TITLE: Is there an analytic solution for this partial differential equation?
QUESTION [5 upvotes]: The Fokker-Planck equation for a probability distribution $P(\theta,t)$:
\begin{align}
\frac{\partial P(\theta,t)}{\partial t}=-\frac{\partial}{\partial\theta}\Big[[\sin(k\theta)+f]P(\theta,t)-D\frac{\partial P(\theta,t)}{\partial\theta}\Big].
\end{align}
where $f$, $k$, $D$ are constants, and the initial distribution is a delta function.

REPLY [4 votes]: Starting from Abhishek Halder's derivation of an ODE, we can indeed get a closed form, involving the solutions of the double-confluent Heun equation.
$\phi \left( \theta \right) = 
C_1 {\it HeunD} \left({\frac{2}{kd}},{\frac { \left( 2\,k-4\,\lambda \right) d+{f}^{2}+1}{k^{2}d^{2}}},{\frac {4\,if}{{k}^{2}{d}^{2}}},{\frac { \left( 2k+4\lambda \right) d-{f}^{2}-1}{{k}^{2}{d}^{2}}},{\frac {i}{\tan \left(k\theta/2 \right) }}\right) 
{{\rm e}^{{\frac {1}{kd} \left( -i\sin\left( k\theta/2 \right) \cos \left( k\theta/2 \right) +f
\arctan \left( {\frac {\sin \left( k\theta/2 \right) }{\cos \left( 
k\theta/2 \right) }} \right) - \left( \cos \left( k\theta/2
 \right)  \right) ^{2} \right) }}} +
C_2\,{\it HeunD} \left({-\frac {2}{kd}},{\frac { \left( 2k-4\lambda \right) d+{f}^{2}+1}{{
k}^{2}{d}^{2}}},{\frac {4\,if}{{k}^{2}{d}^{2}}},{\frac { \left( 2k+4
\lambda \right) d-{f}^{2}-1}{{k}^{2}{d}^{2}}},{\frac {i}{\tan
 \left( k\theta/2 \right) }} \right) {{\rm e}^{{\frac {1}{kd}
 \left( i\sin \left( k\theta/2 \right) \cos \left( k\theta/2
 \right) +f\arctan \left( {\frac {\sin \left( k\theta/2 \right) }{
\cos \left( k\theta/2 \right) }} \right) - \left( \cos \left(
k\theta/2 \right)  \right) ^{2} \right) }}}
$
A change of variables, $\theta \mapsto t/k$, can simplify the results a fair bit.somewhat.