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Kramers equation

  • Thread starter WarnK
  • Start date
31
0
1. Homework Statement
2. Homework 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.
 

Answers and Replies

743
1
I would suggest separation of variables, but that F(x) seems to screw things up a bit...
 
31
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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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