I want to solve the following 4th order PDE with the boundary conditions as written. It would be really helpful if someone could guide me on how to solve this. I have tried implicit finite differencing as well as inbuilt solvers in mathematica and matlab. If there are numerical solvers I can directly use, please let me know.
$y_t = -4y_{xxxx}$
Boundary conditions : $y(x=0) = 0 \\ y_x(x=0) = a_0\sin{\omega t} \\ y_{xx}(x=L) = 0 \\ y_{xxx}(x=L) = 0\\$