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System of PDE's dealing with probability density

  1. Aug 1, 2013 #1

    fluidistic

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    Hi guys,
    I must solve, I believe, 2 simultaneous PDE's where the unknown function that I must find represent a conditional density of probability. It is a function of 3 variables, namely x, y and t. So it is P(x,y,t).
    P(x|y,t) means that the density of probability of a certain function (called the potential function) has the value x at time t, knowing that it had the value y at time ##t_0=0##. Personally I prefer my own notation ##P(x,t|y,0)##.
    The 2 PDE's are:
    (1)##\frac{\partial P}{\partial t} = \frac{\sigma ^2}{2} \frac{\partial ^2 P}{\partial x^2}##
    (2)##\frac{\partial P}{\partial t}=\frac{\sigma ^2}{2} \frac{\partial ^2 P}{\partial y^2}##.
    The boundary conditions are 3 apparently. Namely that -infinity is a reflection point and B is an absorbing point.
    Mathematically they are ##P(-\infty, t |y,0)=0##, ##P(x,0|y,0)=\delta(x-y)## and ##P(x,t|B,0)=0## (where I'm not sure on the 3rd one, as you can see in https://www.physicsforums.com/showthread.php?t=703730).


    The solution is supposed to be either one of these 2 functions (there's a typo in the book and I think the second one should be the correct one but I'm not 100% sure):
    [tex]\frac{1}{\sqrt{2\pi}\sigma} \left [ \exp \{ -\frac{(x-y)^2}{2\sigma ^2 t} \} - \exp \{ - \frac{(x+y-2B)^2}{2\sigma ^2 t} \} \right ][/tex] (1st one)
    [tex]\frac{1}{\sqrt{2\pi t}\sigma} \left [ \exp \{ -\frac{(x-y)^2}{2\sigma ^2 t} \} - \exp \{ - \frac{(x+y-2B)^2}{2\sigma ^2 t} \} \right ][/tex] (2nd one, I think it's the correct solution).

    Now I want to derive the correct solution. But I'm having very hard on how to apply the boundary conditions, especially because the value for y is when t=0 while the value for x is when t=t and there's only 1 t in the equations.

    My first step anyway was to sum up the 2 PDE's. I fall over the heat equation in Cartesian coordinates: ##\frac{\partial P}{\partial t} = \frac{\sigma ^2}{4} \left ( \frac{\partial ^2 P}{\partial x^2} + \frac{\partial ^2 P}{\partial y^2} \right )##. By looking at the solution, separation of variables doesn't look like the way to go. Also I don't know how to apply the boundary conditions... any help is appreciated.
     
  2. jcsd
  3. Aug 2, 2013 #2

    fluidistic

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    Actually none of the answers given in the book satisfy the diffusion equation...
    In other words, if ##P(x,y,t)=\frac{1}{\sqrt{2\pi t}\sigma} \left [ \exp \{ -\frac{(x-y)^2}{2\sigma ^2 t} \} - \exp \{ - \frac{(x+y-2B)^2}{2\sigma ^2 t} \} \right ]##, then I have showed that ##\frac{\sigma ^2}{2} \frac{\partial P ^2}{\partial x^2} \neq \frac{\partial P}{\partial t}##. Same for the other possible solution. None work.
    Any help to solve the diffusion equation with these strange boundary conditions is welcome.


    Edit: NEVERMIND! The 2nd answer works!!!
    I still have to figure out how to show that the boundary conditions are satisfied, but at least the function given indeed satisfies the diffusion equation. Phew!
     
    Last edited: Aug 2, 2013
  4. Aug 4, 2013 #3

    fluidistic

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    Problem solved. I've finally showed that the 2 boundary conditions as well as the initial conditions are satisfied. The book mistakenly called the 3 eq. boundary conditions though.
     
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