Finding exact solution to initial boundary value problem

[Superman]

Hi,

Im going to find the exact solution of the attached IBVP. Im not sure how to solve it. Mabye a Fourer transform will work out.

If some one have a hint, I would appreciate it!

 

[Superman]

Update:

I got the solution, with the "method of characterstic".

 

[topsquark]

Update:

I got the solution, with the "method of characterstic".

Care to share? It might help someone else as well.

-Dan

 

[Superman]

Im going out now, but can share it later this evning!

 

[Superman]

Solution:

We have that the change of u wrt t is:

[MATH]du/dt = u_x dx/dt + u_t [/MATH].

If we compare this to our problem equation we have that

[MATH]du/dt = 0; dx/dt = 2 [/MATH],

wich means that [MATH]du/dt[/MATH] is constant along curves given by [MATH]dx/dt = 2[/MATH].

These curves are given by: [MATH]x = 2t + x_0[/MATH] (1)

By using initial condition, we know the value of u along the x - axis (where t = 0), and hence along all curves (for all t) given by equation (1).

Hence, the solution is

[MATH]u(x,t) = f(x-2t) = sin(\pi(x-2t))[/MATH].