when we use finite difference for descritizing the stationary version of Fokker Planck equation, the result will be a homogeneous equation like A*P=0, where P is value of probability at the meshes, this equation has a trivial solution P=0, so how should we solve Stationary version of FPE with FD?
Not a specific answer to your question but a system like A.x=0 has the trivial solution only if det(A) 0. If it is det(A) =0 the solution is indetermined.
However, you should check the full problem with the prescribed BCs.
Dear Ali,
The point is that you look for a PDF solution whose integral on the domain is equal to 1 , if you have flux zero boundary conditions. If you have homogeneous b.c. then zero is the solution of your problem.
Dear Dr. Alfio Borzì
Even with zero flux boundary condition we get again A*P=0,
do you mean that I have to add Normalization condition (integral of the solution on the domain=1) to my system of equations (in this case the number of equations=number of unknowns+1 )?
" Even with zero flux boundary condition we get again A*P=0 "
Again, try to check the condition on det(A). It is a classical problem in elliptic equations with homogeneous Neumann BC.s that the solution is determined up to an arbitrary function fo time. An example it the pressure equation in incompressible flows.
The non-trivial solution is obtained for non-homogeneous Dirichlet BCs that change the source term.
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