Results 1 to 3 of 3

Math Help - wave equation

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    44

    wave equation

    I have to find the solution to this problem:
    u_{xx}=u_{tt}+f(x,t)
    u(x,0)=0
    u_t(x,0)=0

    where f(x,t) is a step function:
    1 for |x|<= 1 and 0< t <  1
    0 else

    I know I have to use the duhamel's principle, but I have problem with the integration of f(x,t).

    Can anybody help me please?
    thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by sidi View Post
    I have to find the solution to this problem:
    u_{xx}=u_{tt}+f(x,t)
    u(x,0)=0
    u_t(x,0)=0

    where f(x,t) is a step function:
    1 for |x|<= 1 and 0< t < 1
    0 else

    I know I have to use the duhamel's principle, but I have problem with the integration of f(x,t).

    Can anybody help me please?
    thank you
    So using Duhammel's principle you need to find a \phi(x,t;s)
    Such that \phi_{tt}=\phi_{xx} with initial conditions
    \phi(x,s;s)=0 \text{ and } \phi_{t}(x,s;s)=f(x,s)

    Now by D'lamberts we know that

    \phi(x,t,;s)=\frac{1}{2}\int_{x-(t-s)}^{x+(t-s)}f(z,s)dz

    Now by Duhammel's principle the solution is

    u(x,t)=\frac{1}{2}\int_{0}^{t}\int_{x-(t-s)}^{x+(t-s)}f(z,s)dz ds

    Note that the region of integration depends on the x and t varaibles. Also the forcing function is non zero in the rectangle [-1 ,1] \times [0,1] (In red in the diagram). In this diagram the function will be zero in any of the shaded (blue) area's and the wave will travel in the white strips.. I hope this helps.
    wave equation-domain.jpg
    Last edited by TheEmptySet; March 12th 2010 at 10:08 PM. Reason: explain better
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2009
    Posts
    44
    I still don't know how to find the region of integration....
    On your diagram are there the x and t axis? Where is s?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Wave equation help.
    Posted in the Differential Equations Forum
    Replies: 7
    Last Post: May 30th 2011, 10:58 PM
  2. The Wave Equation
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: January 26th 2010, 01:23 PM
  3. Partial differential equation-wave equation(2)
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: September 6th 2009, 08:54 AM
  4. Partial differential equation-wave equation - dimensional analysis
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: August 28th 2009, 11:39 AM
  5. wave equation
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: February 7th 2009, 08:31 AM

Search Tags


/mathhelpforum @mathhelpforum