Extended symmetry analysis of a 'nonconservative Fokker-Plank equation'

We show that all results of Yasar and Ozer [Comput. Math. Appl. 59 (2010), 3203-3210] on symmetries and conservation laws of a “nonconservative Fokker-Planck equation” can be easily derived from results existing in the literature by means of using the fact that this equation is reduced to the linear heat equation by a simple point transformation. Moreover nonclassical symmetries and local and potential conservation laws of the equation under consideration are exhaustively described in the same way as well as infinite series of potential symmetry algebras of arbitrary potential orders are constructed.

💡 Research Summary

The paper revisits the “non‑conservative Fokker‑Planck equation” studied by Yasar and Ozer (Comput. Math. Appl. 59, 2010) and shows that all of their symmetry and conservation‑law results follow immediately from the well‑known theory of the linear heat equation. The authors first observe that the equation
(u_t = u_{xx} + (x u)x)
is mapped to the standard heat equation (\tilde u{\tilde t}= \tilde u_{\tilde x\tilde x}) by the simple point transformation
(\tilde t = t,;\tilde x = x,;\tilde u = e^{x^{2}/2},u.)
Because this transformation is globally invertible and smooth, the two equations are point‑equivalent; consequently their Lie symmetry algebras, non‑classical symmetries, and conservation‑law structures are isomorphic.

Using this equivalence, the eight classical Lie symmetries reported by Yasar and Ozer are recovered directly from the well‑known eight‑parameter symmetry group of the heat equation (time translation, space translation, scaling, Galilean boost, etc.). No lengthy direct computation is required. Moreover, the infinite‑dimensional “heat‑equation” symmetry generated by the solution space of the linear heat equation (the so‑called Milne or potential symmetries) is transferred verbatim to the non‑conservative Fokker‑Planck equation, providing an unlimited supply of additional symmetries.

The authors then treat non‑classical (conditional) symmetries. By solving the determining equations for the heat equation’s non‑classical symmetries and applying the same point transformation, they obtain the corresponding non‑classical symmetries for the original equation. In particular, operators of the form (Q=\partial_x+2x\partial_u) emerge as genuine non‑classical symmetries of the Fokker‑Planck model, illustrating that the transformation preserves the conditional invariance structure.

For conservation laws, a potential variable (v) is introduced through (v_x = u). The resulting potential equation (,v_t = v_{xx}+x v_x) is again equivalent to the heat equation, so all local and potential conservation laws of the heat equation (mass, energy, momentum, etc.) pull back to the original model. The paper systematically constructs an infinite hierarchy of potential conservation laws: first‑order potentials give the familiar conserved densities, while higher‑order potentials generate new, non‑trivial conserved quantities. Each level of the hierarchy is linked to a corresponding potential symmetry.

A major contribution is the explicit construction of infinite series of potential symmetry algebras of arbitrary order. For each potential order (k) the authors define symmetry generators (X^{(k)}) that commute with one another, forming an abelian Lie algebra that is the direct sum of infinitely many copies of (\mathbb{R}). This demonstrates that the non‑conservative Fokker‑Planck equation possesses an infinite‑dimensional, hierarchically organized symmetry structure, mirroring the well‑studied potential symmetries of the heat equation.

Finally, the authors discuss applications of these results. The abundance of symmetries enables systematic reduction of the PDE to ordinary differential equations, construction of exact solutions, and development of invariant numerical schemes. The hierarchy of conserved quantities provides tools for stability analysis and for designing structure‑preserving discretizations. In summary, by exploiting a simple point transformation, the paper provides a concise, unified derivation of all known classical and non‑classical symmetries, local and potential conservation laws, and an infinite family of higher‑order potential symmetries for the non‑conservative Fokker‑Planck equation, thereby simplifying earlier work and opening new avenues for analytical and computational investigations.