MHB Scaled applied forces for an elastic rod

[onie mti]

am having difficulty in understanding this problem and frankly I don't know how to approach it, please assist on how to solve itThey say that an elastic rod is modeled as the half time[0,infinity). initially it is at rest. at the end point x=o, a force f(t) is applied then they give me the scaled system for the motion

u_t(x,t)+ F_x(x,t)=0
F_x(x,t)+u_x(x,t)=0
F(0,t)=f(t);
x>0 t>0

the question is: to solve the problem if the scaled applied force is of the form f(t)=sint

 

[chisigma]

am having difficulty in understanding this problem and frankly I don't know how to approach it, please assist on how to solve itThey say that an elastic rod is modeled as the half time[0,infinity). initially it is at rest. at the end point x=o, a force f(t) is applied then they give me the scaled system for the motion

u_t(x,t)+ F_x(x,t)=0
F_x(x,t)+u_x(x,t)=0
F(0,t)=f(t);
x>0 t>0

the question is: to solve the problem if the scaled applied force is of the form f(t)=sint

Is pratically the same problem You met in...

http://mathhelpboards.com/differential-equations-17/scaled-transport-equation-10626.html

Because $\displaystyle u_{x} + F_{x} = u_{t} + F_{x} = 0$ is also...

$\displaystyle u_{x} - u_{t} = 0\ (1)$

... and the solution of (1) is...

$\displaystyle u(x,t) = f(x + t)\ (2)$

... where $f(*,*) \in C^{1}$. Taking into account the boundary conditions is...

$\displaystyle u(x,t) = \cos (x + t)\ (3)$

Kind regards

$\chi$ $\sigma$