x}}}{x^{3}}\frac{\partial^3x}{\partial y^3}=C$$

Where $x=x(z,y)$ and $A$,$B$ and $C$ are constants. I would also appreciate if the graph is provided.

$\endgroup$
3
  • 3
    $\begingroup$ Why would you even expect that this PDE has closed form solutions? In any case, even if it does, one would expect that a lot more data than the value of $x$ at one single point would be needed to tie down the solution. $\endgroup$ Oct 14, 2016 at 19:36
  • 1
    $\begingroup$ In fact, if you multiply through by $x_{zzz} x_{yy}$ so that you don't have derivatives in the denominator, the resulting equation admits $x \equiv e^3$ as a solution. Alternatively, if you prescribe real analytic data (up to $(\partial^5/\partial y^5) x$ along the slice $\{y = 0\}$), then by Cauchy-Kovalevsky there should be a local solution, which can be found by Taylor expansion and coefficient matching. $\endgroup$ Oct 14, 2016 at 19:43
  • $\begingroup$ @WillieWong Oh I'm terribly sorry the boundary condition that X=e^3 is not correct, I was copying the question from by physics SE edit box as I had posted an equation similar to this one, with the only difference being that it was a third order nonlinear PDE of a similar structure but this is a sixth order one. I will edit the question immediately, thank you . $\endgroup$ Oct 14, 2016 at 19:48

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.