What method can be used for solving this fokker Planck equation and how? Let's have this equation:
\begin{equation}
\frac{\partial p(x,t)}{\partial t} = - a \frac{\partial p(x,t)}{\partial x} + \frac{1}{2} b \frac{\partial^2 p(x,t)}{\partial x^2}
\end{equation}
a and b are constant and $0<x<L$.
boundary conditions 
\begin{equation}
p(0,t)= f(t) \qquad 
p(L,t)= g(t)
\end{equation}
I need to know which method can work? I have no idea how to deal with this boundary value problem 
Many thanks for your help in advance
 A: Your PDE is second order linear parabolic and homogeneous PDE which can be written as:
$$ \partial_t p + a \nabla p = D \, \nabla^2 p, \quad p = p(x,t), \quad 0<x<L, \quad t >0,$$ 
where $a$ and $D = b/2$ are constants. Since you have a bounded domain on $x$ and your equation is linear, separation of variables can do the trick here. Assume you are given an initial condition such that $p(x,0) = p_0(x)$ and you define the non-dimensional variable $\xi = x/L$ (you can do the same with $t$ and $p$) so you have:
$$\partial_t p + k \partial_\xi p = \alpha \partial_{\xi}^2p, \quad p = p(\xi,t), \quad 0  < \xi < 1, \quad t > 0, \tag{1} $$ with $\alpha = D/L^2$, $p(0,t) = f(t), \ p(1,t) = g(t)$ and $p(\xi,0) = p_0(L \xi) \equiv \tilde{p}_0(\xi)$. We cannot proceed further with separation of variables because of the inhomogenous boundary conditions, so we have to make use of the superposition principle defining:
$$p = u+v,$$
where $u$ satisfies homogenous boundary conditions and $v$ is the simplest function that satisfies $v(0,t) = f(t)$ and $v(1,t) = g(t)$. Set $v(x,t) = A(t) x+ B(t)$ and solve for $A$ and $B$. Write the problem for $u$:
$$\partial_t u + k \partial_\xi u  = \alpha \partial_\xi^2 u \underbrace{ -\partial_tv-k\partial_\xi v + \alpha\partial^2_\xi v}_{W(x,t)}, \tag{2}$$ the known term $W$ makes your equation for $u$ non-homogenous so you must solve the following new problem, which I create (Fredholm's alternative):
$$ \theta_t + k \theta_x = \alpha \theta_{xx}, $$
together with $\theta(0,t) = \theta(1,t) = 0$ and I don't care about the initial conditions (which for $u$ turn to be $u(x,0) = p(x,0)-v(x,0)$).
Set now: 
$$\theta(\xi,t) = P(\xi)Q(t), \quad P \neq 0, \ Q \neq 0$$ and substitute back in the PDE for $\theta$ to have:
$$PQ' + k P' Q = \alpha P'' \implies \frac{Q'}{Q} = \frac{\alpha P'' - k P'}{P} = \lambda \in \mathbb{R}^- \cup \{0\} \mathbb{R}^+. \tag{3} $$
Eq. $(3)$ yields to two different problems for $P$ and $Q$. Solve for $P$ and find the so-called eigenfunctions and eigenfunctions associated to the homogenous Dirichlet boundary conditions $P(0) = P(1) = 0$. Expand then the solution $u$ in terms of the eigenfunctions (Sturm-Liouville theory): 
$$u = \sum^\infty_{n} Q_n(t) P_n(x),$$ and find the Fourier coefficients $Q_n(t)$ by introducing this in eq. $(2)$ and making use of the property of orthogonality of $P_n(x)$, i.e.,
$$ \int^1_0 P_n(x) P_m(x) r(x) \, \mathrm{d} x = \delta_{nm}, $$ where $\delta$ is the Kronecker delta and $r(x)$ is the weight function of the self-adjoint problem for $P$ (which is $r(x) = 1$ here).
Hope you find this helpful. Cheers!
A: Is Eq 3 correct?
There is a 'constructive' solution to this where you in-cooperate elements of physics. Use de Broglie's theory and Kac-Moody algebra.
A: Perhaps this solution isn't as easy on the eye but it is possible, in my view, to make sense of it from a physics standpoint. We write a continuity equation;
$$
\frac{\partial}{\partial t}P(x,t) + \frac{\partial}{\partial x}J(x,t) = 0 \,
$$
and allow Fick's law of diffusion to be expressed as $ J = J(x,t) = y^{\partial_{_{x}}}P(x,t)$. 
By substitution...
$$
\frac{\partial}{\partial t}P(x,t) = -\frac{\partial y^{\partial_{_{x}}}}{\partial x}P(x,t)  \,
$$
Equating the RHS of this diffusion equation to the RHS of your expression yields an answer where $ P(x,t) = X(x)T(t)$. If you can give me a job I'd be happy to explain further.
