Non-homogeneous Boundary value Problem

[primaryd]

Hello,

I am trying to solve a vibration problem analytically but I don't understand how to implement the non-homogeneous boundary conditions.

The problem is defined as below:

y$_{t}$$_{t}$(x,t) = a$^{2}$y$_{x}$$_{x}$(x,t)

With

Boundary conditions:
y(0,t) = 0 [ fixed at zero ]
y$_{x}$(L,t) = $\frac{f(t)}{AE}$ [ Force f(t) at free end x=L ]


Initial condition:
y(x,0) = 0 [ initial displacement = 0 ]
y$_{t}$(x,0) = 0 [ initial velocity = 0 ]



My first question > Is the second BC in it's correct form? I am trying to model a time-dependent force at x=L

second question > How is this problem solved? I tried separation of variables and that didn't work.


any help / resources will be appreciated!

Thanks!

 

[HallsofIvy]

Then "homogenize" your boundary conditions.

The function v(x,t)= (f(t)/AE)x satisfies both v(0, t)= 0 and v_x(L, t)= f(t)/AE.

Define u(x,t)= y(x,t)- v(x,t) so that y(x,t)= u(x,t)- (f(t)/AE)x, $y_{xx}= u_{xx}$ and $y_{tt}= u_{tt}- (f''(t)/AE)x$ so your differential equation becomes
$$u_{tt}- (f''(t)'/AE)x= a^2u_{xx}$$
the boundary conditions are $u(0, t)= y(0,t)- (f(t)/AE)(0)= 0$, $u_x(L,t)= y_x(L,t)- f''(t)/AE= 0$, and the initial conditions are [tex]u(x, 0)= y(x, 0)- v(x, 0)= -(f(0)/AE)x[/itex], $u_t(x, 0)= y_t(x, 0)- v_t(x, 0)= -(f(0)/AE)x$.