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Non-homogeneous Boundary value Problem

  1. Apr 12, 2012 #1
    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[itex]_{t}[/itex][itex]_{t}[/itex](x,t) = a[itex]^{2}[/itex]y[itex]_{x}[/itex][itex]_{x}[/itex](x,t)

    With

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


    Initial condition:
    y(x,0) = 0 [ initial displacement = 0 ]
    y[itex]_{t}[/itex](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!
     
  2. jcsd
  3. Apr 14, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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, [itex]y_{xx}= u_{xx}[/itex] and [itex]y_{tt}= u_{tt}- (f''(t)/AE)x[/itex] so your differential equation becomes
    [tex]u_{tt}- (f''(t)'/AE)x= a^2u_{xx}[/tex]
    the boundary conditions are [itex]u(0, t)= y(0,t)- (f(t)/AE)(0)= 0[/itex], [itex]u_x(L,t)= y_x(L,t)- f''(t)/AE= 0[/itex], and the initial conditions are [tex]u(x, 0)= y(x, 0)- v(x, 0)= -(f(0)/AE)x[/itex], [itex]u_t(x, 0)= y_t(x, 0)- v_t(x, 0)= -(f(0)/AE)x[/itex].
     
    Last edited: Apr 14, 2012
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