I want to solve the following differential equation. It is about modeling the vibration of a linearized elastic rod.
$$\frac{\partial^2} {\partial x^2} ( E I \frac{\partial^2 w} {\partial x^2}) + \rho S \frac{\partial^2 w} {\partial t^2} =0$$
In the above formula, $E I = 1, \rho S = 1$, $w(x,t)$ is a binary function of $x$ and $t$.
The boundary and initial conditions are as follows:
$$w(x,t) \Big| _{t=0}=\frac{x^2} {6} (3 - x)$$ $$\frac{\partial w} {\partial t}\Big| _{x=0}=0$$
$$\frac{\partial^2 w} {\partial t^2}\Big| _{x=1}=0$$ $$\frac{\partial^3 w} {\partial t^3}\Big| _{x=1}=0$$
I wrote the following code according to the above conditions:
ClearAll["Global`*"]
tau = 10;
L = 1;
Elastic = 1;
Imoment = 1;
ρ = 1;
S = 1;
sol = NDSolveValue[{D[Elastic*Imoment*D[w[x, t], {x, 2}], {x, 2}] +
S*ρ*D[w[x, t], {t, 2}] == 0, w[x, 0] == x^2/6 (3 - x),
D[w[0, t], {t, 1}] == 0,
D[w[L, t], {t, 2}] == 0 D[w[L, t], {t, 3}] == 0},
w[x, t], {x, 0, L}, {t, 0, tau},
Method -> {"MethodOfLines",
"DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> 100},
"SpatialDiscretization" -> {"TensorProductGrid",
"MaxPoints" -> 100, "MinPoints" -> 100,
"DifferenceOrder" -> 2}}, MaxSteps -> 10^6]
But I can't get the numerical solution of $w(x,t)$, so I can't draw the vibration image of the first 10 seconds.
What can I do to solve this partial differential equation?
D[w[x, t], {t, 1}] == 0 /. t -> 0
is what you should. $\endgroup${w[0, t] == 0, D[w[x, t], {x, 1}] == 0 /. x -> 0}
. I also guess that the right end should be free. I am not 100% sure, but I think the correct boundary conditions for that would be{D[w[x, t], {x, 2}] == 0 /. x -> L, D[w[x, t], {x, 3}] == 0 /. x -> L}
. Other users will know that better than me. $\endgroup$