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I'm working on a research idea, and got stuck on some of the math. Here's a particular case I'm stumped on.

Problem

I'm looking for a function $f(x, y, z, t)$ that satisfies

$$\frac{\partial f}{\partial t} = k(z-xy)\frac{\partial^2f}{\partial x \, \partial y}$$

where $k$ is a constant.

I have an initial condition of $f(x, y, z, 0) = x^{n_1}y^{n_2}z^{n_3}$ for integer constants $n_1, n_2, n_3$.

I also know that $f(1, 1, 1, t) = 1$ for all $t$.

What I've tried and what I know

I started with a simpler problem like:

$$\frac{\partial f}{\partial t} = k(y-x)\frac{\partial f}{\partial x}$$

and was able to solve that using the Method of Characteristics, even if I have more than one first-order term on the rhs.

I know very little about PDEs. I know (some) about ODEs and (a lot about) linear algebra, but PDEs are new to me, so a solution that spells out the details would be helpful.

Ultimately I'm trying to generalize somewhat (to a list of terms on the right with first-and-second-order partials, potentially of more than three non-time variables); I figure it's plausible that once I understand the specific case well, I'll be able to generalize.

Michael Hardy
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Michael
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  • Some thoughts: $z$ only appears as a parameter in the equation. You can separate out the $t$ variable by letting $f(t,x,y)=T(t)F(x,y)$, and $T'/T=\lambda=\text{const}$. The mixed partial derivatives may be turned into the wave equation or Laplacian operator by a change of variable the form $x,y=η±χ$ or $x,y=η±iχ$, then in polar $η,χ$ co-ordinates the equation reads $\lambda F=(1-r^2)\nabla^2F$, which may be solved by separating variables again $F(r,\phi)=R(r)P(\phi)$. The angular equation is simple: $P''/P=\text{const}$, Mathematica solves the radial equation in terms of hypergeometric2F1. – Sal Jul 11 '21 at 21:16

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