# Question about a 3rd order linear PDE

Consider $f(x,y,t)$ whose spatial second order derivatives evolve over time based on the following pair of PDEs:

\begin{align} \frac{\partial^3}{\partial x^2 \partial t}f\,&=\,\frac{1}{2}\left(\frac{\partial^2}{\partial y^2}f-\frac{\partial^2}{\partial x^2}f\right)\\ \\ \frac{\partial^3}{\partial y^2 \partial t}f\,&=\,\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}f-\frac{\partial^2}{\partial y^2}f\right) \end{align}

The process is subject to initial condition $f(x,y,t=0)=f_0(x,y)$ for some given $f_0(x,y)$.

Is there an analytical expression for the solution $f$ in terms of $f_0$?

Help would be greatly appreciated.

Golabi

• You have two equations for the evolution of a single unknown. How is this possible? If you combine the two equations you obtain $\partial_t \nabla^2 f= 0$ and therefore $\nabla^2 f = \nabla^2 f_0$. Is this useful? – Dmoreno Apr 12 '17 at 17:06
• Thanks! You are right. Thank you also for the tip that if I combine the two equations, I can constrain the solution space of $f$ by requiring its Laplacian to match that of $f_0$ for all time $t$. – Golabi Apr 12 '17 at 17:18

Write $v = f_{xx}$ and $w = f_{yy}$, your equation simplifies to the pair of ODE

$$\partial_t (v + w) = 0, \qquad \partial_t(v-w) = - (v-w)$$

So you have that

$$f_{xx}(t,x,y) = \frac12 \left[ f_{xx}(0,x,y)(1 + e^{-t}) + f_{yy}(0,x,y)(1 - e^{-t}) \right]$$

and

$$f_{yy}(t,x,y) = \frac12 \left[ f_{xx}(0,x,y)(1 - e^{-t}) + f_{yy}(0,x,y)(1 + e^{-t}) \right]$$

Now, these would be the solutions assuming that a solution exists. However, necessarily you need to have

$$f_{xxyy} = f_{yyxx}$$

which will require, by our explicit formula, that

$$f_{xxxx}(0,x,y) = f_{yyyy}(0,x,y)$$

so you are not allowed to prescribe data freely for this equation. (I am not sure if there are other constraints also.)