User:TheLarry/Notebook/Larrys Notebook/2010/08/11

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This is the solution to the smoluchowski equation with a wlc force acting on it at t=.06 seconds.

I talked to Igor asking how to solve the smoluchowski equation with a worm like chain force on it. He said i should try a method called finite derivatives. It is a fairly simple concept where you replace the derivatives with their newtonian definition except instead of letting h go to zero you make it very small. This makes the pde a function of only the formula you are looking for.

Using this i can solve the equation.

Interestingly is that the solution leans forward a bit. This is because of the first derivative of the worm like chain. It is positive for x values greater than 0 and negative for values less than zero. When i made both sides of zero positive, I lost this lean.

I still have a problem using finite derivatives. If I allow the time to go a bit past .06 seconds, the function becomes unstable. This is because of the values i chose for delta x and delta t. I can (hopefully) get rid of this problem by switching from FTCS to another scheme possibly Frank-Nicolson. The advantage of FTCS is that it is simpler to solve. but this instability is a pain in the ass so i should probably just change it.