In this paper, previous considerations concerning differential equations involving a para-Grassmann variable are extended by allowing two independent ordinary variables. For the differential equation of second order, the structure of its solutions is determined and several examples representing analogs of well-known second order differential equations are considered in detail. As a warm-up, the differential equation of first order is treated.
REFERENCES
1.
J.
Schwinger
, “The theory of quantized fields. IV
,” Phys. Rev.
92
, 1283
–1299
(1953
).2.
J. L.
Martin
, “The Feynman principle for a Fermi system
,” Proc. R. Soc. A
251
, 543
–549
(1959
).3.
The Supersymmetric World. The Beginning of the Theory
, edited by G.
Kane
and M.
Shifman
(World Scientific Publishing Co., Inc.
, River Edge, NJ
, 2000
).4.
M.
Légaré
, “On a set of Grassmann-valued extensions of systems of ordinary differential equations
,” arXiv:1903.12051 (2019
).5.
J.
Monterde
and O. A.
Sánchez-Valenzuela
, “Existence and uniqueness of solutions to superdifferential equations
,” J. Geom. Phys.
10
, 315
–343
(1993
).6.
H. S.
Green
, “A generalized method of field quantization
,” Phys. Rev.
90
, 270
–273
(1953
).7.
A. J.
Kálnay
, “A note on Grassmann algebras
,” Rep. Math. Phys.
9
, 9
–13
(1976
).8.
Y.
Ohnuki
and S.
Kamefuchi
, Quantum Field Theory and Parastatistics
(Springer
, Berlin
, 1982
).9.
A. T.
Filippov
, A. P.
Isaev
, and A. B.
Kurdikov
, “Para-Grassman analysis and quantum groups
,” Mod. Phys. Lett. A
7
, 2129
–2141
(1992
).10.
A. T.
Filippov
, A. P.
Isaev
, and A. B.
Kurdikov
, “Para-Grassman extensions of the Virasoro algebra
,” Int. J. Mod. Phys. A
8
, 4973
–5003
(1993
).11.
A. T.
Filippov
, A. P.
Isaev
, and A. B.
Kurdikov
, “Para-Grassman differential calculus
,” Theor. Math. Phys.
94
, 150
–165
(1993
).12.
C.
Ahn
, D.
Bernard
, and A.
LeClair
, “Fractional supersymmetries in perturbed coset CFTs and integrable soliton theory
,” Nucl. Phys. B
346
, 409
–439
(1990
).13.
S.
Durand
, “Fractional superspace formulation of generalized super-Virasoro algebras
,” Mod. Phys. Lett.
7
, 2905
–2912
(1992
).14.
S.
Majid
and M. J.
Rodríguez-Plaza
, “Random walk and the heat equation on superspace and anyspace
,” J. Math. Phys.
35
, 3753
–3760
(1994
).15.
D.
Ellinas
and I.
Tsohantjis
, “Brownian motion on a smash line
,” J. Nonlinear Math. Phys.
8
, 100
–105
(2001
).16.
D.
Ellinas
and I.
Tsohantjis
, “Random walk and diffusion on a smash line algebra
,” Infin. Dimens. Anal. Quantum Probab. Relat. Top.
6
, 245
–264
(2003
).17.
A. P.
Isaev
, Z.
Popowicz
, and O.
Santillan
, “Generalized Grassmann algebras and its connection to the extended supersymmetric models
,” arXiv:hep-th/0110246 (2001
).18.
N.
Alvarez-Moraga
, “Coherent and squeezed states of quantum Heisenberg algebras
,” J. Phys. A: Math. Gen.
38
, 2375
–2398
(2005
).19.
M.
Schork
, “Some algebraical, combinatorial and analytical properties of paragrassmann variables
,” Int. J. Mod. Phys. A
20
, 4797
–4819
(2005
).20.
T.
Mansour
and M.
Schork
, “On linear differential equations involving a paragrassmann variable
,” Symmetry, Integrability Geom.: Methods Appl.
5
, 073
(2009
).21.
T.
Mansour
and M.
Schork
, “On linear differential equations with variable coefficients involving a para-Grassmann variable
,” J. Math. Phys.
51
, 043512
(2010
).22.
T.
Mansour
and R.
Rayan
, “On Cauchy-Euler’s differential equation involving a para-Grassmann variable
,” J. Math. Phys.
59
, 103508
(2018
).23.
24.
V. N.
Shander
, “Vector fields and differential equations on supermanifolds
,” Funct. Anal. Appl.
14
, 160
–162
(1980
).25.
E.
Witten
, “An interpretation of classical Yang-Mills theory
,” Phys. Lett. B
77
, 394
–398
(1978
).26.
S.
Li
, E.
Shemyakova
, and T.
Voronov
, “Differential operators on the superline, Berezinians, and Darboux transformations
,” Lett. Math. Phys.
107
, 1689
–1714
(2017
).27.
G.
Salgado
and J. A.
Vallejo-Rodríguez
, “The meaning of time and covariant superderivatives in supermechanics
,” Adv. Math. Phys.
21
, 987524
(2009
).© 2020 Author(s).
2020
Author(s)
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