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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.

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# System of PDE's dealing with probability density

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