The authors investigate the Fokker-Planck equation (FPE). This equation describes many mathematical models of different phenomena and processes in physics – mainly Brownian motion, chemistry, biology, finance, etc. In the one-dimensional case it has the form: \[ (\ast ) \;\;\;\;\partial_t W \;= \;P W, \] where \(W\equiv W(x, t)\), \(P\) is a differential operator defined by \(P\equiv [-\partial_xD^{(1)}(x,t)+ \partial_{xx}D^{(2)}(x,t)]W(x, t)\), \(W (x, t)\) is the probability distribution function normalized by \(\int\limits_{\Omega }W (x, t)dx=1\), \(\Omega\) is some domain, \(t\geq 0\), \(D^{(1)}(x,t)\) is the drift coefficient, and \(D^{(2)}(x,t)\) is the diffusion coefficient. It is known that \(D^{(1)}(x,t)\) represents the external force acting on some particle, and \(D^{(2)}(x,t)\) describes the effect of fluctuation. The similarity solutions (SS) of the FPE would be of interest. These solutions are possible if the FPE is invariant under the scale transformation \(\overline{x}=\varepsilon^ax\), \(\overline{t}= \varepsilon^bt\) (\(\varepsilon >0\)), \(a,b\in\mathbb{R}\). Then the FPE reduces to the form \[ \rho_2(z)y^{\prime\prime }(z)+[2\rho_2^{\prime }(z)- \rho_1(z)+\alpha z]y^{\prime }(z)+ [\rho_2^{\prime\prime }(z)-\rho_1^{\prime }(z)+\alpha ]y(z)=0,\tag{\(**\)} \] where \(y^{\prime }\) is the derivative w.r.t. \(z\), \(z\equiv x/t^{\alpha }\), \(\alpha =a/b\) (\(a,b\neq 0\)), and \(W(x,t)= t^{\alpha c/a}y(z)\); here \(\rho_1 (z)\) and \(\rho_2 (z)\) are scale invariant functions of \(z\). For the considered problem, the authors take impenetrable boundaries. In this case, the probability density and the associated probability current density must vanish on the boundary. Thus the solution of (\(\ast\ast \)) takes the form \[ y(z)=\biggl( C^{\prime }+ C\int\limits_{z_0}^{z}\exp\left(-\int\limits_{s_0}^{s}f(r)dr\right) \rho_2^{-1}(s) ds \biggr)\exp\left(\int\limits_{z_0}^{z}f(r)dr\right)dr ,\tag{\(***\)} \] where \(f(r)\equiv [\rho_1(r)-\rho_2^{\prime }(r)-\alpha r]\rho_2^{-1}(r)\), \(\rho_2(r)\neq 0\), and \(C\), \(C^{\prime }\) are integration constants. Here there are two interesting cases for \(C\neq 0\) and \(C=0\). Further, set \(\rho_1= \lambda z-\mu \) and \(\rho_2=\sigma \), where \(\lambda ,\mu ,\sigma\in\mathbb{R}\). Then the authors discuss the problem for \(\lambda =\alpha \) and \(\lambda \neq \alpha \). It follows from the above stated cases \(C\neq 0\) and \(C=0\), that any two sets of \(\{\rho_1 (z), \rho_2 (z)\}\), and \(\{\tilde{\rho}_1 (z), \tilde{\rho}_2 (z)\}\) that give the same \(f(z)\) define equivalent FPEs as the probability distribution functions are exactly the same. Thus, a general method for obtaining exact solutions of FPEs is established.
Furthermore, it is shown that the class of exact solutions of the FPEs can be extended by exploiting certain properties of the general formula presented in the recent papers by C. L. Ho [J. Math. Phys. 54, No. 4, 041501, 8 p. (2013; Zbl 1292.82029); Ann. Phys. 326, No. 4, 797–807 (2011; Zbl 1213.81125)].
The FPEs related to deformed radial oscillator potential and exceptional Laguerre polynomials, and also to deformed Poschl-Teller potential and exceptional Jacobi polynomials are discussed as well.