atyy said:
If you set m=0 in Eq 9, I think you get the one dimensional heat equation
http://mathworld.wolfram.com/HeatConductionEquation.html
Yes you are right. The problem I'm having is with the boundary conditions; I don't really know what they are. The article you gave me (Gerstein-Mandelbrot) seems to deal with the heat equation at page 52 for when their c is worth 0 (Goel's m if I understood well).
However in Goel, the diffusion equation (eq.9) of page 192 becomes the heat equation indeed and both the eq. 6a and 6b (backward diffusion equation, the Kolmogorov one) also becomes the heat equation.
But P is a function of x, y and t. The 2 PDE's are then ##\frac{\partial P}{\partial t} = \frac{\sigma ^2}{2} \frac{\partial ^2 P}{\partial x^2}## and ##\frac{\partial P}{\partial t} = \frac{\sigma ^2}{2} \frac{\partial ^2 P}{\partial y^2}##. Subject to the boundary conditions of page 192 (8a, 8b and 8c, yet I have still doubts that Goel wrote well the 8b one).
I wanted to kind of "cheat", look up the solution in Goel for the general case m not necessarily 0 and set up m=0 in his solution. However, the resulting solution does not satisfy the diffusion equation.
Here is eq. 10 of page 193 with m=0. ##P(x,y,t)=\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 ]## but as I said, I tried to see if it solves the heat equation and it does not.
I tried with the solution ##\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 ]## but it also fails to solve the heat equation. That's why I'm starting to believe that the solution Goel gives for the general case of m doesn't even work, because it fails for at least m=0.
In the article of Gerstein-Mandelbrot, it gives a solution of ##I(z_0,\tau)=(4\pi )^{-1/2}z_0 \tau ^{-3/2}\exp \{ -\frac{z_0}{4\tau} \}## where I believe G-M's tau is equivalent to Goel's t and G-M's ##z_0## is equivalent to Goel's x-y or something like that, I am not really sure (but the threshold potential B must appear somewhere... maybe in ##z_0##?).
P.S.:Also notice that apparently for the 2 PDE's, the second derivative of P with respect to x is equal to the second derivative of P with respect to y; at least if Goel's solution works (which does not seem to do, but maybe the real solution has this property). So one could just add both equation to fall over a single PDE. Guess what this PDE is? The heat equation for either x or y. That is, ##\frac{\sigma ^2}{2} \frac{\partial P ^2}{\partial x^2} = \frac{\partial P}{\partial t}##. (I have started a thread on this topic at
https://www.physicsforums.com/showthread.php?p=4461916#post4461916).
1 more comment: by looking at either G-M's solution or Goel, it doesn't seem like the solution is separable. So separation of variables might not be the way to go. That may due to the weird boundary conditions, I'm not really sure.
