2

I'm dealing with the following PDE:

\begin{equation} k\frac{\partial^2 w}{\partial x^2} + k\frac{\partial^2 w}{\partial y^2} - \frac{\partial^4 w}{\partial x^2 y^2} = 0 \end{equation}

I have unsuccessfully flipped through the entire Handbook of Linear PDE by Polyanin, but I can't seem to find a "name" nor a solution. The closest thing I found is the Rayleigh bar vibration equation,

\begin{equation*} \frac{\partial^2 w}{\partial x^2} - \frac{\partial^2 w}{\partial y^2} - \frac{\partial^4 w}{\partial x^2 y^2} = 0 \end{equation*}

Although my equation resembles a Laplace operator rather than a d'Alembert one. So far I haven't been able to find a suitable change of coordinates to convert the first equation into a more familiar one, but I'm wondering if someone with more experience may point me in some direction (or instantly recognize the equation as some well-known PDE).

jackphen
  • 117
  • I found a link to the equation in a paper "Rayleigh model of vibrations of N-Stepped bar", but I'd still like to know where you read it first, along with if you've seen any solutions to it. The solution in the paper uses the Fourier method, and a Lagrangian formulation. I'd imagine it extends to your case, but some work might be required. – Sarvesh Ravichandran Iyer Mar 02 '21 at 13:14
  • I found it in the Handbook of Linear Partial Differential Equations by Polyanin and Nazaikinskii (2 ed.), section 11.2.6, where it is also referred to as one-dimensional wave equation with strong dispersion. – jackphen Mar 02 '21 at 15:19
  • What a book! I've never read a book with 1623 pages. Did you try fitting it into 11.2.6.4? – Sarvesh Ravichandran Iyer Mar 02 '21 at 15:28
  • It was my first guess, but I stopped when I read that the coefficient b(x) should be positive. I will try to set b(x) = -k and see what happens. – jackphen Mar 02 '21 at 16:15
  • Sure, try to have a go. (+1 on the question, and for recommending me this book, I will forward it to professors who may need it) – Sarvesh Ravichandran Iyer Mar 02 '21 at 16:45

1 Answers1

1

Your equation is equivalent to the system $$ kw-w_{yy} = v, \ \ \ kv-v_{xx} = k^2 w $$ in the sense that a smooth solution of either one gives a smooth solution of the other. But the system seems somewhat unusual in that it is sort of two linked ODEs, where $x$ is a parameter in the equation that has $y$ derivatives, and vice versa. I think your equation is hyperbolic due to the 4th derivative, in spite of having the Laplacian terms.

Bob Terrell
  • 3,812