# System of PDE's dealing with probability density

1. Aug 1, 2013

### fluidistic

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):
$$\frac{1}{\sqrt{2\pi}\sigma} \left [ \exp \{ -\frac{(x-y)^2}{2\sigma ^2 t} \} - \exp \{ - \frac{(x+y-2B)^2}{2\sigma ^2 t} \} \right ]$$ (1st one)
$$\frac{1}{\sqrt{2\pi t}\sigma} \left [ \exp \{ -\frac{(x-y)^2}{2\sigma ^2 t} \} - \exp \{ - \frac{(x+y-2B)^2}{2\sigma ^2 t} \} \right ]$$ (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. Aug 2, 2013

### fluidistic

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
3. Aug 4, 2013

### fluidistic

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.