I don't understand your notation. Is "D[f(x,t), {x,4}]" the fourth derivative of f with respect to x?
I have a problem with decoupling this PDE. The solutions of time unknown function are natural frequencies. Is it possible t obtain it. I believe that the calculating w is smaller problem then obtaining the solutions of time function. If somebody from the physics can help for decoupling PDE and then calculating code for values.
Everything known constants. Unknown x, t.
I1[x_] := I0/(1 + C0/E^((2*\[Alpha]*(x - xc))/h));
Conditions:
w[0, t] = 0;
w[L, t] = 0;
D[w[0, t], {x, 2}] = 0;
D[w[L, t], {x, 2}] = 0;
D[w[x, 0], {t, 2}] = 0;
D[w[x, 0], {t, 2}] = 0;
PDE
D[E0*(I0 - I1[x])*w[x, t], {x, 4}] + \[Rho]*A*D[w[x, t], {t, 2}] == 0