In this paper, we study a new generalization of the kinetic equation emerging in run-and-tumble models [see, e.g., Angelani et al., J. Stat. Phys. 191, 129 (2024) for a time-fractional version of the kinetic equation]. We show that this generalization leads to a wide class of generalized fractional kinetic (GFK) and telegraph-type equations that depend on two (or three) parameters. We provide an explicit expression of the solution in the Laplace domain and show that, for a particular choice of the parameters, the fundamental solution of the GFK equation can be interpreted as the probability density function of a stochastic process obtained by a suitable transformation of the inverse of a subordinator. Then, we discuss some particularly interesting cases, such as generalized telegraph models, fractional diffusion equations involving higher order time derivatives, and fractional integral equations.
REFERENCES
1.
F. J.
Sevilla
, G.
Chacón-Acosta
, and T.
Sandev
, “Anomalous diffusion of self-propelled particles
,” J. Phys. A: Math. Theor.
57
, 335004
(2024
). 2.
L.
Angelani
, A.
De Gregorio
, R.
Garra
, and F.
Iafrate
, “Anomalous random flights and time-fractional run-and-tumble equations
,” J. Stat. Phys.
191
, 129
(2024
).3.
4.
M. J.
Schnitzer
, “Theory of continuum random walks and application to chemotaxis
,” Phys. Rev. E
48
, 2553
–2568
(1993
). 5.
E. W.
Montroll
and G. H.
Weiss
, “Random walks on lattices. II
,” J. Math. Phys.
6
, 167
–181
(1965
). 6.
J.
Masoliver
, K.
Lindenberg
, and G. H.
Weiss
, “A continuous-time generalization of the persistent random walk
,” Phys. A: Stat. Mech. Appl.
157
, 891
–898
(1989
). 7.
J.
Klafter
and I. M.
Sokolov
, First Steps in Random Walks: From Tools to Applications
(OUP
, Oxford
, 2011
).8.
J.
Masoliver
and K.
Lindenberg
, “Continuous time persistent random walk: A review and some generalizations
,” Eur. Phys. J. B
90
, 1
–13
(2017
). 9.
K.
Gorska
, F.
Sevilla
, G.
Chacon-Acosta
, and T.
Sandev
, “Fractional telegrapher’s equation under resetting: Non-equilibrium stationary states and first-passage times
,” Entropy
26
, 665
(2024
). 10.
L.
Angelani
and R.
Garra
, “On fractional Cattaneo equation with partially reflecting boundaries
,” J. Phys. A: Math. Theor.
53
, 085204
(2020
). 11.
A.
Compte
and R.
Metzler
, “The generalized Cattaneo equation for the description of anomalous transport processes
,” J. Phys. A: Math. Gen.
30
, 7277
(1997
). 12.
R. L.
Schilling
, R.
Song
, and Z.
Vondracek
, Bernstein Functions
(De Gruyter
, Berlin
, 2012
).13.
M. M.
Meerschaert
and H.-P.
Scheffler
, “Triangular array limits for continuous time random walks
,” Stoch. Process. Their Appl.
118
, 1606
–1633
(2008
). 14.
G. H.
Weiss
, “Some applications of persistent random walks and the telegrapher’s equation
,” Phys. A: Stat. Mech. Appl.
311
, 381
–410
(2002
). 15.
K.
Martens
, L.
Angelani
, R.
Di Leonardo
, and L.
Bocquet
, “Probability distributions for the run-and-tumble bacterial dynamics: An analogy to the Lorentz model
,” Eur. Phys. J. E
35
, 1
–6
(2012
). 16.
A. A.
Kilbas
, H. M.
Srivastava
, and J. J.
Trujillo
, Theory and Applications of Fractional Differential Equations
(Elsevier
, 2006
), Vol. 204.17.
F.
Mainardi
, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models
, 2nd ed. (World Scientific
, Singapore
, 2022
).18.
M. M.
Meerschaert
, E.
Nane
, and P.
Vellaisamy
, “Inverse subordinators and time fractional equations,” Handbook of Fractional Calculus with Applications (De Gruyter, 2019), Vol. 1.19.
F.
Cinque
and E.
Orsingher
, “Analysis of fractional Cauchy problems with some probabilistic applications
,” J. Math. Anal. Appl.
536
, 128188
(2024
). 20.
E.
Awad
, “Dual-phase-lag in the balance: Sufficiency bounds for the class of Jeffreys’ equations to furnish physical solutions
,” Int. J. Heat Mass Transf.
158
, 119742
(2020
). 21.
E.
Awad
, T.
Sandev
, R.
Metzler
, and A.
Chechkin
, “From continuous-time random walks to the fractional jeffreys equation: Solution and properties
,” Int. J. Heat Mass Transf.
181
, 121839
(2021
). 22.
C.
Cattaneo
, “Sur une forme de l’equation de la chaleur eliminant la paradoxe d’une propagation instantantee
,” C. R. Acad. Sci.
247
, 431
–433
(1958
).23.
O. G.
Bakunin
, “Mysteries of diffusion and labyrinths of destiny
,” Phys.-Usp.
46
, 309
(2003
). 24.
P.
Vernotte
, “Les paradoxes de la theorie continue de l’equation de la chaleur
,” C. R. Acad. Sci.
246
, 3154
–3155
(1958
).25.
K.
Gorska
, A.
Horzela
, E. K.
Lenzi
, G.
Pagnini
, and T.
Sandev
, “Generalized Cattaneo (telegrapher’s) equations in modeling anomalous diffusion phenomena
,” Phys. Rev. E
102
, 022128
(2020
). 26.
D.
Applebaum
, Lévy Processes and Stochastic Calculus
(Cambridge University Press
, 2009
).27.
G.
Ascione
, M.
Savov
, and B.
Toaldo
, “Regularity and asymptotics of densities of inverse subordinators,” Trans. Lond. Math. Soc.
11
(1), e70004
(2024
). 28.
F.
Mainardi
, “The fundamental solutions for the fractional diffusion-wave equation
,” Appl. Math. Lett.
9
, 23
–28
(1996
). 29.
M. M.
Meerschaert
, R. L.
Schilling
, and A.
Sikorskii
, “Stochastic solutions for fractional wave equations
,” Nonlinear Dyn.
80
, 1685
–1695
(2015
). 30.
Y.
Fujita
, “Cauchy problems of fractional order and stable processes
,” Jpn. J. Appl. Math.
7
, 459
–476
(1990
). 31.
E.
Orsingher
and L.
Beghin
, “Time-fractional telegraph equations and telegraph processes with brownian time
,” Probab. Theory Relat. Fields
128
, 141
–160
(2004
). 32.
R.
Gorenflo
, A. A.
Kilbas
, F.
Mainardi
, and S. V.
Rogosin
, Mittag-Leffler Functions, Related Topics and Applications
(Springer
, 2020
).33.
O.
Vilk
, E.
Aghion
, T.
Avgar
, C.
Beta
, O.
Nagel
, A.
Sabri
, R.
Sarfati
, D. K.
Schwartz
, M.
Weiss
, D.
Krapf
, R.
Nathan
, R.
Metzler
, and M.
Assaf
, “Unravelling the origins of anomalous diffusion: From molecules to migrating storks
,” Phys. Rev. Res.
4
, 033055
(2022
). 34.
O. G.
Bakunin
, “Diffusion equations and turbulent transport
,” Plasma Phys. Rep.
29
, 955
–970
(2003
). 35.
A.
Monin
and A.
Yaglom
, Statistical Fluid Mechanics, Volume 2: Mechanics of Turbulence
(Dover Publications
, 2007
).36.
J.
Ignaczak
, “Modeling heat transfer in metal films by a third-order derivative-in-time dissipative and dispersive wave equation
,” J. Therm. Stresses
32
, 847
–861
(2009
). 37.
L.
Kuśmierz
and E.
Gudowska-Nowak
, “Subdiffusive continuous-time random walks with stochastic resetting
,” Phys. Rev. E
99
, 052116
(2019
). © 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
Author(s)
You do not currently have access to this content.
