Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation

@article{78264,
   author = {Koval, Serhii D and Bihlo, Alexander and Popovych, Roman},
   article_location = {New York (USA)},
   article_number = {5},
   doi = {http://dx.doi.org/10.1017/S0956792523000074},
   keywords = {(1+2)-dimensional ultraparabolic Fokker-Planck equation; complete point-symmetry pseudogroup; Lie symmetry; Lie reductions; exact solutions; Kramers equations},
   language = {eng},
   issn = {0956-7925},
   journal = {European Journal of Applied Mathematics},
   title = {Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation},
   url = {https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/abs/extended-symmetry-analysis-of-remarkable-12dimensional-fokkerplanck-equation/C825941B001CE386DC5A1D96F86CA101},
   volume = {34},
   year = {2023}
}

TY  - JOUR
ID  - 78264
AU  - Koval, Serhii D - Bihlo, Alexander - Popovych, Roman
PY  - 2023
TI  - Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
JF  - European Journal of Applied Mathematics
VL  - 34
IS  - 5
SP  - 1067-1098
EP  - 1067-1098
PB  - Cambridge University Press
SN  - 09567925
KW  - (1+2)-dimensional ultraparabolic Fokker-Planck equation
KW  - complete point-symmetry pseudogroup
KW  - Lie symmetry
KW  - Lie reductions
KW  - exact solutions
KW  - Kramers equations
UR  - https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/abs/extended-symmetry-analysis-of-remarkable-12dimensional-fokkerplanck-equation/C825941B001CE386DC5A1D96F86CA101
N2  - We carry out the extended symmetry analysis of an ultraparabolic Fokker–Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker–Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker–Planck equation using the direct method, analyse its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalised reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker–Planck equation, in particular, those parameterised by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker–Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.
ER  - 

KOVAL, Serhii D, Alexander BIHLO a Roman POPOVYCH. Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation. \textit{European Journal of Applied Mathematics}. New York (USA): Cambridge University Press, 2023, roč.~34, č.~5, s.~1067-1098. ISSN~0956-7925. Dostupné z: https://dx.doi.org/10.1017/S0956792523000074.