Solving Kramers Equation PDE - Find a Solution

  • Thread starter Thread starter WarnK
  • Start date Start date
WarnK
Messages
30
Reaction score
0

Homework Statement


Homework Equations



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

The Attempt at a Solution



I have no idea how to even guess a solution to this.
 
Physics news on Phys.org
I would suggest separation of variables, but that F(x) seems to screw things up a bit...
 
Assumeing F(x) = -V_x and makeing an ansatz
P(x,v) = C_1 exp(C_2 x + C_3 V(x) + C_4 v + C_5 v^2)
I get these conditions on the constants C_i
(2BC_5+A)C_5=0
(4BC_5+A)C_4-C_2+(C_3+2C_5)F(x)=0
(BC_4-F(x))C_4+2BC_5=0
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?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Solving modified heat equation
Replies
5
Views
1K
PDE and the separation of variables
Replies
11
Views
1K
Solve the given first order Partial differential equation.
Replies
1
Views
874
How should I show that all solutions of this equation are bounded?
Replies
4
Views
2K
How should I show that ## B ## is given by the solution of this?
Replies
2
Views
1K
Solving a Second Order Non-Linear PDE with Undetermined Coefficients
Replies
2
Views
1K
Finding Integer Solutions to Polynomial Equations: Can it be Done Easily?
Replies
9
Views
2K
Why is this not a general solution to this nonlinear DE?
Replies
5
Views
1K
Proving a formula for # of edges
Replies
8
Views
2K
Given unit impulse response, obtain differential equation
Replies
7
Views
1K

Hot Threads

Recent Insights

Back
Top