Well your original worksheet (ie code1.mw ), has a PDE whose solution requires four boundary conditions and two initial conditions, but you only supplied four boundary constions and one initial condition (most of which were syntactically incorrect!) so, rather obviously, this problem cannot be solved,
You latest worksheet ( ie code.mw ) has a PDE whose solution requires four boundary conditions and two initial conditions, but contains no initial/boundary conditions at all, so, rather obviously,, this problem cannot be solved either. However the text accompanying, this worksheet "specifies" (I use the word loosely),
the boundary conditions are:
and the initial conditions are:
If I interpret these correclty, and insert them into your worksheet, then the PDE can be solved (numerically) as shown in the attached.
Do I believe this solution???? Probably no!. The value w(x,t) grows so rapidly as a function of time, that it is diffciult to envisage this PDE+conditions reperesenting any "real-world" phenomenon. With this caveat, the attached shows a plot of w(x,t), over the range x=0..L, t=0..1, as well as a couple of plots for x=10, t=0..1 and x=20, t=0..1 just to demonstrtae the "explosive" growth in the function w(x,t).
Obviously all sorts of other plots/values could be generated - but since I have no ide what you want........ anyhow for what it is worth, check the attached