I'd apply Dirichlet boundary conditions on both ends, the driven end with the prescribed displacement, the other one with homogeneous b.c., namely:

• $u(0,t) = u_0(t)$ at the driven end in $x=0$
• $u(L,t) = 0$ at the fixed end in $x=L$.

I'm assuming that the string has pre-stress since string without it can't hold compression forces, and that assumptions of small strain and displacement holds.

I'd apply Dirichlet boundary conditions on both ends, the driven end with the prescribed displacement, the other one with homogeneous b.c., namely:

• $u(0,t) = u_0(t)$ at the driven end in $x=0$
• $u(L,t) = 0$ at the fixed end in $x=L$.

I'm assuming that the string has pre-stress since string without it can't hold compression forces, and that assumptions of small strain and displacement holds.

Meaning of the boundary condition In this model,

• Dirichlet boundary conditions model prescribed displacements $u_D(t) = u(x_D,t)$

• Neumann boundary conditions model prescribed axial force $N_N(t) = N(x_N,t)$, being $N(x,t) = EA(x,t) \frac{\partial u}{\partial x}(x,t)$,

  $EA(x_N,t) \frac{\partial u}{\partial x}(x,t) = N_N(t)$

  At a free end (even if it's quite hard to me to think at a free end in a string with the pre-stress needed by the string, not so hard if we're dealing with beam or rod structural elements that hold compression stress as well as tension stress), you can prescribe

  $EA(x_N,t) \frac{\partial u}{\partial x}(x,t) = 0$

  but it should not be the case of your set-up.