Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$ a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE
$$A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$$
for $(t, x) \in (0, \infty) \times (0, L)$. Moreover he told me that he has $$\partial_x^2 w(0,0) = \partial_x^2 w(0,L) = 0, \\
\partial_x^3 w(0,0) = \partial_x^3 w(0,L) = 0.$$
If we had $x \in \mathbb{R}$ and a starting value, i.e. $w(0,x) = f(x), \partial_t w(0,x) = g(x)$, I think i could solve this with the fouriertransform. But I don't know how to handle this type of PDE.
So I have a few questions about this.


*

*Is this enough to determine a solution uniquely?

*If not, what else do I need?

*How to actually solve the PDE?


Any help would be really nice!
 A: The given initial conditions are probably not enough: specifying some derivatives at just $2$ points is unlikely to be a boundary condition that guarantees uniqueness. You would probably need boundary conditions on at least an open subset of the boundary, like the conditions you are suggesting, and you may need more; consider the example of boundary-value problem for the heat equation, which requires you to impose boundary conditions on the parabolic boundary and not just the initial surface.
Assuming that by $\partial_x^2$ you mean $(\partial_x^2)u = u_{xx}$ and not $(\partial_x^2)u = (u_x)^2$, this is a linear homogeneous equation in spacetime variables and a transform technique or power series solution is probably the first thing I'd try. Since the spatial coordinates are in an interval $[0,L]$, you could try taking a Fourier series in the spatial variable to get an ODE and solve it symbolically. You could also take a Laplace transform in the temporal variable, and again get an ODE. Or combine the two approaches to get an algebraic equation.
Without more information on the coefficients, it's not clear what else you can try, or even what class of PDE this equation lies in.
