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Homework Help: Kramers equation

  1. Apr 4, 2008 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations

    Find a solution to the PDE
    [tex] B P_{vv} - v P_x + (A v - F(x)) P_v + A P = 0 [/tex]
    where A and B are constants, P = P(x,v)

    3. The attempt at a solution

    I have no idea how to even guess a solution to this.
     
  2. jcsd
  3. Apr 4, 2008 #2
    I would suggest separation of variables, but that F(x) seems to screw things up a bit...
     
  4. Apr 5, 2008 #3
    Assumeing [tex]F(x) = -V_x[/tex] and makeing an ansatz
    [tex] P(x,v) = C_1 exp(C_2 x + C_3 V(x) + C_4 v + C_5 v^2) [/tex]
    I get these conditions on the constants C_i
    [tex](2BC_5+A)C_5=0[/tex]
    [tex](4BC_5+A)C_4-C_2+(C_3+2C_5)F(x)=0[/tex]
    [tex](BC_4-F(x))C_4+2BC_5=0[/tex]
    in the second eq we can put C_3=-2C_5 and get rid of F(x) there, but it's still in the third eq. Suggestions?
     
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