I'm currently researching a data driven algorithm and I want to express the PDE results in a different manner. This is the kind of form I would like to get, mostly I want to express the t partial derivative as a second order derivative, I don't mind much about the x partial derivative terms:
If the image doesn't load, here's the equation I would LIKE to get:
$$ EI\frac{d^4w}{dx^4} + ρA\frac{d^2w}{dt^2} = 0 $$
However, my algorithm returns this:
$$ \frac{dw}{dt} = 0.002w\frac{dw}{dx} + 0.003w\frac{d^2w}{dx^2} + 0.001w\frac{d^3w}{dx^3} $$
Clearly, they are very different, but I would at least like to get it in terms of $\frac{d^2w}{dt^2}$ not $\frac{dw}{dt}$. Please let me know how to go about expressing this and what assumptions might have to be made. In this case we can assume $t=0$ and $x=0$ initially at least.
Differentiate your equation with respect to $t$. To eliminate mixed derivatives on the right, also differentiate with respect to $x$. For example you get $$w_{tx}=.002ww_{xx}+\cdots$$ and your result gives $w_{tt}$ expressed in terms of $w$ and its $x$ derivatives up to order 6.